INSTITUTE OF PHYSICS PUBLISHINGREPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 69 (2006) 759–796

doi:10.1088/0034-4885/69/3/R05

Time-dependent single-electron transport throughquantum dots
Toshimasa Fujisawa1,2 , Toshiaki Hayashi1 and Satoshi Sasaki1
1

NTT Basic Research Laboratories, NTT Corporation 3-1 Morinosato-Wakamiya,
Atsugi 243-0198, Japan
2
Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro, Tokyo 152-8550, Japan
Received 21 November 2005, in final form 16 January 2006
Published 20 February 2006
Online at stacks.iop.org/RoPP/69/759

Abstract
We describe time-dependent single-electron transport through quantum dots in the Coulomb
blockade regime. Coherent dynamics of a single charge qubit in a double quantum dot
is discussed with full one-qubit manipulation. Strength of decoherence is controlled withthe applied voltage, but uncontrolled decoherence arises from electron–phonon coupling andbackground fluctuations. Then energy-relaxation dynamics is discussed for orbital and spin
degree of freedom in a quantum dot. The electron–phonon interaction and spin–orbit coupling
can be investigated as the dissipation problem. Finally, charge detection measurement ispresented for statistical analysis of single-electron tunnelling transitions and for a sensitive
qubit read-out device.
(Some figures in this article are in colour only in the electronic version)

0034-4885/06/030759+38$90.00© 2006 IOP Publishing Ltd

Printed in the UK

759

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T Fujisawa et alContents

1. Introduction
1.1. Charge qubit in a DQD
1.2. Electron spin qubit
1.3. Entanglement state
2. Charge qubit in a DQD
2.1. DC transport characteristics of a DQD
2.2. Dynamics of a charge qubit in a DQD
2.3. Coherent control of charge qubit in a DQD
2.4. Decoherence mechanisms
2.5. Towards two-qubit manipulation
3. Relaxation dynamics in a QD
3.1. Electrical pump–probe measurement
3.2. Momentum relaxation
3.3. Spin relaxation
3.4. Spin–orbit coupling effect
4. Statistics of single-electron tunnelling
4.1. Charge detection techniques
4.2. Charge detection of a quantum dot
4.3. Single shot readout device for qubits
5. Summary
Acknowledgments
References

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1. Introduction
Time-dependent phenomena in nanostructures are of growing interest for understanding andcontrolling their dynamic behaviour. One of the applications is quantum computing, in whichsome kinds of information processing can be performed efficiently in a parallel fashion by
manipulating particles (qubits) in a programmable manner [1,2]. Various quantum algorithms
have been successfully demonstrated in some physical systems [3], and there has been progressin integrating large number of qubits required for practical quantum computers, especially insolid state systems [4–9]. Although the realization of a practical quantum computer may require
enormous research activity, quantum information research has already succeeded in the sensethat a common language is provided for communicating with interdisciplinary researchers.
Quantum dynamics can now be discussed in terms of quantum information theory, which indeed
facilitates discussions between physicists, chemists, mathematicians and quantum engineers.
Generally, any quantum process that changes an initial state (density operator) to a final
state can be described by a completely-positive trace-preserving mapping [1]. Knowledge ofthe mapping suffices for defining a quantum information process. Identical mapping provides
the same process regardless of the microscopic origin of the interactions. Quantum computation
is based on an assembly of unitary operations, which can be decomposed into some fundamental
unitary operations on one- or two- qubit subsystems. Therefore, the problem can be reduced
to a few kinds of unitary operators. Actually, realistic operations are influenced and degradedby the environment coupled to the qubit system, and thus the mapping becomes a non-unitary
quantum process [10]. Two important quantum processes that degrade the coherence of thesystem are dissipation, in which the energy of the quantum system is exchanged with theenvironment, and dephasing, in which the phase of the quantum system is randomized by
the environment. The former is often characterized by longitudinal relaxation time (T1 ), andthe latter by transverse relaxation time (T2 ). In addition, the measurement process can also
be considered a quantum process under coupling to the measurement apparatus. Quantum
computing requires a full set of quantum processes for initializing all qubits, performing oneand two-qubit unitary operations, measuring each qubit state and avoiding errors from nonunitary operations [11].
Quantum dynamics in nanometre-scale solid-state devices is attractive for controllingsome single quanta in a tailored structure with a programmable sequence of quantum processes.
Superconducting circuits with Josephson junctions have successfully demonstrated one andtwo-qubit operations with a high degree of coherency [12,13]. The system can be well isolated
from the environment by designing device parameters and a proper pulse sequence. Anothersystem is the semiconductor quantum dot (QD), which provides artificial electronic states thatcan be controlled with external voltages. Since atomic-like electronic states can be designed
and actually formed in a semiconductor device, QDs are often referred to as artificial atoms[14–17]. High controllability of electronic states is useful for studying the dynamic behaviourof artificial quantum systems as well as that of quantum computing systems. There are two
major choices of the qubit bases: the spin degree of freedom in a single QD and the charge(orbital) degree of freedom in a double quantum dot (DQD). In this paper, we shall review someresearch on spin and charge qubits in QDs in connection to quantum information processing
and physical phenomena underlying realistic devices.
1.1. Charge qubit in a DQD
A single-electron charge qubit can be defined in a DQD, in which one electron occupies
the ground state of one dot or the other [18]. There are two types of coupling between the

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two QDs: electrostatic coupling, which prevents two electrons occupying the same QD; andtunnelling, which allows an electron to occupy two QDs [19]. This allows us to describe the
DQD with a simplified two-level system. One-qubit unitary operation has been demonstrated
with Larmor precession, as will be shown in section 2.3 [20, 21]. Two-qubit unitary operation
can be designed with dipole coupling between neighbouring charge qubits (section 2.5) [22].
Controlled-NOT (CNOT) gate operation and entanglement state have been proposed anddiscussed theoretically for different geometries of DQD arrays [23]. The charge state canbe read out with a sensitive charge detector, such as a single electron transistor (SET) or
a quantum point contact (PC) device, which will be discussed in section 4.3. A charge
qubit is advantageous for all-electric control with state-of-the-art high-speed electronics for
programmable pulse sequences. Actually, superconducting charge qubits have been used forthe successful demonstration of CNOT operation [24] as well as single-shot measurement [25].
Therefore, similar measurements are also expected for semiconductor QD systems.
However, decoherence is a serious problem as described in section 2.4. Most singleelectron devices suffer from background charge fluctuations, in which electron occupationof impurities in the device is thermally activated [26]. Although the thermally activatednoise should decrease with decreasing temperature, the noise can also be excited during qubit
operation even at zero temperature [25]. Background charge fluctuation has 1/f frequency
dependence, and its low-frequency part randomizes the phase of the qubit (decoherence).
Echo techniques developed in magnetic spin resonance experiments are useful in recovering
the coherency of the system [12]. However, the high-frequency part of the fluctuation cannot besafely neglected and gives rise to significant decoherence and dissipation [25]. Understanding
and eliminating the fluctuation will require much effort in device fabrication. Electron–phonon
interaction is another decoherence source important to semiconductor systems. Piezoelectric
coupling, which appears in polar materials such as GaAs, is dominant for low-energy
excitations and brings about super-Ohmic coupling to qubit systems [27, 28]. Actually,
resonant tunnelling current through a DQD has revealed a spectral function of electron–phonon
coupling [29].
1.2. Electron spin qubit
Electron spin is a natural two-level system and thus ideal for a qubit [30–32]. An electron
confined in a QD is a convenient way to handle individual electron spins. According to Loss and
DiVincenzo [31], unitary operation of a single spin can be performed by Rabi oscillation withoscillating magnetic field at the electron spin resonance (ESR) condition. Arbitrary unitary
operation for one qubit can be designed using two kinds of ESR oscillations, for instance, witha different phase of the oscillating field. Two-qubit operation can be performed by applying
voltage-controlled exchange interaction between electrons in neighbouring QDs [32]. The
two-electron wavefunction with the spin singlet and triplet configurations experiences different
Coulomb repulsion, resulting in energy splitting between the singlet and triplet states [33]. The
opening of this splitting for a short period can be used to swap the spin information of thetwo-electron system (swap operation). The combination of the swap operation and one-qubit
operation generates an arbitrary unitary operation including CNOT gate and more complicatedoperations.
Much experimental progress has been made in this direction. Zeeman splitting, which
determines the Larmor precession frequency, has been measured with spin-dependent transport
through the spin-up or spin-down branch of single-electron state in a QD [34, 35]. Such spindependent tunnelling under large Zeeman splitting can be used to prepare an initial state in
the ground state. A spin-up electron (ground state for GaAs) can be selectively injected into

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a QD. One can also determine the spin state of a QD using spin-dependent tunnelling, inwhich only a spin-down electron with higher energy is allowed to escape from the QD [36].
High-speed charge detection determines whether the electron has escaped or remains; thus
single-electron spin state can be determined (section 4.3). This single-shot read out techniqueis quite important for the statistical analysis of the multiple-qubit system and essential forstudying correlation in an entanglement state.
In contrast to an electron in a vacuum, the electron g-factor largely deviates from 2 becauseof spin–orbit coupling in a compound semiconductor (e.g. −0.44 for bulk GaAs) [37]. Despitelarge spin–orbit coupling, electron spin in a QD is expected to have a relatively long relaxation
time [38, 39]. Actually, the dissipation time (T1 ) of an electron spin can be made longer than
1 ms, which is about 107 times longer than the typical spin precession time [36, 40, 41], as
discussed in section 3.3. However, decoherence time (T2 ) is strongly affected by the magnetic
field fluctuation caused by nuclear spin fluctuation. The ESR signal of conduction electrons in
GaAs shows significant broadening and hysteresis due to hyperfine coupling to the nuclear spinsof the host material, implying T2 of about 10 ns as estimated from the width of the ESR peak.
Recent spin-dependent transport experiments indicate that the nuclear spin fluctuation gives rise
to a fluctuation in the local magnetic field, resulting in a short dephasing time of about 10 ns [42].
For coherent control of an electron spin qubit, ESR has been demonstrated for an
isolated electron trapped in a defect in silicon [43]. This technique could be applied for
one-qubit manipulation. However, ESR Rabi oscillation would remain relatively slow whenthe experimentally available magnetic field is considered. Other techniques, such as thevoltage-controlled spin–orbit coupling effect, can be very efficient for spin manipulationin semiconductors [44]. Two-electron exchange splitting for two-qubit operation has been
extensively studied in DQDs. In contrast to the relatively large exchange energy in a single
QD [45], the exchange spitting of spatially separated electrons in a DQD can be too small
to resolve in the conventional transport measurement [33, 46]. Extremely small exchange
splitting can be measured with the aid of hyperfine coupling to nuclei. Recently, coherent
oscillations between spin singlet and triplet states have been successfully demonstrated with
voltage-controlled exchange splitting [47]. Significant decoherence from the nuclear spin
fluctuation can be removed by using an echo technique to cancel the low-frequency part of thefluctuations. Nevertheless, for the control of individual spins, controlling nuclear spins may
be required in order to improve the spin relaxation time. The nuclear spin fluctuation can be
minimized at full nuclear-spin polarization. It should be noted that hyperfine coupling may be
useful in a positive way in that the electron spin information can be transferred to nuclear spin
information, as theoretically predicted in [48].
1.3. Entanglement state
In addition to these standard approaches to quantum computing, there are various techniquesthat are particularly important for quantum information science. For instance, successful
generation of an entangled photon pair relies on a non-linear optical process that generates
two visible photons from an ultraviolet photon [2]. Each photon of an entangled pair can beanalysed with a polarizing beam splitter and photon detectors. The coincidence statistics ofphoton detection at various angles indicates a violation of Bell’s inequality, which tells us thequantum mechanical nature of a non-local photon pair. The generation of entangled states canbe used in various kinds of quantum communication devices, like quantum teleportations andquantum repeaters.
How can we perform such experiments for electrons in QDs? We have to develop an
electronic version of each optical component, such as an entanglement generator, polarizing
beam splitter/analyser and photon counting device. Various techniques for ballistic electron

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optics have been developed since the 1990s. It is known that an electron can travel ballisticallyfor more than 100 µm in a high-mobility semiconductor heterostructure [49,50]. The electron
trajectory can be modulated locally by gate electrodes, and tuneable electron refraction in
an electron prism as well as controllable reflection for electron mirrors have actually been
demonstrated [51]. Electron transport also shows interferometric effects, like the Aharonov–Bohm effect and the Al’tshuler–Aronov–Spivak effect in a ring geometry [52–54]. It has
been demonstrated that interference patterns in a double-slit interferometer are degraded bydetecting which path (slit) an electron transports through, as quantum mechanics predicts [55].
Furthermore, the two-electron collision noise in an electron beam splitter indicates antibunching characteristics of Fermionic symmetry [56]. When the spin degree of freedom
is considered, bunching (anti-bunching) is expected for spin singlet (triplet) correlation. These
measurements have motivated further studies on electron flying qubits in semiconductors. The
charge detection measurement described in section 4 may be useful for developing an electron
counting device.
Some theoretical studies have proposed generation and detection schemes forentanglement spin pair in mesoscopic electron transport [57]. Bell-state measurement can be
performed with electron interferometers, beam splitters and local spin rotations. The selection
rule in the relaxation process may be useful for selecting one of the entanglement states as
described in section 3.4. In addition, it has been theoretically predicted that the single-charge
measurement is essential for the exponential speedup of quantum computation with electron
flying qubits [58].
These quantum information techniques with electron charge and spin can be developedby understanding fundamental electron dynamics in quantum dots. In this paper, we describe
some recent progress in time-dependent single-electron transport through quantum dots. In
section 2, coherent dynamics of a single charge qubit in a DQD is described. Full onequbit manipulation with controlled decoherence is demonstrated. Energy relaxation dynamics
is discussed in section 3. The electron–phonon interaction and spin–orbit coupling can beinvestigated as the dissipation problem in QDs. A possible scheme for generating the spin
entanglement state is also described. Finally, charge detection measurement is presented in
section 4. Statistical analysis of single electron tunnelling transitions is discussed with possible
applications for electron counting and a sensitive qubit read-out device.
2. Charge qubit in a DQD
2.1. DC transport characteristics of a DQD
The transport properties of a semiconductor DQD have been studied extensively (for reviews,
see [14, 59, 60]). The DQD considered in this section consists of two lateral QDs, which arecoupled to each other through a tunnel barrier. Each QD is also connected to an electron
reservoir via a tunnel junction, as shown by the equivalent circuit in figure 1(a) and theenergy diagram in figure 1(b). Each tunnel barrier has a small coupling capacitance as well as
tunnelling coupling, and single-electron transport through the DQD can be measured. Here, wedenote a tunnel coupling between localized states in the two dots as Tc and tunnelling rates forthe left and right tunnel barriers as L and R , respectively. In addition, the DQD is connected
to gate voltages Vl and Vr via capacitors Cl and Cr , respectively, so that the local electrostatic
potential of each dot can be controlled independently. The difference of the electrochemical
potentials of the two dots is expressed as ε = εL − εR . The energy difference between theelectrochemical potentials µL and µR of the left and right reservoirs corresponds to the applied
source–drain voltage Vsd .

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765

Figure 1. (a) Electric circuit model of a DQD containing m and n electrons in the left and rightdot, respectively. The two QDs are coupled to each other via a tunnel junction, and each dot isconnected to an electron reservoir via a tunnel junction. The electrostatic potential of the left (right)
quantum dot is controlled by the gate voltage Vl (Vr ) through the capacitance Cl (Cr ). (b) Energy
diagram of a double quantum dot. Tc is a tunnel coupling between the two quantum dots. L andR are tunnelling rates for the left and right tunnel barriers. The difference of the electrochemical
potentials between the left (µL ) and right (µR ) reservoirs corresponds to the source–drain voltage
eVsd . The relative position of the electrochemical potential of each quantum dot can be controlled
by the gate voltage Vl (Vr ). ε is the difference in the electrochemical potentials between the twoquantum dots. (c) SEM image of a double quantum dot structure.

A DQD can be routinely fabricated using various techniques. High-quality QDs areoften fabricated from a two-dimensional electron gas in a GaAs/AlGaAs modulation-doped
heterostructure using standard semiconductor fabrication processes, such as electron beamlithography, dry etching and gate metallization [29,61,62]. As shown in the scanning electron
micrograph of figure 1(c), a narrow conductive channel is formed between the upper and lower
etched grooves (dark regions). Three tunnelling barriers are formed by applying negative
voltages to the gate electrodes (the bright vertical lines), leaving the left and right QDs (whitecircles) between the source and drain electrodes. A very simple system where just one electron
occupies a DQD can also be made by optimizing the device structure [36].
In these devices, all DQD parameters (Tc , L , R , ε and eVsd ) can be controlled withexternal voltages almost independently. The total energy of the system is given by its enthalpy,
which is the electrostatic charging energy in all capacitors subtracted by the work that has been
done by the voltage sources [59]. The stable charge configuration (m, n) with m electrons
in the left QD and n electrons in the right QD is determined to minimize the total energy.
A schematic stability diagram of a DQD is depicted in the Vl –Vr plane in figure 2(a). Whenthe tunnelling coupling (Tc ) is negligibly small, the boundaries of the stable charge states
appear as a honeycomb pattern, a part of which is shown by dashed lines. The triple points, E
and H, of three charge states are separated by a length corresponding to the inter-dot Coulomb
energy U . Electrons pass through three tunnel barriers sequentially in the vicinity of triple
points. For example, transport at E is illustrated in figure 2(b), where an electron travels fromthe left to the right lead. On the other hand, the tunnelling process at H can be viewed as holetransport, as the unoccupied state (hole) moves from the right to the left (not shown in the
diagram).
When the tunnelling coupling (Tc ) is significantly large, the chemical energy of the DQD
has to be considered in describing the stable charge configuration. The charging diagram

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T Fujisawa et al

Figure 2. (a) Schematic stability diagram of a DQD in a linear transport regime. (m, n) represents
a stable charge state of the DQD. U and Tc correspond to the inter-dot Coulomb energy and thetunnel coupling, respectively. (b) Sequential electron transport for forward direction at the triple
point E. (c) Coherent electron transport through the bonding state at B.

deviates from the honeycomb pattern as shown by solid lines in figure 2(a). The minimum
distance between A and B is increased by the coupling energy  ≡ 2h̄Tc from its original value
U . The ground state on the line between A and B is now a bonding state (superposition) of the(0, 1) and (1, 0) charge states. In this case, linear transport is allowed only on the solid lines
close to points A and B. Coherent electron transport through the bonding state is a suitablepicture, as shown in figure 2(c).
When a large source–drain bias voltage Vsd is applied, the conductive regions change
from triple points to triangular shaped regions as shown in figure 3(a), where the sequential
tunnelling regime is considered for simplicity. The electron-like transport around the triple
point E is allowed when the electrochemical potential of the left QD is below that of the left
reservoir (εL < µL bounded on the lines, l1 ), that of the right QD below that of the left one(εR < εL bounded on the lines, lres ), and that of the right reservoir below that of the right
QD (εR < ER bounded on the lines, l4 ). These conditions determine the lower-left triangle infigure 3(a). Similarly, hole-like transport is allowed in the upper-right triangle. These triangles
for electron- and hole-transport are separated by the inter-dot Coulomb energy U .
Since the electron energy is quantized in a QD, transport between the dots is allowed
basically only for resonant condition (ε = εL − εR = 0), which is indicated by the thick line
in figure 3(a). The current peak profile is expected to be the Lorentzian form
I (ε) =e
R |Tc |2,h̄ ε 2 + R2 /4 + |Tc |2 (1 + R / L )(1)

for electron flow from the left to the right. Figure 3(b) shows a typical resonant tunnelling
peak fitted with equation (1) [63, 64]. The current spectrum often involves inelastic current
at the off-resonant condition as seen on the positive ε side of the traces in figure 3(b) andthe corresponding grey region in figure 3(a). This inelastic current is attributed to electron–
phonon coupling, in which the electron can tunnel from a higher-energy state in one dot toa lower-energy state in the other dot by spontaneously emitting an acoustic phonon. This
electron–phonon coupling will be discussed with the decoherence problem in section 2.4 andmomentum relaxation in section 3.2.

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Figure 3. (a) A schematic stability diagram of a double quantum dot at a finite bias voltage Vsd .
Resonant tunnelling current can flow through the device along the bottom thick line of the triangles
where the chemical potentials of the two quantum dots are aligned. Inelastic tunnelling current
flows inside the grey triangles. (b) Typical resonant current. The gate voltages are converted to theenergy difference ε. Dashed lines are Lorentzian curves fitted to the data.

Figure 4. (a) Bloch sphere representation of a state of the two-level system. (b) Larmor precessionof a pseudospin about the effective magnetic field parallel to the z-axis ( = 0). (c) Larmor
precession of a pseudospin about the general effective magnetic field ( = 0).

2.2. Dynamics of a charge qubit in a DQD
Here, we describe idealized dynamics of a two-level system composed of a single energy levelin each dot. The Hamiltonian of a closed system, i.e. an isolated DQD without any reservoirs
can be written in the form
HTLS = εL |LL| + εR |RR| + Tc (|LR| + |RL|),(2)
1
1= εσz + σx + const,(3)
2
2
with the same notation defined in figure 1(b). σx and σz are the x and z components of the Paulimatrices, respectively, on the basis of localized states |L and |R in the DQD. This can beseen as a spin–1/2 system subjected to an external magnetic field B = (, 0, ε): H = 21 σ · B.
Therefore, the mathematical structure of the DQD is the same as that of a spin–1/2 system.
An arbitrary state |ψ in the two-dimensional (one-qubit) Hilbert space can be expressed asθθ|ψ = cos |L + eiφ sin |R,
(4)
2
2
where the coupling angle θ and the phase φ can be identified by a point on a sphere, known as
Bloch-sphere representation, as shown in figure 4(a). The North and South poles correspondto |L and |R, respectively.

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T Fujisawa et al

Figure 5. (a) Schematic diagram of a two-state system. Dotted lines are the energy levels for|L and |R. Solid lines are the energy levels for the bonding (|ψB ) and antibonding (|ψA )
states. With a finite coupling  between |L and |R, the energy levels of the two states |ψB  and|ψA  are anticrossed. The gap energy of the anticrossing corresponds to the coupling energy, .
(b) Photon-assisted tunnelling current spectra. Each line is offset such that the right vertical axis
gives the applied frequency. The diagrams show photon-assisted tunnelling between the two QDs.

The eigenstates for the Hamiltonian of equation 2 are the bonding (|ψB ) and anti-bonding(|ψAB ) states:
 
 
β
β
|ψAB  = cos|L + sin
|R,
(5)
2
2
 
 
β
β
|ψB  = sin|L − cos
|R,
(6)
2
2
where β = arctan(/ε) is the coupling angle. Any state on the longitudinal circle at φ = 0
can be prepared√ as eigenstates of the system by adjusting β. The corresponding eigenenergies,± 21 h̄ = ± 21 ε 2 + 2 , show anti-crossing behaviour, as shown in figure 5(a). With increasing
ε (|ε|  ), the eigenstates approach the base ket vectors |L and |R.
This energy splitting has been clearly observed in photon-assisted tunnelling transportmeasurements on a strongly coupled DQD [65]. Microwave voltage of frequency f applied
to a gate electrode modulates the energy bias ε = ε0 + ε1 cos(2πf t), which allows transition
between the bonding and the anti-bonding state by microwave absorption or emission at theresonant condition,= 2πf [19, 66]. This transition is measured with a current fromthe source to the drain electrode weakly coupled to the DQD. As shown in figure 5(b), theobtained current spectra exhibit a hyperbolic dependence of peak position, which clearly
indicates the formation of the bonding and anti-bonding states. It should be noted thatmicrowave spectroscopy probes the coherent coupling only when the microwave excitation
causes weak perturbation (ε1  hf ). When an intense microwave field is applied, the system
is described by quasi-eigenstates (Floquet state), where an electron and photon are strongly
coupled [67]. The resonant condition changes with the microwave amplitude approximately
as J0 (ε1 / hf ) = 2πf , where Jn (x) is the nth order Bessel function of the first kind. Such
dynamic coupling between an electron and photons has actually been demonstrated using
microwave spectroscopy [68].

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Now, we turn to the coherent dynamics of a DQD in the absence of a microwave field.
Suppose non-stationary superposition of eigenstates are prepared by some means, for instance,by changing the system Hamiltonian instantly from one to another, which do not commute
with each other. Non-stationary superposition shows Larmor precession about the fictitious
magnetic field B under the time-independent Hamiltonian. The time-evolution operator isgiven by




−iH t
−iσ · Bt
U (t) = exp
= exp
.
(7)
h̄
2h̄
The motion in the Bloch sphere is schematically shown in figure 4(b) for the  = 0 caseand figure 4(c) for the general case of ε ∼  > 0. Therefore, any state on the Bloch spherecan be prepared by appropriate Larmor precession. Arbitrary one-qubit operation can be
designed by combining two Larmor precessions in, for instance, the Euler angle decomposition,
Uy (γ )Uz (β)Uy (α), with precession operator Uy/z along the y/z axis.
In contrast, practical DQDs are often coupled to some electrodes. Consider the situation
where both eigenstates are in the source–drain transport window (µL > ± 21 h̄ > µR ) under alarge energy bias, µL −µR = eVsd  kB T . We neglect double occupancy where two electrons
occupy the DQD. An electron can enter the DQD only from the left electrode at rate L andcan escape only to the right lead at rate R . These tunnelling processes induce decoherence
to the charge qubit in the DQD. The system can be described by a density matrix, ρ, which
can be obtained by considering coupling to the electronic states in the electrodes. Under the
Born–Markov approximation, master equations for the reduced density matrix readd(8)
ρLL = − i (ρRL − ρLR ) + L (1 − ρLL − ρRR ),dt
2dρRR = i (ρRL − ρLR ) − R ρRR ,
(9)
dt
2
d

R
(10)
ρLR = − iερLR − i (ρRR − ρLL ) −
ρLR ,
dt
2
2
d

R
ρRL = iερRL + i (ρRR − ρLL ) −
ρRL .
(11)
dt
2
2
Note that the tunnelling rates L and R do not enter the equations in a symmetric way.
This is because the escaping process (R ) kills the coherence of the system (the last termin equations (10) and (11)) but the incoming process (L ) just creates an electron in theleft dot (the last term in equation (8)). Figure 6(b) shows the time evolution of the density
matrix element, ρLL , calculated for the initial state, ρ(t = 0) = |LL|. As R increases,
the coherent oscillation degrades more quickly and finally disappears. The total number ofelectrons escaping to the right electrode is obtained by
 T
nt (T ) =
ρRR (t)R dt,(12)0

and the time-averaged current It = ent (T )/T corresponds to the measurable current throughthe device. This current It depends on the electron occupation in the right dot, ρRR (t = 0),
providing projective measurement of the charge qubit.
2.3. Coherent control of charge qubit in a DQD
A DQD device may contain a few tens of electrons basically forming many-body states. In a
simplified picture, even if the dots contain more than one electron, one excess electron added
to the DQD occupies either the right or the left dot. Each charge state involves a ground state

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Figure 6. (a) Schematic energy diagram in the transport regime of a DQD. (b) Time evolution ofdensity matrix element, ρLL , under initial condition, ρLL = |LL|.  = L = R .

and excited states corresponding to the orbital degree of freedom in each dot. Therefore, thetwo-level system (qubit) given by equation (2) is a good approximation when only one welldefined state from each dot is considered as |L or |R. This can be justified under a condition
where excitation energies, like the thermal energy (2.5 µeV at 30 mK for the measurement
described below), are much smaller than the characteristic energies of the dot, i.e. the additionenergy (2–3 meV for typical DQD), the single-particle excitation energy (50–200 µeV), andthe electrostatic coupling energy (about 100 µeV).
To observe the Larmor precession of charge qubit in a DQD, we introduce a rectangularvoltage pulse to the drain electrode, as shown in figure 7(a). The fast voltage pulse switches thesource–drain bias voltage, Vsd , between VH in the transport regime and VL (∼0) in the Coulomb
blockade regime, as shown in figure 7(b). At the same time, it also switches the energy bias εabruptly between ε0 (for VH ) and ε1 (for VL ), since the DQD is also capacitively coupled to theelectrodes. This simple rectangular pulse is sufficient for performing initialization, coherent
manipulation and read-out processes sequentially.
When the DQD is in the Coulomb blockade region (Vsd = VL ∼ 0), tunnelling processes
between the DQD and the electrodes are energetically forbidden, as shown in the energy
diagram of figure 7 (d). Therefore, to the lowest approximation, the coupling to the electrodecan be ignored and the closed two-level system discussed in section 2.2 can be applied. Other
decoherence will be discussed in section 2.4. In contrast, when a large source–drain voltage(Vsd = VH ) is applied, single-electron tunnelling current flows through a DQD and the allowed
tunnelling processes induce significant decoherence to the charge qubit. The corresponding
energy diagrams at a large positive voltage are shown in figures 7(c) and (e). In this case, anelectron in the right dot may escape to the right electrode, while another electron may enter
the left dot from the left lead. Under the condition, L , R  Tc , the stationary solution of themaster equation (equations (8)–(11)) becomes a density matrix with element ρLL ∼ 1, which
describes an electron confined in the left dot [21].
Therefore, the decoherence of the qubit can be switched on (in the transport regime) andoff (in the Coulomb blockade regime) by changing the voltage. Our approach is to first prepare
a confined state |L in the transport regime and switch the system into the Coulomb blockade
regime by applying a rectangular voltage pulse of an adjustable length, tp . Since |L is nolonger an eigenstate in the Coulomb blockade regime, coherent time evolution is expected.
After the pulse, the DQD is again set in the transport regime, so that the electron, if it occupies
the right dot, contributes to the current. With this scheme, we expect one electron tunnelling per

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Figure 7. (a) Measurement setup and a SEM image of a DQD sample. A pulse generator isconnected to the drain electrode through a coaxial cable. (b) Typical pulse waveform used forcoherent control of the qubit. (c)–(e) Schematic energy diagrams of the DQD during initialization(c), manipulation (d) and measurement (e).

pulse at most. By repeating the pulse with repetition frequency, frep = 100 MHz, a reasonablepumping current, Ip = eρRR (tp )frep , is obtained, where ρRR (tp ) is the probability of finding anelectron in the right dot after the pulse length tp , and this provides a projective measurement.
In practice, inelastic transition contributes to an additional current during the initializationand measurement period (when Vsd = VH ). In order to reduce the effect of inelastic current
on the measurement, we employ a lock-in technique to measure modulation current, Imod , byturning on and off the pulse sequence with low frequency (100 Hz). Then,
Imod = [eρRR (tp ) − tp Iinel ]frep(13)

should be measured.
Full one-qubit operation can be obtained by combining a rotation gate for controlling θand a phase-shift gate for φ in equation (4). The rotation gate, which changes the occupationof the electron, can be performed with finite  at ε = 0, where the state rotates about the xaxis as shown in the inset of figure 8(a). The observed pulse-modulated current Imod (circles)in this situation clearly shows an oscillating behaviour as a function of tp and is fitted wellwith an exponentially damping sinusoidal function (solid line in figure 8(a)). From the fitting,the oscillation frequency in this particular case is estimated to be fosc = 2.3 GHz. The
dashed line shows the inelastic current contribution, the second term of equation (13). The
oscillation frequency can also be controlled by tuning the centre gate voltage, VC , which mainly
changes the central barrier height and thus the coupling energy, . As shown in figure 8(b),the oscillation frequency is an exponential function of VC , as expected from the tunnellingprobability for realistic potential.
The amplitude of the oscillation is approximately half of the ideal case of efrep = 16 pA.
This degradation might have arisen from the population of other excited states duringinitialization. This could happen when the applied source–drain voltage (VH ) is higher than the
single particle excitation energy. Or non-ideal pulse waveform (finite rising time or jittering)

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Figure 8. (a) Pulse-modulated current, Imod , as a function of pulse length, tp . The dots are theexperimental data. The solid line is a fitting function. The dotted line is the offset of the oscillation.
The inset shows the schematic trajectory on the Bloch sphere. (b) Dependence of the coherent
oscillation frequency on the central gate voltage. The line is an exponential function of the centergate voltage, VC . Note that the vertical axis is on a logarithmic scale.

could have influenced the rotation gate operation. The decay of the oscillation is attributed todecoherence, which is discussed in the next subsection.
An ideal phase-shift gate can be realized with finite ε at  = 0, but this cannot be achievedin the experiment since  is always positive. Therefore, we approximate the phase-shift gate
by applying ε  , where the state precesses more or less about the z-axis. To demonstrate thephase-shift operation that does not change electron population, we apply the pulse waveformshown in figure 9(a). A sharp tipping pulse of length tt , which works as phase shift gate, isadded at the center of the square pulse for the π -rotation gate (NOT gate) [20].
√ As a result,the first π/2 pulse rotates the qubit state from |L to a superposition state, (1/ 2)(|L − i|R)
on the equator, as shown by arrow (i) in the inset of√figure 9(b). The sharp tipping pulse
induces a phase difference φ, resulting in a state (1/ 2)(|L − i exp(iφ)|R), as shown byarrow (ii). Then, the final state after the second π /2 pulse becomes sin(φ/2)|L + cos(φ/2)|R(arrow (iii)), whose probability for |R, cos2 (φ/2), should be measured as the corresponding
pulse induced current, Imod . The phase shift φ is approximately given by the area of the pulse(hatched region) as φ εt tt /h̄. Actually, we change the height of the tipping pulse, εt , while
keeping the pulse width constant at tt ∼ 55 ps. Figure 9(b) shows Imod obtained in this way, andthe obtained oscillation indicates the phase-shift operation on the qubit. Since the oscillation
is obtained with a fixed pulse width, the amplitude decay at higher Vt implies an increased
decoherence rate at large Vsd .
2.4. Decoherence mechanisms
One of the most important quality factors for qubits is the phase coherence time, T2 , which is
the time a qubit remains in a superposition state. In spite of the successful manipulation ofa single-charge qubit, the qubit is actually influenced by uncontrolled decoherence, which ispresent even in the Coulomb blockade regime. Some possible decoherence mechanisms are
summarized here.
First, background charge noise (1/f noise) in the sample and electrical noise in the gate
voltages cause fluctuation of the qubit parameters ε and Tc , which gives rise to decoherence ofthe system [18,69,70]. The amplitude of low-frequency fluctuation in ε is estimated to be about
1.6 µeV, which is obtained from low-frequency noise in the single-electron current, or 3 µeV,Time-dependent single-electron transport

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Figure 9. (a) Pulse shape used for phase-shift gate operation. A shape tipping pulse (hatchedregion) is added at the middle of the π pulse. (b) Dependence of the pulse-induced current Imod on
the tipping-pulse height Vt , demonstrating the phase-shift operation. The insets show schematic
trajectories in the Bloch sphere. The left and right inset corresponds to phase shifts φ = π and 2π ,respectively.

which is estimated from the minimum line width of an elastic current peak at the weak coupling
limit [29]. Low-frequency fluctuation in  is relatively small and estimated to be about 0.1 µeV
at h̄Tc = 10 µeV, assuming local potential fluctuation in the device [71]. Actually, the εfluctuation explains the decoherence rate observed at the off-resonant condition (|ε|  h̄Tc ).
The 1/f charge fluctuations are usually considered to be an ensemble of bistable fluctuators,
like electron traps, each of which produces a Lorentzian frequency spectrum [26]. The
microscopic origin of the charge fluctuators is not well understood, and their magnitude differsfrom sample to sample, even when samples are fabricated in the same batch. Understanding
the fluctuators is a practical and important issue in developing quantum information devices.
A recent noise measurement has indicated that the fluctuation in ε can be reduced by decreasing
the temperature as suggested by a simple phenomenological model where the activation energyof the traps is uniformly distributed in the energy range of interest [71]. Cooling samples very
slowly with positive gate voltage is sometimes effective in reducing charge fluctuation at low
temperature [72].
In contrast, the decoherence at the resonant condition (ε = 0) is dominated by othermechanisms. Although the first-order tunnelling processes are forbidden in the Coulomb
blockade regime, higher-order tunnelling, namely co-tunnelling, processes can take place anddecohere the system [73]. Actually, the co-tunnelling rate estimated from the tunnelling rates
is close to the observed decoherence rate and may thus be a dominant mechanism in thepresent experiment [18]. However, since we can reduce the co-tunnelling effect by making
the tunnelling barrier less transparent, we should be able to eventually eliminate it in future
measurements.
The electron–phonon interaction is an intrinsic decoherence mechanism in semiconductor
QDs. Spontaneous emission of an acoustic phonon persists even at zero temperature and causes
an inelastic transition between the two states [29]. Actually, the negative background slope
shown by the dashed line in figure 8(a) corresponds to the inelastic tunnelling transition at
the off-resonant condition (ε = ε0 ) during the initialization/measurement sequence. Theenergy relaxation time in this case is about 10 ns, but becomes shorter at the resonant condition(ε = 0). The phonon emission rate at the resonant condition cannot be directly estimated from774

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Figure 10. (a) Schematic diagram of two charge qubits (Q1 and Q2 ) in a parallel configuration. A
round square represents a charge qubit with two QDs (◦), one of which accommodates an electron( ). (b) Two charge qubits in a perpendicular configuration. (c) Energy diagram of a parrarel
two-qubit system. Dashed lines are obtained under zero tunnelling coupling (1 = 2 = 0); solidlines are for finite coupling (1 = 2 = 0.3δ).

•this data, but it may be comparable to the observed decoherence rate. It should be noted thata measurement from a single vertical QD has also shown a phonon emission rate of the orderof 10 ns (described in section 3.2) [41]. Strong electron–phonon coupling is related to thefact that the corresponding phonon wavelength is comparable to the size of the QD [27, 29].
In this sense, electron–phonon coupling may be reduced by using much smaller or muchlarger QD structures. In addition, polar semiconductors, such as GaAs, exhibit a piezoelectric
type of electron–phonon coupling, which is significant for low-energy excitations (<0.1 meVfor GaAs) [74]. Non-polar semiconductors, such as Si or carbon-based molecules, may be
preferable for reducing the phonon contribution to the decoherence.
Other mechanisms, such as the electromagnetic environment, have to be considered to fully
understand the decoherence. It should be noted that the quality of the coherent oscillation has
actually been improved by reducing high-frequency noise from the gate voltages and a coaxial
cable. We expect that further studies will exploit ways to reduce some decoherence effects.
2.5. Towards two-qubit manipulation
The CNOT gate is a nontrivial two-qubit gate, which flips the target qubit only when the control
qubit is in logical one state. The experimental implementation of the CNOT gate for a charge
qubit was motivated by Barenco et al [22], and some realistic device geometries have been
proposed [23, 75]. For two qubits arranged parallel to each other as in figure 10(a), the stateof the control qubit Q1 modifies the energy bias, ε2 , of the target qubit, Q2 , via the Coulombinteraction. The coupling term, δσ1z σ2z , appears in the system Hamiltonian, where δ is thecoupling energy and σnz is the z-component of the Pauli matrix of the nth qubit. When the two
qubits are arranged perpendicularly as in figure 10(b), the coupling term is given by δ σ1z σ2x ,where the state of the control qubit, Q1 , modifies the tunnelling coupling of the target qubit, Q2 .
Realistic devices may contain both terms in the Hamiltonian, but it is instructive to consider
only one term in order to understand how the CNOT gate can be implemented. Figure 10(c)shows an energy diagram of two parallel charge qubits with HamiltonianH =1(ε1 σ1z + 2 σ1x + ε2 σ2z + 2 σ2x + δσ1z σ2z ) .
2(14)

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Figure 11. (a) Schematic diagram of an asymmetric DQD. (b) Energy diagram of the three-levelsystem formed in asymmetric DQD. (c) Coherent oscillations taken at different resonant conditions(i) between the ground state of the left dot to the first excited state of the right dot and (ii) between
the ground state of the left dot to the second excited state of the right dot.

Here, tunnelling coupling is assumed to be identical for the two qubits (1 = 2 = ), andeigenenergies are plotted for the energies normalized to the coupling energy δ. Anticrossing
behaviour between different logical qubit states can be controlled by tuning the energy bias, ε1and ε2 , of the two qubits. When the system is adjusted at the anticrossing point of |10 and |11states (shown by the arrow), coherent oscillation between these states is expected. The half
period of the oscillation swaps states |10 and |11, providing a controlled-rotation (CROT)gate. A CNOT gate can also be realized by combining additional one-qubit operations.
However, the coupling term is always present in the geometries shown in figures 10(a) and(b). Therefore, the one-qubit operation in the two-qubit system is constrained to combine a
couple of pulses for decoupling two qubits. For instance, combination of coherent oscillations
between |10 and |11 states and between |00 and |01 states can be used for a one-qubit
rotation gate for the second qubit. Although constant coupling is not an intrinsic problem forthe quantum computing scheme, electrical tuning of the coupling may be useful for performing
one- and two-qubit operations in more complicated systems.
Tuneable coupling can be achieved by introducing an excited state into one of the two
QDs. Consider an asymmetric DQD consisting of a small QD with its ground state |0 anda larger QD with its ground state |1 and the first excited state |2, as shown in figures 11(a)and (b). There are tunnelling couplings, 01 , between |0 and |1 and 02 between |0 and|2, providing a three-level system. Such a three-level system is available in a realistic DQD
device [76]. Actually, coherent oscillations have been observed for two resonant tunnelling
conditions at slightly different gate voltages as shown in figure 11(c) [18]. The two oscillations
correspond to coupling 01 and 02 in the three-level system.
Then, we can use the coherent oscillation between |0 and |1 for a one-qubit manipulation
at the resonant condition where excitation to |2 is negligible. A superposition of |0 and |1 isuseful for constructing a CROT gate with a neighbouring qubit, as described above. π -rotation
at the other resonant condition exchanges probability amplitudes of |0 and |2, resulting in a
local superposition of |0 and |2 states. Since this local superposition occupies the right dot
only, its Coulomb interaction to the neighbouring qubit is almost independent of the qubit state.
Therefore, a three-level system in a DQD is of interest for tuneable coupling in an integrated
system.

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Figure 12. (a) Schematic diagram of the pulse measurement setup. (b) The pulse waveform forthe single-step pulse measurement. (c),(d) Schematic energy diagrams for the low-voltage and thehigh-voltage states, respectively.

3. Relaxation dynamics in a QD
In this section, we discuss energy relaxation dynamics in quantum dots, which is related to theT1 of qubit dynamics. After electrical pump–probe methods for measuring the relaxation time
are described, some important relaxation mechanisms of orbital- and spin-degree of freedom
in QDs are discussed based on some experiments.
3.1. Electrical pump–probe measurement
We start with single-step pulse measurement as schematically shown in figures 12(a) and (b)where a rectangular-shaped voltage pulse, Vg (t), is applied to the gate electrode [77]. Consider
the relaxation process from a first excited state (ES) to the ground state (GS) of N0 -electron
QD. First, the QD is set in the N = N0 − 1 CB state when the gate voltage is in the low-voltage
state, Vg = Vl , as shown in figure 12(c). We also tune the two tunnel barriers such that thetunnel rate through the left entrance barrier, L , is much larger than that through the right exit
barrier, R . After a sufficiently long time period, tl  L−1 , the QD is prepared in N = N0 − 1
GS. In this situation, both N0 -electron GS and ES of interest are above the chemical potentialof the two electrodes, µL and µR . Next, Vg is switched to Vh (high-voltage state) within a shorttime, which should be comparable to or shorter than L−1 . We adjust Vh such that only the ES
is located in the transport window between µL and µR , as shown in figure 12(d). Here, theQD is in the N0 -electron CB region. The applied voltage, eVsd = µR − µL , should be larger
than h̄L to prevent the back-flow of the electron to the left electrode. Then, an electron canenter the GS (ES) with a probability proportional to the rate, L,g + R,g (L,e ). If an electron
enters the ES, it can either tunnel out to the right electrode to give a net current, or relax to the
GS. When the barriers are highly asymmetric (L  R ), the electron can stay in the ES for a

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Figure 13. (a) Pulse waveform for the double-step pulse measurement. (b), (c), (d) Schematicenergy diagrams for the low-, high- and medium-voltage states, respectively.

relatively long time, ∼R−1 , during which relaxation may take place. If the electron in the EStunnels out to the right electrode, another electron enters the ES or GS. However, this cycle isterminated once the GS is occupied, either by relaxation from the ES or by direct tunnellingfrom the electrodes.
The average number of tunnelling electrons per pulse, nt , is given asnt = nmax [1 − exp(−Dth )](15)

by solving rate equations [78, 79]. Here, nmax is the maximum number of tunnelling electronsobtained with sufficiently long th and tl , and D is the decay rate of the transient current. This
D gives the relaxation rate W of interest provided that L  W  R . One can determine
W from this relation. The pulse length dependence of nt shown in figure 16 is obtained inthis way.
If the bias is reversed (thicker emitter barrier), on the other hand, the decay rate isinsensitive to W , as given by D ∼ tot,g = L,g + R,g ∼ L,g . This situation can beused to determine the total tunnelling rate, tot,g . Another way to obtain tot,g is to vary tl withth fixed at a long-enough value, th  R−1 , corresponding to the escaping process from the
GS to either electrode as shown in figure 12(c). In this case, the average number of tunnellingelectrons per pulse is given by
nt = nmax [1 − exp(−tot,g tl )],(16)

from which tot,g can be determined by fitting the data to this equation. Since an asymmetricbarrier is not required, this method can be used for various situations.
The above single-step pulse method is unable to determine the relaxation rate if W < R,e(long relaxation time), as in the case of the relaxation involving a spin flip. Such a caserequires an improved method involving the application of a double-step voltage pulse, where
Vg is switched between three voltages, Vl , Vh and Vm , as shown in figure 13(a).

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Figure 14. (a) Schematic diagram of a vertical QD together with the pulse measurement set-up.
(b) Scanning electron micrograph of the vertical dot device. (c) dIsd /dVg for N = 1 (first stripe)and 2 (second stripe) measured with Vsd = 2.8 mV at temperature T ∼ 100 mK.

First, when Vg = Vl , the N = N0 GS and ES are above the chemical potential as shownin figure 13(b). The GS and ES are emptied by maintaining this condition for a sufficiently
long time, tl (initialization). Next, by instantly changing Vg to Vh , both states are pulled down
below the electrochemical potential of the electrode as shown in figure 13(c). Now, an electron
enters either ES or GS (arrows). Once an electron enters the dot, it cannot leave the dot because
large energy is required in order to excite the electron to the electrode, nor can another electron
enter the dot (Coulomb blockade). This electron, if it populates the ES with a probability P , isallowed to relax to the GS while the gate voltage is kept at Vh during the wait time th . Finally,
the pulse height is adjusted so that only the ES is within the transport window between µL
and µR (read-out). Then, the electron can contribute to the current only if it remains in the ES
after th . This read-out pulse width, tm , is made sufficiently longer than R−1 . Actually, several
electrons [1/(1 − P )] flow during this time for the unrelaxed case. Therefore, the average
number of tunnelling electrons per pulse cycle, nt , follows an exponential decay,P(17)exp(−W th ),1−Pfrom which the relaxation rate W can be determined [41]. Thus, the double-step pulse scheme
clearly separates pumping and probing sequences and is applicable to a wide range of W  L .
The lower limit is practically determined by current sensitivity. Spin relaxation measurement
data shown in figure 17 are obtained with this technique.
nt =3.2. Momentum relaxation
The QD devices discussed in section 2 are called ‘lateral’ dot device because current
flows laterally. There is another type of QD called a ‘vertical’ dot, which is shown infigures 14(a) and (b). It is in the form of a submicron-sized circular mesa fabricated froman AlGaAs/InGaAs/AlGaAs double barrier structure. Current flows vertically through themesa, and the electron number, N , in the dot formed in the InGaAs layer is changed using aTime-dependent single-electron transport

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Figure 15. (a) Energy difference between 1s and 2p states, ε1s–2p in the N = 1 QD. (b) Size of
the QD (a, lx and ly ) and the phonon wavelength, λ1s–2p , corresponding to the energy spacing. (c)Decay time of the transient current for forward bias ( ) and reverse bias (◦), and energy relaxation
times calculated from the piezoelectric potential (· · · · · ·), or the deformation potential (- - - -), andboth potentials (——).

•

gate electrode wrapped around the mesa. Electronic states (spin S and total angular momentumM) and N can be unambiguously determined in a vertical QD owing to the well-defined lateralconfinement potential (approximately harmonic potential) and to a built-in tunnel rate via the
AlGaAs barriers [16]. The first few electrons occupy 1s (M = 0) and 2p (M = ±1) orbitals,
which have two-fold and four-fold degeneracy including spin, respectively. This results in a‘shell structure’ in the Coulomb blockade oscillation characteristics, where large energy gaps
appear at N = 2 (N = 6) when the 1s (2p) orbital is fully occupied by electrons, just as inreal atoms.
Although the nominal dot shape is circular, a specific QD used for the relaxation
measurement has non-circular lateral confinement potential in the x–y plane with confinement
strength, h̄ωx = 2.5 meV and h̄ωy = 5.5 meV. Therefore, degeneracy of the two 2p states with
(M = ±1) is already lifted at B = 0. Despite this non-circularity, we still use the terms ‘1s’
and ‘2p’ to label states for convenience.
Figure 14(c) shows excitation spectra, dIsd /dVg , as a function of magnetic field with alarge source–drain bias, Vsd = 2.8 meV [17]. Positive peaks in the 1st (2nd) stripe correspondto electron injection to ground and excited states of the N = 1 (N = 2) QD. GS and ES are
designated by arrows together with schematic electron configurations in the inset, in which
the lower (upper) horizontal bar represents the 1s (2p) orbital. Here, we are not able toresolve small Zeeman splitting (∼0.1 meV at 5 T). The energy difference between the 1s and2p states, ε1s–2p , estimated from the 1st stripe of the excitation spectra in figure 14(c) is plottedin figure 15(a). The solid line shows a fitted theoretical calculation based on the asymmetricparabolic confinement potential (confinement energies h̄ωx and h̄ωy ) in the x–y plane [80].
The corresponding characteristic size of the QD in the x/y direction is given by
2
lx/y = h̄/m∗ (ωx/y+ ωc2 /4)−1/4 ,(18)
where m∗ and h̄ωc are the effective mass and cyclotron energy, respectively. The magnetic
field dependence of these characteristic lengths is plotted in figure 15(b).
We apply the single-pulse measurement scheme to extract momentum relaxation time in
the N = 1 QD, which can be regarded as an artificial hydrogen atom. In this case, an electron

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Figure 16. Average number of tunnelling electrons per pulse, nfor , in the forward bias for different
magnetic fields.

relaxes from the 2p state to the 1s state, preserving the spin. The dot is coupled to the top
and bottom electrode with t−1 ∼ 100 ns and b−1 ∼ 3 ns, reflecting the built-in asymmetric
AlGaAs barriers. Figure 16 shows the average number of tunnelling electrons per pulse, nfor in
the forward bias for different magnetic fields, i.e. different ε1s–2p . Each data point is fitted withequation (16) (solid line), from which the decay time, τfor , in the 3–10 ns range is estimated.
Although there is some scatter of data points, τfor decreases systematically with B as shownby the solid circles in figure 15(c). We also show the decay time in reverse bias, τrev , by open
circles for comparison. The increase of τrev with B is consistent with the decrease of tunnellingrate, tot , with B.
In this situation, with an energy difference of a few millielectronvolts and at low
temperature, a dominant momentum relaxation mechanism is the spontaneous emission ofan acoustic phonon [29]. The corresponding phonon wavelength is given by λ1s–2p =hvphonon /ε1s–2p , where vphonon = 5100 ms−1 is the phonon velocity in GaAs. λ1s–2p iscomparable to or even smaller than the characteristic size of the QD as compared in figure 15(b).
The observed increase of the relaxation time with the decrease of λ1s–2p is in line with theinefficient phonon emission rate known as phonon bottleneck effect [81, 82]. In figure 15(c),we also plot calculated relaxation times based on Fermi’s golden rule for the piezoelectricmechanism and deformation mechanism and for the sum of the two mechanisms [41, 83, 84].
The reasonable agreement with the experimental data in spite of the lack of fitting parameters
confirms that the momentum relaxation in the N = 1 QD occurs via emission of a phonon.
3.3. Spin relaxation
Next, we consider the spin relaxation process in an N = 2 QD, which mimics a helium
atom. At low magnetic field, the many-body GS is a spin singlet (S) with total spin, S = 0,
while the first ES is a triplet state (T) with S = 1, as schematically illustrated in the inset
of figure 14(c). Energy relaxation from the triplet ES to the singlet GS not only involves
momentum relaxation as in the N = 1 QD, but also spin relaxation. Therefore, some kindof spin–flip mechanism is necessary for relaxation. Spin is generally considered to be a goodquantum number, which suggests that the spin–flip process occurs over a much longer timeTime-dependent single-electron transport

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Figure 17. Relaxation time measurement for the triplet ES in the N = 2 QD. nt is the average
number of tunnelling electrons per pulse.

cotunnelling,

Figure 18. Spin relaxation time as a function of the high-state voltage of the double-step pulse.

scale compared with a simple phonon-emission process. Therefore, we use the double-steppulse technique explained above. Figure 17 shows the average number of tunnelling electrons
per pulse, nt , as a function of wait time, th , in the high-voltage state of the double-step pulse. By
fitting the exponential curve (equation (17)) to the data, the spin relaxation time τ = 200 µs is
estimated. This is four to five orders of magnitude longer than the momentum relaxation timeobserved for the N = 1 QD [85, 86]. Recently, relaxation times of longer than milli seconds
have been reported in QDs between Zeeman sublevels, as well as between a spin triplet and asinglet state [34, 36, 40]. Electron spin is therefore considered to be a promising candidate fora quantum bit.
Now, the question is: what is the spin relaxation mechanism? Figure 18 shows the observed
spin relaxation time as a function of the high state voltage, Vh . The energy required to excite
the N = 2 triplet ES to N = 1 GS, 1 , increases with gate voltage, Vh , while the excitation

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Figure 19. (a) Evolution of the two Coulomb oscillation peaks under magnetic field showing a
spin-pair behaviour, taken at Vsd = 0.15 mV. Transition fields are denoted by triangles. (b) dI /dVL
at Vsd = 1.2 mV taken from the dashed rectangular region in (a).

energy to N = 3 GS, 3 , decreases with increasing Vh , as shown in the subordinate scalesin figure 18. The observed spin relaxation time strongly depends on Vh , especially aroundthe small 1 regime. This implies a strong influence from electrodes other than the thermal
excitation effect, which predicts much sharper Vh dependence, as shown by the dashed line
in figure 18. Although the dot is kept at the N = 2 CB condition during the relaxation
stage, higher-order tunnelling, or co-tunnelling, is quite effective in causing an exchange ofelectrons having opposite spins between the dot and the lead electrodes [87]. This results inspin relaxation from the triplet ES to the singlet GS. According to the second-order perturbation−1
theory, the co-tunnelling rate, τcot
, is approximately given by
−1 2
−1
τcot
= ST (h̄tot )2 (−1
1 + 3 ) / h,(19)where tot = L + R is the total tunnelling rate and ST the singlet–triplet energy difference[41, 88]. The solid line in figure 18(a) shows τco calculated from equation (19) without any
fitting parameters (tot is obtained from a separate single-pulse measurement). The reasonableagreement with the experimental data indicates that inelastic co-tunnelling is a dominant spin
relaxation mechanism in this experiment.
In a vertical QD, the tunnelling rate through the barriers is predetermined by thesemiconductor crystal growth. In a lateral QD, on the other hand, we have more freedom
in tuning the tunnelling rate with the gate voltages, which allows us to study spin dynamics ina wider regime where different relaxation mechanisms compete. In a lateral QD similar to theone shown in figure 1(c), tunnelling rates, L and R , can be changed by changing the gate
voltages, VL and VR . Although we cannot determine the exact number of electrons, we candetermine whether it is even or odd from the electron-addition spectrum. Figure 19(a) shows anobserved dc current, I , through the dot as a function of left gate voltage, VL , and magnetic field,
B. An overall pair-wise motion of the two stripes with B reflects spin degeneracy. The electron
orbitals in a few-electron QD can be classified by the Landau level (LL) index [89, 90]. The
lower stripe, corresponding to odd N , involves a level crossing (denoted by a solid triangle),
associated with two different LLs. As for the upper stripe with even N , the ground state ismostly spin singlet. However, direct and exchange Coulomb interactions favour a spin triplet
GS via rearrangement of electrons between different LLs [45]. In our data, the triplet GS
is realized between the two open triangles. These spin states can also be observed in the
excitation spectrum of figure 19(b), where dI /dVL , with a large Vsd = 1.2 mV, is plotted as afunction of VL and B. Some ESs as well as the GS that fall within the source–drain transportTime-dependent single-electron transport

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Figure 20. (a) Log–log plot of τs as a function of tot for both the inter-LL and intra-LL cases.
The solid lines are fitted to the data. (b) The coefficient α for the co-tunnelling component as afunction of ST . β is a ratio between the effective tot and the experimentally determined one.

window are observed. The spin relaxation denoted by the arrow involves an orbital change
between different LLs (inter-LL transition).
We performed double-step pulse measurement to obtain spin relaxation time, τs , and total
tunnelling rate, tot . By changing some gate voltages, we measured how τs changes with
tot in the range from 1×108 to 3×109 s−1 , as shown in figure 20(a). The data points at−1
2
large tot are consistent with co-tunnelling theory, as shown by the dashed line tcot
= αtot(see equation (19)). The proportional factor α depends on ST as shown in figure 20(b),which is also consistent with the theory, as shown by the solid line. Therefore, spin relaxationin the large tot regime can be well explained by the standard co-tunnelling theory. It isseen in figure 20(a) that, when tot is reduced, τs increases and eventually saturates. In thisregime, the co-tunnelling process is well suppressed and intrinsic spin relaxation time, which is
independent of tot , can be investigated. Theoretically, spin–orbit interactions are predicted tomake the dominant contribution to spin relaxation in GaAs QD systems [39,91]. By assuming
this prediction, the spin–orbit relaxation time, τso , can be obtained by fitting the data with
(1/τso +1/τcot )−1 as shown by the solid line in figure 20(a).
3.4. Spin–orbit coupling effectSpin–orbit interaction in III–V semiconductors originates from the absence of crystal inversion
symmetry (the Dresselhaus effect), asymmetric confinement potential (the Rashba effect) andinterface or surface effects [37]. For a one-electron system in a QD, these spin–orbit interaction
couples a spin-up (down) state of one orbital to a spin-down (up) state of another orbital. Whenspin–orbit coupling is considered between the two orbitals, say a and b with matrix element
so = |ψa,↑ |Hso |ψb,↓ | = |ψa,↓ |Hso |ψb,↑ |, spin-up and spin-down components are mixed
and finite phonon emission probability is expected between Zeeman sublevels under a magnetic
field [77, 92]. This spin-mixing and electron–phonon coupling between Zeeman sublevels
give the B −5 dependence of the relaxation time, as shown by optical excitation/detection
measurement for self-organized QDs [39, 40].
In the case of a two-electron state, which can be approximated by the Slater determinantof the one-electron orbitals, spin–orbit interaction gives selective mixing between different
spin states. The spin singlet state (|S) is coupled to two of the triplet sublevels (|T+  and|T− ) having SZ = ±1 but not to the other sublevel (|T0 ) having SZ = 0 [39, 93]. Therefore,
relaxation from the triplet to the singlet state should be governed by a selection rule in which

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T Fujisawa et al

Figure 21. (a) ST dependence of the τs for the inter-LL relaxations. Data represented by open
symbols are obtained by fitting a single exponential function, while the solid circle represents thefast component of the double exponential function. The dotted line is a guide for the eye. (b)Allowed and forbidden transitions from the triplet sublevels to the singlet state.

|T0  state is still free from the spin–orbit relaxation mechanism as shown in figure 21(b).
This simple argument applies when the singlet-triplet energies are so close to each other that
coupling with other states is negligible.
Figure 21(a) shows the magnetic field dependence of τs measured in the small-tot regime
where spin–orbit interaction is a dominant spin relaxation mechanism. The data are actuallyplotted against ST . τs is almost constant in a wide ST regime (except at a dip aroundST ∼ 380 µeV) and tends to increase when ST < 200 µeV, which might arise from thephonon emission spectra in a QD as discussed in section 3.2. The sharp dip observed atST ∼ 380 µeV is as narrow as about 20 µeV in ST (∼ 10 mT in B). It may be related to400 µeV), where the triplet state (T) crosses withthe level crossing at B
0.52 T (STanother state (X) as seen in figure 19(b). The state X can be identified as spin singlet, sinceunresolved anti-crossing behaviour (<100 µeV) is observed between X and T [80, 94, 95].
This situation suggests that strong spin–orbit coupling around the crossing point enhances
the spin relaxation rate, as theoretically predicted in [92]. It should be noted that the
decay of the pulse-induced current shows a non-single exponential behaviour around the
dip as shown in figure 22(b), while single exponential decay is always observed at otherconditions, e.g. at ST = 300 µeV shown in figure 22(a). The decay characteristic
in figure 22(b) can be expressed well by a double exponential function (the solid line),2exp(−th /τso ) + 13 exp(−th /τcot ) with a 2 : 1 ratio consistent with the spin–orbit selection
3
rule. Thus, the fast component with rate τso can be assigned to the relaxation from |T+  and|T−  via spin–orbit coupling and the slow component to the relaxation from |T0  via remaining
co-tunnelling contribution, τcot .
Although the above observations agree well with the selection rule for spin–orbit coupling,
we cannot safely rule out other possibilities. Hyperfine coupling to nuclear spins might appear
at the level coincidence of different spin states [42, 96]. In this case, one of the triplet stateswith finite Zeeman splitting can couple to the singlet state via flip–flop interaction, leading
to a 1 : 2 selectivity rather than the observed 2 : 1 ratio. However, our measurement does
not provide sufficient accuracy for identifying the origin of the selection rule. Identifying
the mechanism requires further investigations. Nevertheless, the observed double exponential
behaviour should be related to some selection rule for spin-triplet and unknown (probably
singlet) states.

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785

Figure 22. (a) Logarithmic plot of nt versus th at ST = 300 µeV (B = 0.6 T). The solid line isan exponential function fitted to the data. (b) A logarithmic plot of nt versus th at ST = 380 µeV(B = 0.55 T). The solid line is a double-exponential function fitted to the data. The dotted anddashed lines are the fast and slow components, respectively. Each inset is a linear plot of thesame data.

Now, let us assume spin–orbit coupling dominates the singlet-triplet relaxation process.
Then, we can think of an ‘entanglement generator’ using the selection rule. The singlet
ground state |S = |↑a |↓a holds a spin pair in an orbital a, while the triplet state
contains non-entangledstates, |T+  = |↑a |↑b and |T−  = |↓a |↓b , and entangled state,√|T0  = (1/ 2)(|↑a |↓b + |↓a |↑b ), with an electron in orbitals a and b. After a proper
waiting time after electron injection into one of the four states, the system is left in theentangled triplet state |T0  or the singlet ground state |S. Our pulse measurement is based on
the extraction of an electron only from the triplet excited state, and thus this scheme can beused to generate or analyse an entangled spin pair by detecting the extracted electron with asensitive electrometer [35, 36].
4. Statistics of single-electron tunnelling
Statistics of a transition from one state to another reflects the symmetry of the particle (boson or
fermion), interaction between the particles and coupling to the environment. Single-electron
tunnelling is a tunnelling transition that is influenced by Coulomb interaction inside a QD [59].
The transition is forbidden if the Coulomb energy cost for injecting an electron into the
QD exceeds the excitation energy of the transition (Coulomb blockade). Tunnelling is also
influenced by spin correlation (e.g. spin singlet or triplet), as shown by the Pauli spin blockade
effect in transport through a DQD [97].
Such single-electron tunnelling has been discussed with conventional current
measurement. However, current measurement has very low sensitivity in the sense that millions
of electrons are required to obtain a meaningful current signal. In contrast, charge detection
measurement allows direct detection of electron occupation in a QD and provides a sensitive
measurement for single-electron transport.
4.1. Charge detection techniques
The single-electron transistor (SET), whose transport is influenced by Coulomb interactionin the QD, is known as a sensitive electrometer with expected sensitivity of about

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T Fujisawa et al

Figure 23. (a) Schematic diagram of the measurement setup. The sample shown in the scanningelectron micrograph contains two GaAs QDs (white circles) made by a dry etching (upper, central
and lower dark regions) and Schottky gates (vertical bright lines). The white bar is a scale of 200 nm.
The RF carrier signal is transmitted through the lower QD and the LC resonator. The measurement
was done in a dilution refrigerator (T ∼ 0.1 K). (b) Energy diagram of a single-electron tunnelling
process.

√10−6 e/ Hz [98, 99]. The radio-frequency (RF) technique allows us to operate such a SETwith carrier signal at gigahertz frequency, and the charge detection bandwidth can exceed
100 MHz [100, 101]. Another type of electrometer can be made from a semiconductor
point contact (PC) structure, which exhibits quantized conductance corresponding to thenumber of occupied one-dimensional channels [102–104]. Tunnelling probability for anunoccupied channel depends on the potential barrier at the contact, which is sensitive tocharge distribution around the contact. Thus, these electrometers provide a charge detection
scheme for QDs electrostatically coupled to the electrometer. SET electrometer has essentiallybetter sensitivity than PC electrometer, since a large change in the transmission probability is
expected. However, the practical signal-to-noise ratio remains more or less the same, when
it is determined by background charge fluctuation in the device or other extrinsic noise in
the measurement system. In this case, the PC electrometer has the advantage of a simpler
structure, and it can be operated with fewer gate electrodes in a wide gate-voltage range.
Figure 23(a) shows a typical circuit diagram along with a scanning electron micrographfor charge detection measurement. Two electrically isolated channels are integrated in an
AlGaAs/GaAs heterostructure [101, 105]. A QD in the lower channel works as a SETelectrometer. One end of the SET is connected to an LC resonator and a low-noise highfrequency amplifier. An RF carrier signal at the resonant frequency of 650 MHz is sent fromthe other end of the SET, and the transmission signal is detected using a phase-sensitive
detection scheme. The output voltage Vdet is proportional to the conductance of the SET, sothat the SET serves as an RFSET electrometer. This SET is coupled to the other QD formed in
the upper channel. The coupling coefficient η (∼0.005 in this sample) is given by the coupling
capacitance between the two QDs relative to the total capacitance of the QD in the electrometer.
Charging an electron to the upper QD induces a fractional charge of about 0.005e on the lower
QD, which can be measured from a change in the electrometer output.
4.2. Charge detection of a quantum dot
Consider the two-level charge fluctuation in a QD in the single-electron tunnelling regime.
Only one excess electron can occupy the QD. By assuming a single spinless energy state in the
QD, as schematically shown in the energy diagram of figure 23(b), transport is characterizedby tunnelling rates L and R across the left and right barriers, respectively [59]. The current

Time-dependent single-electron transport

787

Figure 24. (a) Frequency spectra of the RFSET signal, Vdet , for CB condition and singleelectron transport conditions (#3–#6). Each spectrum is obtained by averaging data for about 1 h.
Sharp-peak artefacts from 50-Hz harmonics are removed for better visibility. (b) Time-domain
measurements of Vdet for CB (the lower trance) and single-electron tunnelling regime (the upper
trace for #5). No averaging is performed. (c) Histograms, H (Vdet ), of the two traces of (b) but fora longer time domain.

is defined by the net charge transfer from the left lead to the right lead. For instance, under arelatively large voltage that ensures forward tunnelling direction, the averaged current is given
by I = eL R /(L + R ). In contrast, an electrometer measures the occupation of the charge
state, regardless of which lead an electron was injected from or escaped to. Therefore, it isconvenient to characterize the charge fluctuation using the incoming rate (in ) and the outgoingrate (out ), respectively, for increasing and decreasing the electron number in the QD. Assuming
a purely random stochastic process for tunnelling transitions, the power spectrum of the charge
fluctuation is given by the Lorentzian form:
4e2(20)
Sq (ω) = −1.
−1(in + out) ω2 + (in + out )2
The cutoff frequency is determined by the total tunnelling rate  ≡ in + out , while the−1−1+ out) 2 . Therefore, in principle, onelow(zero)-frequency magnitude is given by 4e2 /(incan determine in and out from the power spectrum.
Figure 24(a) shows the power spectrum of the detector output, which is proportional
to Sq (ω) [106]. The bottom curve, labelled CB, is measured when the upper QD is in the
Coulomb blockade region, where no charge fluctuation is expected in the QD. This noise
floor arises from a background charge fluctuation (1/f spectrum) of the device in the lowfrequency region (1 kHz) and from white noise of the amplifier in the high-frequency region(1 kHz). When the QD is adjusted in the single-electron tunnelling regime, the excess power
spectrum is observed as shown by curves #3–#6, in figure 24(a). The curves can be fitted

788

T Fujisawa et alwell with the Lorentzian profile (dashed line for #5), and the characteristic frequency (f0 )shifts in accordance with the gate voltage (corresponding to the tunnelling rate changing from∼100 kHz (#3) to ∼10 Hz (#6)).
When a large bias voltage is applied, electrons transport from the left lead through the dot
to the right lead (in = L and out = R ). This is the case where the average current is related
to the charge fluctuation. Figure 24(b) shows typical real-time traces of the SET electrometer
output. The histogram of the detector voltage is shown in figure 24(c). The trace measured
in the single-electron tunnelling regime (#5) involves two-level fluctuations, as shown by the
double-peak structure in the histogram. This contrasts with the other trace taken in the CB
region (just noise with a single peak in the histogram). In the upper trace, represented by open
circles for incoming and solid circles for outgoing processes, five electrons pass through the
QD in a 10-ms period. The corresponding current (0.1 fA) is much smaller than the noise levelof conventional current meters, which suggests possible application of the charge detection
scheme for extremely sensitive current meter [100, 106, 107].
Real-time charge detection should work in the frequency range where the signal level is atleast a couple of decibels higher than the noise level. As seen in the frequency spectrum (e.g. #5)
of figure 24(a), the high-frequency limit is determined by the white noise of the measurement
system, while the low-frequency limit is determined by 1/f noise of the device. The frequency
range can be expanded by optimizing the device geometry to increase the coupling efficiency
or by improving the measurement setup.
Single electron tunnelling is a correlated tunnelling process: an electron that has entered
the small island leaves it before another electron is allowed to enter [59, 85]. Two-level
fluctuations seen in the above measurement indeed originate from the single-electron tunnellingcharacteristics, excluding double occupancy. In addition, single-electron tunnelling provides
intriguing time-correlated transport. In a one-dimensional array of small islands, Coulombinteraction (charge soliton) regulates the electron transport more periodically rather thanrandom Poisson statistics [108]. Recently, a single peak in the frequency spectrum of charge
fluctuation has been observed as a hallmark of single-electron tunnelling oscillation [109].
4.3. Single shot readout device for qubits
An important application of sensitive charge detection is a readout device for various kinds ofqubits. It is desirable to determine a single qubit state with a reasonably high fidelity (singleshot readout). An important criterion for single-shot readout is that the measurement time (tm )has to be much shorter than the dissipation time (T1 ) of the qubit [110]. Since a back action on
the qubit is inevitable, decoherence time (T2 ) during the measurement process cannot exceed
tm . In order to minimize the influence during quantum computation, it is desirable to be able
to switch on the measurement device only for the measurement process.
Naturally, charge measurement should be applied for readout of charge qubits [111].
Figure 25(a) shows a device structure containing a DQD in the upper electrical channel anda PC charge detector in the lower channel (The dotted regions are the conductive regions).
The two channels are well isolated (10 G ) by applying sufficiently large voltage Viso to
the isolation gate. The PC electrometer is placed on the left side of the structure so that theelectrometer is more sensitive to the left dot and less sensitive to the right one. Thus, one
electrometer is sufficient for detecting arbitrary charge states in the DQD.
First, we investigate a strongly (coherently) coupled DQD by conventional conductancemeasurement on the PC electrometer. The PC conductance is adjusted at the maximal
sensitivity condition (about half the quantized conductance). In order to improve the signalto-noise ratio, the charge state of the DQD is modulated by applying low-frequency (100 Hz)

Time-dependent single-electron transport

789

Figure 25. (a) Schematic measurement setup and a scanning electron micrograph of a double
quantum dot coupled with a QPC electrometer. (b) Schematic energy diagram of eigenstates. (c)Current through the QPC. (d) The derivative of the QPC current. (e) Experimental charge detection
signal. The dashed and dotted lines are fitted to the data by assuming thermal distribution andcoherent tunnelling coupling, respectively.

sinusoidal voltage Vmod to the electrode as shown in figure 25(a), and the corresponding
modulated current Idet through the PC is measured with a lock-in amplifier.
Figure 25(b) shows a schematic energy diagram of eigenenergies Ea and Eb when theenergy bias ε (same definition as in section 2.1) is changed. The PC current should follow the
charge occupation, Idet ∼ |R|ϕ|2 − |L|ϕ|2 , as schematically shown for the bonding statein figure 25(c). The derivative curve, dIdet /dε, is shown in figure 25(d) and is given byh̄2 T 2dIdet
∼ 2 2 C 2 3/2 .
dε(h̄ Tc + ε )(21)Figure 25(e) shows experimental traces of dIdet /dVmod around the charge degeneracy
point ε ≡ EL − ER = 0. The energy scale in the figure was determined from photon
assisted tunnelling spectroscopy with microwave irradiation (not shown) [19]. The peak in
the topmost trace at the central gate voltage, VC = −494 mV, is the narrowest one observed inthis measurement. The peak width is probably related to the external noise or ac modulation
voltage used. The profile can be fitted well by assuming Fermi-Dirac distribution of charges(dashed line),
dIdet
(22)
∼ cosh−2 (ε/kB Teff ),dεwith the effective thermal energy of kB Teff ∼ 30 µeV as a parameter. As the central gate voltage
VC is made less negative to increase the tunnelling coupling, the peak broadens, suggestingthat the two QDs are coherently coupled by the tunnelling coupling. The peaks for the lowest
two traces in figure 25(e) can be fitted well with equation 21 (see the dotted line fitted withh̄Tc ∼ 60 µeV for VC = −482 mV). The disagreement on the left-hand side arose from the

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T Fujisawa et al

Figure 26. (a) PC charge detection of a DQD. The PC current (colour) is plotted as a function of
two gate voltages VL and VR . Schematic charging diagram is attached. (b) Real time detection of
charge fluctuation measured at several points a–h in (a).

broad positive signal from another peak on the left. Therefore, the experiment demonstrates
that the PC detector can determine the charge distribution in a DQD even when the system isin a bonding state of the two localized charge states [112].
In contrast, when the DQD is weakly (incoherently) coupled, we can resolve timedependent fluctuation between localized charge states. Figure 26(a) shows a charging diagramof the DQD obtained from the PC current profile (colour plot) when gate voltages VL andVR for the DQD are swept. When these gate voltages are fixed at several points (labelleda–h) in the diagram, we see charge fluctuation between some charge states as shown infigure 26(b). Three-level fluctuation is observed at points c and f, where three charge states areenergetically degenerated, while two-level fluctuation is seen at d and e, where two charge states
are degenerated. No significant fluctuation is observed at a and h, where the DQD is Coulomb
blockaded. Observation of these fluctuations suggests the readout of the instantaneous charge
state of a DQD, which is an encouraging single-shot readout. In this setup, charge states canbe measured with a time resolution of about 500 µs.
However, there is a large gap between the measurement time and the typical T1 (∼10 ns,
discussed in section 2.3) of the charge qubit in a DQD. This gap can be overcome by introducing
an auxiliary island that traps an electron for a sufficiently long time for reading, as demonstrated
for a superconducting charge qubit [25]. Another strategy is to make T1 sufficiently long
only during the measurement period. Since T1 is determined by phonon emission process
between the charge states, one can control its rate by changing the tunnelling barrier [113,114].
Therefore, single-shot readout for a charge qubit is feasible by switching the interdot couplingfrom the strong regime for coherent control to the weak regime for readout.
The electron spin state in a QD can also be measured with a charge detector by converting
spin information to a charge state [36, 115]. Figure 27 illustrates spin-up and spin-down
sublevels of a ground state of a one-electron QD connected to a lead. By adjusting thechemical potential of the lead between the Zeeman split sublevels in a magnetic field as shownin figure 27(c), only a spin-down electron (for negative g-factor material) is allowed to escapefrom the QD, and subsequently a spin-up electron is injected from the lead (figure 27(d)).
Thus, single-spin measurement can be performed with an electrometer. The measurement has
been demonstrated by applying a two-level voltage pulse to a gate electrode to empty the QD

Time-dependent single-electron transport

791

Figure 27. Schematic charging diagram for single-shot charge detection. (a) Emptying a quantum
dot. (b) Injection of an electron with random spin orientation. The figure illustrates the case whenspin-down electron is injected. (c) Reading out spin-down state. (d) Relaxing into spin-up groundstate.

(figure 27(a)), inject an electron with random spin orientation (figure 27(b) for spin-down case)and read out the spin state (figures 27(c) and (d)). The charge state of the QD is measured
with a PC electrometer. Since the pulse also changes the PC current, the current displays the
charge state superimposed on the pulse waveform. The appearance of a signal in the read-out
sequence indicates that a spin-down electron was injected into the QD. Measurement error
arises from finite noise and limited bandwidth in the current measurement. The demonstration
yields 65% visibility, which is high enough to say that a single-shot readout of an electron spin
qubit has succeeded [36].
5. Summary
We have reviewed some time-dependent phenomena in semiconductor quantum dots. In theexperiments shown in section 2, a high-speed voltage pulse is applied to induce coherent
oscillations of a charge qubit in a double quantum dot. Tailoring the pulse waveform provides
arbitrary unitary transformation in the one-qubit space. However, coupling to the environment
causes decoherence (dissipation) problem. Although some decoherence mechanisms can be
reduced by developing better fabrication techniques and refining measurement conditions,intrinsic electron–phonon interaction, especially piezoelectric coupling, would dominate
the dissipation processes in semiconductors. In this sense, quantum dots in non-polar
semiconductors (Si, carbon based material) may be preferable. Relaxation dynamics has also
been focused in section 3, where the electronic pump–probe technique is used to determine
the dissipation time. One can significantly reduce the relaxation rate by making use of spin
selection rules for the relaxation. The spin degree of freedom is more isolated from thephonon bath, as we demonstrated in a quantum dot. This evidence has motivated many people
to consider electron spin qubit as one of the best candidates for many-qubit systems. We havealso discussed the spin selection rule for spin–orbit coupling in the two-electron system, inwhich the spin-correlated triplet state is expected to have a much longer relaxation time. A
full understanding of the dissipation and decoherence mechanism may provide a clever way
to overcome the decoherence problem. Moreover, the selection rule in the spin–orbit coupling
can be used to create or determine spin entanglement state by combining with the charge
detection scheme.
On the other hand, high-speed and high-precision measurement of a quantum state isdesired to obtain a reliable outcome from a single qubit. In section 4, charge detection
measurements with a single electron transistor or a quantum point contact device were describedin the context of single-shot readout for charge and spin qubit. The technique is rapidly

792

T Fujisawa et aldeveloping for better visibility and higher speed. Interaction of a qubit with a measurement
apparatus, which is one of the fundamental problems in quantum mechanics, can be investigatedand designed in further studies. Moreover, charge detection scheme is expected to provide asingle-electron counting device, in which individual electron tunnelling processes are recorded
in a real time scale. Electron counting statistics will be a new tool to investigate correlated
electron transport of a small current.
Finally, these techniques can be combined with other mesoscopic electron devices, suchas electron interferometer and beam splitter, by means of nanotechnology. In analogy withthe optical technique used in quantum optics, integration of these components would provide
a novel measurement concept for nontrivial quantum states. We believe these techniques canbe applied for a better understanding of quantum transport in nanostructures as well as fordeveloping quantum engineering in solid state devices.

Acknowledgments
We thank T Brandes, Y Hirayama, T Itakura, Y Nakamura, S Tarucha, and Y Tokura forstimulating discussions. Some of the presented works were supported by the Strategic
Information and Communications R&D Promotion Programme (SCOPE) of the Ministry ofInternal Affairs and Communications of Japan, Grant-in-Aid for Scientific Research from theJapan Society for the Promotion of Science and Solution Oriented Research for Science andTechnology (SORST Program) of the Japan Science and Technology Agency.

References[1] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge
University Press)[2] Bouwmeester D, Ekert A and Zeilinger A (ed) 2000 The Physics of Quantum Information (Berlin: Springer)[3] Vandersypen L M K, Steffen M, Breyta G, Yannoni C S, Sherwood M H and Chuang I L 2001 Experimental
realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance Nature 414 883[4] Nakamura Y, Pashkin Yu A and Tsai J S 1999 Coherent control of macroscopic quantum states in a singleCooper-pair box Nature 398 786[5] Chiorescu I, Bertet P, Semba K, Nakamura Y, Harmans C J P M and Mooij J E 2004 Coherent dynamics of aflux qubit coupled to a harmonic oscillator Nature 431 159[6] Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R-S, Majer J, Kumar S, Girvin S M and Schoelkopf R J
2004 Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics
Nature 431 162[7] Kamada H, Gotoh H, Temmyo J, Takagahara T and Ando H 2001 Exciton Rabi oscillation in a single quantum
dot Phys. Rev. Lett. 87 246401[8] Jelezko F, Gaebel T, Popa I, Domhan M, Gruber A and Wrachtrup J 2004 Observation of coherent oscillationof a single nuclear spin and realization of a two-qubit conditional quantum gate Phys. Rev. Lett. 93 130501[9] Yusa G, Muraki K, Takashina K, Hashimoto K and Hirayama Y 2005 Controlled multiple quantum coherencesof nuclear spins in a nanometre-scale device Nature 434 1001
[10] Weiss U 2001 Quantum Dissipative Systems (Singapore: World Scientific)[11] DiVincenzo D P 1997 Mesocopic Electron Transport (NATO ASI Series E 345) ed L L Sohn et al (Dordrecht:
Kluwer) pp 657–77[12] Nakamura Y, Pashkin Yu A, Yamamoto T and Tsai J S 2002 Charge echo in a cooper-pair box Phys. Rev. Lett.
88 047901[13] Vion D, Aassime A, Cottet A, Joyez P, Pothier H, Urbina C, Esteve D and Devoret M H 2002 Manipulating
the quantum state of an electrical circuit Science 296 886[14] Kouwenhoven L P, Marcus C M, McEuen P L, Tarucha S, Westervelt R M and Wingreen N S 1997 Mesoscopic
Electron Transport (NATO ASI Series E 345) ed Sohn L L et al (Dordrecht: Kluwer) pp 105–214[15] Kouwenhoven L P, Austing D G and Tarucha S 2001 Few-electron quantum dots Rep. Prog. Phys. 64 701

Time-dependent single-electron transport793[16] Tarucha S, Austing D G, Honda T, van der Hage R J and Kouwenhoven L P 1996 Shell filling and spin effects
in a few electron quantum dot Phys. Rev. Lett. 77 3613
[17] Kouwenhoven L P, Oosterkamp T H, Danoesastro M W S, Eto M, Austing D G, Honda T and Tarucha S 1997
Excitation spectra of circular, few-electron quantum dots Science 278 1788
[18] Hayashi T, Fujisawa T, Cheong H D and Hirayama Y 2003 Coherent manipulation of electronic states in a
double quantum dot Phys. Rev. Lett. 91 226804[19] Oosterkamp T H, Fujisawa T, van der Wiel W G, Ishibashi K, Hijman R V, Tarucha S and Kouwenhoven L P
1998 Microwave spectroscopy of a quantum-dot molecule Nature 395 873[20] Fujisawa T, Hayashi T, Cheong H D, Jeong Y H and Hirayama Y 2004 Rotation and phase-shift operations fora charge qubit in a double quantum dot Physica E 21 1046
[21] Fujisawa T, Hayashi T and Hirayama Y 2004 Controlled decoherence of a charge qubit in a double quantum
dot J. Vac. Sci. Technol. B 22 2035
[22] Barenco A, Deutsch D, Ekert A and Jozsa R 1995 Conditional quantum dynamics and logic gates Phys. Rev.
Lett. 74 4083
[23] Fedichkin L, Yanchenko M and Valiev K A 2000 Coherent charge qubits based on GaAs quantum dots with abuilt-in barrier Nanotechnology 11 387[24] Yamamoto T, Pashkin Yu A, Astafiev O, Nakamura Y and Tsai J S 2003 Demonstration of conditional gate
operation using superconducting charge qubits Nature 425 941[25] Astafiev O, Pashkin Yu A, Nakamura Y, Yamamoto T and Tsai J S 2004 Quantum noise in the Josephson
charge qubit Phys. Rev. Lett. 93 267007[26] Kirton M J and Uren M J 1989 Noise in solid-state microstructures—a new perspective on individual defects,
interface states and low-frequency (1/f ) noise Adv. Phys. 38 367[27] Brandes T and Kramer B 1999 Spontaneous emission of phonons by coupled quantum dots Phys. Rev. Lett. 83
3021[28] Brandes T and Vorrath T 2002 Adiabatic transfer of electrons in coupled quantum dots Phys. Rev. B 66 075341[29] Fujisawa T, Oosterkamp T H, van der Wiel W G, Broer B W, Aguado R, Tarucha S and Kouwenhoven L P
1998 Spontaneous emission spectrum in double quantum dot devices Science 282 932[30] Bandyopadhyay S and Roychowdhury V P 1997 Switching in a reversible spin logic gate Superlatt. Microstruct.
22 411[31] Loss D and DiVincenzo D P 1998 Quantum computation with quantum dots Phys. Rev. A 57 120[32] Burkard G, Loss D and DiVincenzo D P 1999 Coupled quantum dots as quantum gates Phys. Rev. B 59 2070
[33] Hatano T, Stopa M and Tarucha S 2005 Single-electron delocalization in hybrid vertical-lateral double quantum
dots Science 309 268[34] Hanson R, Witkamp B, Vandersypen L M K, Willems van Beveren L H, Elzerman J M and Kouwenhoven L P
2003 Zeeman energy and spin relaxation in a one-electron quantum dot Phys. Rev. Lett. 91 196802[35] Hanson R, Vandersypen L M K, Willems van Beveren L H, Elzerman J M, Vink I T and Kouwenhoven L P
2004 Semiconductor few-electron quantum dot operated as a bipolar spin filter Phys. Rev. B 70 241304[36] Elzerman J M, Hanson R, Willems van Beveren L H, Witkamp B, Vandersypen L M K and Kouwenhoven L P
2004 Single-shot read-out of an individual electron spin in a quantum dot Nature 430 431[37] Winkler R 2003 Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Berlin: Springer)[38] Frenkel D M 1991 Spin relaxation in GaAs/Alx Ga1−x As heterostructures in high magnetic fields Phys. Rev. B
43 14228[39] Khaetskii A V and Nazarov Yu V 2000 Spin relaxation in semiconductor quantum dots Phys. Rev. B 61 12639[40] Kroutvar M, Ducommun Y, Heiss D, Bichler M, Schuh D, Abstreiter G and Finley J J 2004 Opticallyprogrammable electron spin memory using semiconductor quantum dots Nature 432 81[41] Fujisawa T, Austing D G, Tokura Y, Hirayama Y and Tarucha S 2002 Allowed and forbidden transitions in
artificial hydrogen and helium atoms Nature 419 278[42] Koppens F H L, Folk J A, Elzerman J M, Hanson R, Willems van Beveren L H, Vink I T, Tranitz H P,
Wegscheider W, Kouwenhoven L P and Vandersypen L M K 2005 Control and detection of singlet-triplet
mixing in a random nuclear field Science 309 1346
[43] Xiao M, Martin I, Yablonovitch E and Jiang H W 2004 Electrical detection of the spin resonance of a single
electron in a silicon field-effect transistor Nature 430 435[44] Kato Y, Myers R C, Driscoll D C, Gossard A C, Levy J and Awschalom D D 2003 Gigahertz electron spin
manipulation using voltage-controlled g-tensor modulation Science 299 1201
[45] Tarucha S, Austing D G, Tokura Y, van der Wiel W G and Kouwenhoven L P 2000 Direct Coulomb andexchange interaction in artificial atoms Phys. Rev. Lett. 84 2485
[46] Liu H W, Fujisawa T, Hayashi T and Hirayama Y 2005 Pauli spin blockade in cotunnelling transport througha double quantum dot Phys. Rev. B 72 161305

794

T Fujisawa et al[47] Petta J R, Johnson A C, Taylor J M, Laird E A, Yacoby A, Lukin M D, Marcus C M, Hanson M P and Gossard A C
2005 Coherent manipulation of coupled electron spins in semiconductor quantum dots Science 309 2180
[48] Taylor J M, Marcus C M and Lukin M D 2003 Long-lived memory for mesoscopic quantum bits Phys. Rev.
Lett. 90 206803[49] Beenakker C W J and van Houten H 1991 Quantum transport in semiconductor nanostructure Solid State Phys.
Adv. Res. Appl. 44 1
[50] Hirayama Y, Saku T, Tarucha S and Horikoshi Y 1991 Ballistic electron-transport in macroscopic 4-terminal
square structures with high mobility Appl. Phys. Lett. 58 2672
[51] Spector J, Stormer H L, Baldwin K W, Pfeiffer L N and West K W 1990 Refractive switch for 2 dimensional
electrons Appl. Phys. Lett. 56 2433
[52] Aharonov Y and Bohm D 1959 Significance of electromagnetic potentials in the quantum theory Phys. Rev.
115 485[53] Al’tshuler B L, Aronov A G and Spivak B Z 1981 The Aharonov-Bohm effect in disordered conductors JETP
Lett. 33 94
Al’tshuler B L, Aronov A G and Spivak B Z 1981 Pis’ma Zh. Teor. Fiz. 33 101[54] Timp G, Chang A M, DeVegvar P, Howard R E, Behringer R, Cunningham J E and Mankiewich P 1988
Quantum transport in one-dimensional GaAs/AlGaAs microstructures Surf. Sci. 196 68[55] Schuster R, Buks E, Heiblum M, Mahalu D, Umansky V and Shtrikman H 1997 Phase measurement in aquantum dot via a double-slit interference experiment Nature 385 417[56] Liu R C, Odom B, Yamamoto Y and Tarucha S 1998 Quantum interference in electron collision Nature
391 263[57] Recher P, Saraga D S and Loss D 2004 Creation and detection of mobile and non-local spin-entangled electrons
Preprint cond-mat/0408526[58] Beenakker C W J, DiVincenzo D P, Emary C and Kindermann M 2004 Charge detection enables free-electron
quantum computation Phys. Rev. Lett. 93 020501[59] Grabert H and Devoret M H (ed) 1991 Single Charge Tunneling, Coulomb Blockade Phenomena in
Nanostuctures (NATO ASI Series B 294) (New York: Plenum)[60] van der Wiel W G, De Franceschi S, Elzerman J M, Fujisawa T, Tarucha S and Kouwenhoven L P 2003 Electrontransport through double quantum dots Rev. Mod. Phys. 75 1
[61] Waugh F R, Berry M J, Mar D J, Westervelt R M, Campman K L and Gossard A C 1995 Single-electron
charging in double and triple quantum dots with tunable coupling Phys. Rev. Lett. 75 705[62] Livermore C, Crouch C H, Westervelt R M, Campman K L and Gossard A C 1996 The Coulomb blockade incoupled quantum dots Science 274 1332
[63] Nazarov Yu V 1993 Quantum interference, tunnel-junctions and resonant-tunnelling interferometer Physica B
189 57[64] van der Vaart N C, Godijn S F, Nazarov Yu V, Harmans C J P M and Mooij J E 1995 Resonant-tunnellingthrough 2 discrete energy levels Phys. Rev. Lett. 74 4702
[65] Fujisawa T and Tarucha S 1997 Photon assisted tunnelling in single and coupled quantum dot systems Superlatt.
Microstruct. 21 247[66] Stoof T H and Nazarov Yu V 1996 Time-dependent resonant tunnelling via two discrete states Phys. Rev. B 53
1050
[67] Stafford C A and Wingreen N S 1996 Resonant photon-assisted tunnelling through a double quantum dot: anelectron pump from spatial rabi oscillations Phys. Rev. Lett. 76 1916
[68] van der Wiel W G, Fujisawa T, Oosterkamp T H and Kouwenhoven L P 1999 Microwave spectroscopy of adouble quantum dot in the low- and high-power regime Physica B 272 31[69] Itakura T and Tokura Y 2003 Dephasing due to background charge fluctuations Phys. Rev. B 67 195320[70] Paladino E, Faoro L, Falci G and Fazio R 2002 Decoherence and 1/f noise in Josephson qubits Phys. Rev.
Lett. 88 228304[71] Jung S W, Fujisawa T, Hirayama Y and Jeong Y H 2004 Background charge fluctuation in a GaAs quantum
dot device Appl. Phys. Lett. 85 768[72] Pioro-Ladrière M, Davies J H, Long A R, Sachrajda A S, Gaudreau L, Zawadzki P, Lapointe J, Gupta J,
Wasilewski Z and Studenikin S A 2005 The origin of switching noise in GaAs/AlGaAs lateral gated devices
Preprint cond-mat/0503602[73] Eto M 2001 Electronic states and transport phenomena in quantum dot systems Japan. J. Appl. Phys.
40 1929[74] Seeger K 1985 Semiconductor Physics: An Introduction (Berlin: Springer) pp 153–213[75] Hollenberg L C L, Dzurak A S, Wellard C, Hamilton A R, Reilly D J, Milburn G J and Clark R G 2004
Charge-based quantum computing using single donors in semiconductors Phys. Rev. B 69 113301

Time-dependent single-electron transport

795[76] Brandes T, Renzoni F and Blick R H 2001 Adiabatic steering and determination of dephasing rates in double-dot
qubits Phys. Rev. B 64 035319[77] Fujisawa T, Austing D G, Tokura Y, Hirayama Y and Tarucha S 2003 Electrical pulse measurement, inelastic
relaxation and non-equilibrium transport in a quantum dot J. Phys.: Condens. Matter. (Topical Review) 15
R1395[78] Fujisawa T, Tokura Y and Hirayama Y 2001 Transient current spectroscopy of a quantum dot in the Coulomb
blockade regime Phys. Rev. B 63 081304(R)[79] Fujisawa T, Tokura Y and Hirayama Y 2001 Energy relaxation process in a quantum dot studied by DC current
and pulse-excited current measurements Physica B 298 573[80] Tokura Y, Sasaki S, Austing D G and Tarucha S 2001 Excitation spectra and exchange interactions in circularand elliptical quantum dots Physica B 298 260–4[81] Benisty H 1995 Reduced electron–phonon relaxation rates in quantum-box systems—theoretical analysis Phys.
Rev. B 51 13281[82] Weinmann D, Häusler W and Kramer B 1995 Spin blockades in linear and nonlinear transport through quantum
dots Phys. Rev. Lett. 74 984[83] Bockelmann U and Bastard G 1990 Phonon-scattering and energy relaxation in 2-dimemsional, onedimensional and zero-dimensional electron gases Phys. Rev. B 42 8947
[84] Bockelmann U 1994 Phonon-scattering between zero-dimensional electronic states—spatial versus Launau
Phys. Rev. B 50 17271[85] Fujisawa T, Austing D G, Tokura Y, Hirayama Y and Tarucha S 2002 Nonequilibrium transport through avertical quantum dot in the absence of spin-flip energy relaxation Phys. Rev. Lett. 88 236802[86] Fujisawa T, Tokura Y, Austing D G, Hirayama Y and Tarucha S 2002 Spin-dependent energy relaxation inside
a quantum dot Physica B 314 224[87] De Franceschi S, Sasaki S, Elzerman J M, van der Wiel W G, Tarucha S and Kouwenhoven L P 2001 Electron
cotunnelling in a semiconductor quantum dot Phys. Rev. Lett. 86 878[88] Averin D V and Nazarov Yu V 1991 Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostuctures(NATO ASI Series B 294) ed H Grabert and M H Devoret (New York: Plenum) pp 217–48[89] Sachrajda A S et al 2001 Spin polarized injection into a quantum dot by means of the spatial separation of
spins Physica E (Amsterdam) 10 493[90] Ciorga M et al 2001 Readout of a single electron spin based quantum bit by current detection Physica E(Amsterdam) 11 35[91] Khaetskii A V and Nazarov Yu V 2000 Spin-dephasing processes in semiconductor quantum dots Physica E
6 470[92] Bulaev D V and Loss D 2005 Spin relaxation and anticrossing in quantum dots: Rashba versus Dresselhaus
spin-orbit coupling Phys. Rev. B 71 205324[93] Dickmann S and Hawrylak P 2003 Spin-singlet-spin-triplet transitions in quantum dots J. Supercond. 16 387[94] Drouvelis P S et al 2004 Effects of anisotropy and magnetic fields on two-electron parabolic quantum dots
J. Phys.: Condens. Matter. 16 3633
[95] Maksym P A 1998 Quantum states of interacting electrons in a 2D elliptical quantum dot Physica B (Amsterdam)
249–251 233[96] Ono K and Tarucha S 2004 Nuclear-spin-induced oscillatory current in spin-blockaded quantum dots Phys.
Rev. Lett. 92 256803[97] Ono K, Austing D G, Tokura Y and Tarucha S 2002 Current rectification by Pauli exclusion in a weakly coupled
double quantum dot system Science 297 1313
[98] Devoret M H and Schoelkopf R J 2000 Amplifying quantum signals with the single-electron transistor Nature
406 1039[99] Korotkov A N and Paalanen M A 1999 Charge sensitivity of radio frequency single-electron transistor Appl.
Phys. Lett. 74 4052
[100] Schoelkopf R J, Wahlgren P, Kozhevnikov A A, Delsing P and Prober D E 1998 The radio-frequency singleelectron transistor (RF-SET): a fast and ultrasensitive electrometer Science 280 1238
[101] Fujisawa T and Hirayama Y 2000 Charge noise analysis of an AlGaAs/GaAs quantum dot using transmissiontype radio-frequency single-electron transistor technique Appl. Phys. Lett. 77 543[102] van Wees B J, Kouwenhoven L P, van Houten H, Beenakker C W J, Mooij J E, Foxon C T and Harris J J 1988
Quantized conductance of magnetoelectric subbands in ballistic point contacts Phys. Rev. B 38 3625
[103] Field M, Smith C G, Pepper M, Ritchie D A, Frost J E F, Jones G A C and Hasko D G 1993 Measurements of
Coulomb blockade with a noninvasive voltage probe Phys. Rev. Lett. 70 1311
[104] Rushforth A W, Smith C G, Godfrey M D, Beere H E, Ritchie D A and Pepper M 2004 Noninvasive detectionof the evolution of the charge states of a double dot system Phys. Rev. B 69 113309

796

T Fujisawa et al[105] Cheong H D, Fujisawa T, Hayashi T, Hirayama Y and Jeong Y H 2002 Impedance analysis of a radio-frequency
single-electron transistor Appl. Phys. Lett. 81 3257
[106] Fujisawa T, Hayashi T, Hirayama Y, Cheong H D and Jeong Y H 2004 Electron counting of single-electron
tunnelling current Appl. Phys. Lett. 84 2343
[107] Lu W, Ji Z, Pfeiffer L, West K W and Rimberg A J 2003 Real-time detection of electron tunnelling in a quantum
dot Nature 423 422[108] Delsing P, Likharev K K, Kuzmin L S and Claeson T 1989 Time-correlated single-electron tunnelling in
one-dimensional arrays of ultrasmall tunnel-junctions Phys. Rev. Lett. 63 1861[109] Bylander J, Duty T and Delsing P 2005 Current measurement by real-time counting of single electrons Nature
434 361[110] Makhlin Yu, Schön G and Shnirman A 2000 Statistics and noise in a quantum measurement process Phys. Rev.
Lett. 85 4578 (2000)[111] Aassime A, Johansson G, Wendin G, Schoelkopf R J and Delsing P 2001 Radio-frequency single-electrontransistor as readout device for qubits: charge sensitivity and backaction Phys. Rev. Lett. 86 3376
[112] DiCarlo L, Lynch H J, Johnson A C, Childress L I, Crockett K, Marcus C M, Hanson M P and Gossard A C
2004 Differential charge sensing and charge delocalization in a tunable double quantum dot Phys. Rev. Lett.
92 226801[113] Fujisawa T, van der Wiel W G and Kouwenhoven L P 2000 Inelastic tunnelling in a double quantum dot coupled
to a bosonic environment Physica E 7 413[114] Hayashi T, Fujisawa T, Tomita R and Hirayama Y Real-time observation of charge states and energy relaxation
in a double quantum dot Japan. J. Appl. Phys. at press[115] Kane B E, McAlpine N S, Dzurak A S, Clark R G, Milburn G J, Sun H B and Wiseman H 2000 Single-spinmeasurement using single-electron transistors to probe two-electron systems Phys. Rev. B 61 2961