Quantum mechanical approach to decoherence and relaxation generated by
fluctuating environment
S.A. Gurvitz∗
Department of Particle Physics, Weizmann Institute of Science,
Rehovot 76100, Israel and Theoretical Division and CNLS,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
D. Mozyrsky
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: November 4, 2018)
We consider an electrostatic qubit, interacting with fluctuating charge of a single electron tran-
sistor (SET) in the framework of an exactly solvable model. The SET plays role of an environment
affecting the qubits’ parameters in a controllable way. We derive the rate equations describing the
dynamics of the entire system for an arbitrary qubit-SET coupling. Solving these equations we
obtain decoherence and relaxation rates of the qubit, as well as the spectral density of qubit param-
eters’ fluctuations. We found that in a weak coupling regime decoherence and relaxation rates are
directly related to the spectral density taken at either zero or Rabi frequency, depending on which
qubit parameter is fluctuating. In the latter case our result coincides with that of the spin-boson
model in the weak coupling limit, despite different origin of the fluctuations. We show that this
relation holds also in the presence of weak back-action of the qubit on the environment. In case of
strong back-action such a simple relationship no longer holds, even if qubit-SET coupling is small.
It does not hold also in the strong coupling regime, even in the absence of the back-action. In
addition, we found that our model predicts localization of the qubit in the strong-coupling regime,
resembling that in the spin-boson model.
PACS numbers: 03.65.Yz, 05.60.Gg, 73.23.-b, 73.23.Hk
I. INTRODUCTION
The influence of environment on a single quantum sys-
tem is the issue of crucial importance in quantum in-
formation science. It is mainly associated with decoher-
ence, or dephasing, which transforms any pure state of
a quantum system into a statistical mixture. Despite a
large body of theoretical work devoted to decoherence, its
mechanism has not been clarified enough. For instance,
how decoherence is related to environmental noise, in
particular in the presence of back-action of the system
on the environment (quantum measurements). More-
over, decoherence is often intermixed with relaxation. Al-
though each of them represents an irreversible process,
decoherence and relaxation affect quantum systems in
quite different ways.
In order to establish a relation between the fluctua-
tion spectrum and decoherence and relaxation rates one
needs a model that describes the effects of decoherence
and relaxation in a consistent quantum mechanical way.
An obvious candidate is the spin-boson model1,2 which
represents the environment as a bath of harmonic oscil-
lators at equilibrium, where the fluctuations obey Gaus-
sian statistics3. Despite its apparent simplicity, the spin-
boson model cannot be solved exactly2. Also, it is hard
to manipulate the fluctuation spectrum in the framework
of this model. In addition, mesoscopic structures may
couple only to a few isolated fluctuators, like spins, lo-
cal currents, background charge fluctuations, etc. This
would require models of the environment, different from
Electrodes
FIG. 1: Electrostatic qubit, realized by an electron trapped
in a coupled-dot system (a), and its schematic representation
by a double-well (b). Ω0 denotes the coupling between the
two dots.
the spin-boson model (see for instace4,5,6,7,8,9,10,11,12). In
general, the environment can be out of equilibrium, like
a steady-state fluctuating current, interacting with the
qubit13,14,15,16. This for instance, takes place in the con-
tinuous measurement (monitoring) of quantum systems17
and in the “control dephasing” experiments18,19,20. All
these types on non-Gaussian and non-equilibrium envi-
ronments attracted recently a great deal of attention21.
In this paper we consider an electrostatic qubit, which
can be viewed as a generic example of two-state systems.
It is realized by an electron trapped in coupled quantum
dots22,23,24, Fig. 1. Here E1 and E2 denote energies of
the electron states in each of the dots and Ω0 is a cou-
pling between these states. It is reasonable to assume
that the decoherence of a qubit is associated with fluctu-
http://arxiv.org/abs/0704.0194v3
Electrostatic Qubit
Single Electron Transistor
(a) (b)
FIG. 2: Qubit near Single Electron Transistor. Here El,r and
E0 denote the energy levels in the left (right) reservoirs and in
the quantum dot, respectively, and µL,R are the correspond-
ing chemical potentials. The electric current I generates fluc-
tuations of the electrostatic opening between two dots (a), or
it fluctuates the energy level of the nearest dot (b).
ations of the qubit parameters, E1,2 and Ω0, generated
by the environment. Indeed, a stochastic averaging of
the Schrödinger equation over these fluctuations param-
eters results in the qubit’s decoherence, which transfers
any qubit state into a statistical mixture25,26. In general,
one can expect that the fluctuating environment should
result in the qubit’s relaxation, as well, as for instance in
the phenomenological Redfield’s description of relaxation
in the magnetic resonance27.
As a quantum mechanical model of the environment
we consider a Single Electron Transistor (SET) capac-
itively coupled to the qubit, e.g., Fig. 2. Such setup
has been contemplated in numerous solid state quantum
computing architectures where SET plays role of a read-
out device16,17,28,29 and contains most of the generic fea-
tures of a fluctuating non-equilibrium environment. The
discreteness of the electron charge creates fluctuations
in the electrostatic field near the SET. If the electro-
static qubit is placed near the SET, this fluctuating field
should affect the qubit behavior as shown in Fig. 2. It
can produce fluctuations of the tunneling coupling be-
tween the dots (off-diagonal coupling) by narrowing the
electrostatic opening connecting these dots, as in Fig. 2a,
or make the energy levels of the dots fluctuate, as shown
schematically in Fig. 2b. Note that while in some regimes
the SET operates as a measuring device16,17, in other
regimes it corresponds purely to a source of noise. In-
deed, if the energy level E0, Fig. 2, is deeply inside
the voltage bias – the case we consider in the begin-
ning, the SET current is not modulated by the qubit
electron. In this case the SET represents only the fluc-
tuating environment affecting the qubit behavior (“pure
environment”30).
A similar model of the fluctuating environment has
been studied mostly for small bias (linear response) or
for the environment in an equilibrium. Here, however,
we consider strongly non-equilibrium case where the bias
voltage applied on the SET (V = µL − µR) is much
larger than the levels widths and the coupling between
the SET and the qubit. In this limit our model can be
solved exactly for both weak and strong coupling (but
is still smaller than the bias voltage). This constitutes
an essential advantage with regard to perturbative treat-
ments of similar models. For instance, the results of our
model can be compared in different regimes with phe-
nomenological descriptions used in the literature. Such
a comparison would allow us to determine the regions
where these phenomenological models are valid.
Since our model is very simple in treatment, the deco-
herence and relaxation rates can be extracted from the
exact solution analytically, as well as the time-correlator
of the electric charge inside the SET. This would make
it possible to establish a relation between the frequency-
dependent fluctuation spectrum of the environment and
the decoherence and the relaxation rates of the qubit,
and to determine how far this relation can be extended.
We expect that such a relation should not depend on
a source of fluctuations. This point can be verified by
a comparison with a similar results obtained for equi-
librium environment in the framework of the spin-boson
model1,2.
It is also important to understand how the decoher-
ence and relaxation rates depend on the frequency of
the environmental fluctuations. This problem has been
investigated in many phenomenological approaches for
“classical” environments at equilibrium. Yet, there still
exists an ambiguity in the literature related to this point
for non-equilibrium environment. For instance, it was
found by Levinson that the decoherence rate, generated
by fluctuations of the energy level in a single quantum dot
is proportional to the spectral density of fluctuations at
zero frequency31. The same result, but for a double-dot
system has been obtained by Rabenstein et al.32. On
the other hand, it follows from the Redfied’s approach
that the corresponding decoherence rate is proportional
to the spectral density at the frequency of the qubit’s
oscillations (the Rabi frequency)27. Since our model is
the exactly solvable one, we can resolve this ambiguity
and establish the appropriate physical conditions that
can result in different relations of decoherence rate to
the environmental fluctuations.
The most important results of our study are related to
the situation when back-action of the qubit on the envi-
ronment takes place. This problem did not receive such
a considerable amount of attention in the literature as,
for example, the case of “inert” environment. This is in
spite of a fact that the back-action always takes place in
the presence of measurement. There are many questions
related to the effects of a back-action. For instance, what
would be a relation between decoherence (relaxation) of
the qubit and the noise spectrum of the environment?
Or, how decoherence is affected by a strong response of
environment? We believe that our model appears to be
more suitable for studying these and other problems re-
lated to the back-action than most of the other existing
approaches.
The plan of this paper is as follows: Sect. II presents a
phenomenological description of decoherence and relax-
ation in the framework of Bloch equations, applied to the
electrostatic qubit. Sect. III contains description of the
model and the quantum rate-equation formalism, used
for its solution. Detailed quantum-mechanical derivation
of these equations for a specific example is presented in
Appendix A. Sect. IV deals with a configuration where
the SET can generate only decoherence of the qubit. We
consider separately the situations when SET produces
fluctuations of the tunneling coupling (Rabi frequency)
or of the energy levels. The results are compared with
the SET fluctuation spectrum, evaluated in Appendix B.
Sect. V deals with a configuration where the SET gener-
ates both decoherence and relaxation of the qubit. Sect.
VI is summary.
II. DECOHERENCE AND RELAXATION OF A
QUBIT
In this section we describe in a general phenomeno-
logical framework the effect of decoherence and relax-
ation on the qubit behavior. Although the results are
known, there still exists some confusion in the literature
in this issue. We therefore need to define precisely these
quantities and demonstrate how the corresponding deco-
herenece and relaxation rates can be extracted from the
qubit density matrix.
Let us consider an electrostatic qubit, realized by an
electron trapped in coupled quantum dots, Fig. 1. This
system is described by the following tunneling Hamilto-
Hqb = E1a
1a1 + E2a
2a2 − Ω0(a
2a1 + a
1a2) (1)
where a
1,2, a1,2 are the creation and annihilation opera-
tors of the electron in the first or in the second dot. For
simplicity we consider electrons as spinless fermions. In
addition, we assume that a
1a1 + a
2a2 = 1, so that only
one electron is present in the double-dot. The electron
wave function can be written as
|Ψ(t)〉 =
b(1)(t)a
1 + b
(2)(t)a
〉 (2)
where b(1,2)(t) are the probability amplitudes for find-
ing the electron in the first or second well, obtained
from the Schrödinger equation i∂t|Ψ(t)〉 = Hqb|Ψ(t)〉 (we
adopt the units where ~ = 1 and the electron charge
e = 1). The corresponding density matrix, σjj′ (t) =
b(j)(t)b(j
′)∗(t), with j, j′ = {1, 2}, is obtained from the
equation i∂t σ = [H,σ]. This can be written explicitly as
σ̇11 = iΩ0(σ21 − σ12) (3a)
σ̇12 = −iǫσ12 + iΩ0(1− 2σ11) , (3b)
where σ22(t) = 1 − σ11(t), σ21(t) = σ∗12(t) and ǫ =
E1 − E2. Solving these equations one easily finds that
the electron oscillates between the two dots (Rabi oscil-
lations) with frequency ωR =
4Ω20 + ǫ
2. For instance,
for the initial conditions σ11(0) = 1 and σ12(0) = 1, the
probability of finding the electron in the second dot is
σ22(t) = 2(Ω0/ωR)
2(1− cosωRt). This result shows that
for ǫ≫ Ω0 the amplitude of the Rabi oscillations is small,
so the electron remains localized in its initial state.
The situation is different when the qubit interacts with
the environment. In this case the (reduced) density ma-
trix of the qubit σ(t) is obtained by tracing out the en-
vironment variables from the total density matrix. The
question is how to modify Eqs. (3), written for an isolated
qubit, in order to obtain the reduced density matrix of
the qubit, σ(t). In general one expects that the environ-
ment could affect the qubit in two different ways. First, it
can destroy the off-diagonal elements of the qubit density
matrix. This process is usually referred to as decoherence
(or dephasing). It can be accounted for phenomenolog-
ically by introducing an additional (damping) term in
Eq. (3b),
σ̇12 = −iǫσ12 + iΩ0(1− 2σ11)−
σ12 (4)
where Γd is the decoherence rate. As a result the qubit
density-matrix σ(t) becomes a statistical mixture in the
stationary limit,
t→∞−→
1/2 0
0 1/2
. (5)
This happens for any initial conditions and even for large
level displacement, ǫ ≫ Ω0,Γd (provided that Ω0 6= 0).
Note that the statistical mixture (5) is proportional to
the unity matrix and therefore it remains the same in
any basis.
Secondly, the environment can put the qubit in its
ground state, for instance via photon or phonon emis-
sion. This process is usually referred to as relaxation.
For a symmetric qubit we would have
t→∞−→
1/2 1/2
1/2 1/2
. (6)
In contrast with decoherence, Eq. (5), the relaxation pro-
cess puts the qubit into a pure state. That implies that
the corresponding density matrix can be always written
as δ1iδ1j in a certain basis (the basis of the qubit eigen-
states). This is in fact the essential difference between
decoherence and relaxation. With respect to elimination
of the off-diagonal density matrix elements, note that re-
laxation would eliminate these terms only in the qubit’s
eigenstates basis. In contrast, decoherence eliminates the
off-diagonal density matrix element in any basis (Eq. (5)).
In fact, if the environment has some energy, it can put
the qubit into an exited state. However, if the qubit is
finally in a pure state, such excitation process generated
by the environment affects the qubit in the same way as
relaxation: it eliminates the off-diagonal density matrix
elements only in a certain qubit’s basis. Therefore exci-
tation of the qubit can be described phenomenologically
on the same footing as relaxation.
It is often claimed that decoherence is associated with
an absence of energy transfer between the system and
the environment, in contrast with relaxation (excitation).
This distinction is not generally valid. For instance, if the
initial qubit state corresponds to the electron in the state
|E2〉, Fig. 1, the final state after decoherence corresponds
to an equal distribution between the two dots, 〈E〉 =
(E1 +E2)/2. In the case of E1 ≫ E2, this process would
require a large energy transfer between the qubit and the
environment. Therefore decoherence can be consistently
defines as a process leading to a statistical mixture, where
all states of the system have equal probabilities (as in
Eq. (5)).
The relaxation (excitation) process can be described
most simply by diagonalizing the qubit Hamiltonian,
Eqs. (1), to obtain Hqb = E+a
+a+ + E−a
−a−, where
the operators a± are obtained by the corresponding ro-
tation of the operators a1,2
30. Here E+ and E− are the
ground (symmetric) and excited (antisymmetric) state
energies. Then the relaxation process can be described
phenomenologically in the new qubit basis |±〉 = a†±|0¯〉
σ̇−−(t) = −Γrσ−−(t) (7a)
σ̇+−(t) = i(E− − E+)σ+−(t)−
σ+−(t) , (7b)
where σ++(t) = 1 − σ−−(t), σ−+(t) = σ∗+−(t) and Γr is
the relaxation rate.
In order to add decoherence, we return to the orig-
inal qubit basis |1, 2〉 = a†1,2|0¯〉 and add the damping
term to the equation for the off-diagonal matrix elements,
Eq. (4). We arrive at the quantum rate equation describ-
ing the qubit’s behavior in the presence of both decoher-
ence and relaxation30,33,
σ̇11 = iΩ0(σ21 − σ12)− Γr
(σ12 + σ21)−
(2σ11 − 1) + Γr
σ̇12 = −iǫσ12 +
iΩ0 + Γr
(1− 2σ11) + Γr
σ12 − κ2(σ12 + σ21)
σ12 , (8b)
where ǫ̃ = (ǫ2 + 4Ω20)
1/2 and κ = Ω0/ǫ̃. In fact, these
equations can be derived in the framework of a particu-
lar model, representing an electrostatic qubit interacting
with the point-contact detector and the environment, de-
scribed by the Lee model Hamiltonian33.
Equations (8) can be rewritten in a simpler form by
mapping the qubit density matrix σ = {σ11, σ12, σ21} to
a “polarization” vector S(t) via σ(t) = [1 + τ · S(t)]/2,
where τx,y,z are the Pauli matrices. For instance, one
obtains for the symmetric case, ǫ = 0,
Ṡz = −
Sz − 2Ω0 Sy (9a)
Ṡy = 2Ω0 Sz −
Γd + Γr
Sy (9b)
Ṡx = −
Γd + 2Γr
(Sx − S̄x) (9c)
where S̄x = Sx(t→∞) = 2Γr/(Γd+2Γr). One finds that
Eqs. (9) have a form of the Bloch equations for spin-
precession in the magnetic field27, where the effect of
environment is accounted for by two relaxation times for
the different spin components: the longitudinal T1 and
the transverse T2, related to Γd and 2Γr as
T−11 =
Γd + 2Γr
, and T−12 =
Γd + Γr
, (10)
The corresponding damping rates, the so-called “depolar-
ization” (Γ1 = 1/T1) and the “dephasing” (Γ2 = 1/T2)
are used for phenomenological description of two-level
systems34. However, neither Γ1 nor Γ2 taken alone would
drive the qubit density matrix into a statistical mixture
Eq. (5) or into a pure state Eq. (6).
In contrast, our definition of decoherence and relax-
ation (excitation) is associated with two opposite effects
of the environment on the qubit: the first drives it into
a statistical mixture, whereas the second drives it into
a pure state. We expect therefore that such a natural
distinction between decoherence and relaxation would be
more useful for finding a relation between these quantities
and the environmental behavior than other alternative
definitions of these quantities existing in the literature.
In general, the two rates, Γd,r, introduced in phe-
nomenological equations (8), (9), are consistent with our
definitions of decoherence and relaxation. The only ex-
ception is the case of Γr = 0 and Ω0 = 0, where are no
transitions between the qubit’s states even in the pres-
ence of the environment (“static” qubit). One easily finds
from Eqs. (3a), (4) that σ12(t) → 0 for t → ∞, whereas
the diagonal density-matrix elements of the qubit remain
unchanged (so-called “pure dephasing”5,34):
t→∞−→
σ11(0) 0
0 σ22(0)
. (11)
Thus, if the initial probabilities of finding the qubit in
each of its states are not equal, σ11(0) 6= σ22(0), then the
final qubit state is neither a mixture nor a pure state, but
a combination of the both. It implies that Γd in Eqs. (8)
would also generate relaxation (excitation) of the qubit.
Note that in this case the off-diagonal density-matrix el-
ements, absent in Eq.(11), would reappear in a different
basis. This implies that the “pure dephasing”5,34 occurs
only in a particular basis.
Let us evaluate the probability of finding the electron
in the first dot, σ11(t). Solving Eqs. (9) for the initial
conditions σ11(0) = 1, σ12(0) = 0, we find
σ11(t) =
e−Γrt/2
−e−t + C2e
where e± =
(Γd ± Ω̃), Ω̃ =
Γ2d − 64Ω20 and C1,2 =
1±(Γd/Ω̃). Solving the same equations in the limit of t→
∞, we find that the steady-state qubit density matrix is
t→∞−→
1/2 Γr/(Γd + 2Γr)
Γr/(Γd + 2Γr) 1/2
. (13)
Thus the off-diagonal elements of the density matrix
can provide us with a ratio of relaxation to decoherence
rates33.
III. DESCRIPTION OF THE MODEL
Consider the setup shown in Fig. 2. The entire system
can be described by the following tunneling Hamiltonian,
represented by a sum of the qubit and SET Hamiltonians
and the interaction term, H = Hqb +HSET +Hint. Here
Hqb is given by Eq. (1) and describes the qubit. The sec-
ond term, HSET, describes the single-electron transistor.
It can be written as
HSET =
l cl +
rcr + E0c
l c0 +Ωrc
rc0 +H.c.) , (14)
where c
l,r and cl,r are the creation and annihilation elec-
tron operators in the state El,r of the right or left reser-
voir; c
0 and c0 are those for the level E0 inside the quan-
tum dot; and Ωl,r are the couplings between the level E0
and the level El,r in the left (right) reservoir. In order to
avoid too lengthy formulaes, our summation indices l, r
indicate simultaneously the left and the right leads of the
SET, where the corresponding summation is carried out.
As follows from the Hamiltonian (14), the quantum dot
of the SET contains only one level (E0). This assumption
has been implied only for the sake of simplicity for our
presentation, although our approach is well suited for a
case of n levels inside the SET, E0c
0c0 →
n Enc
and even when the interaction between these levels is in-
cluded (providing that the latter is much less or much
larger than the bias V )35,36. We also assumed a weak
energy dependence of the couplings Ωl,r ≃ ΩL,R.
The interaction between the qubit and the SET, Hint,
depends on a position of the SET with respect to the
qubit. If the SET is placed near the middle of the qubit,
Fig. 2a, then the tunneling coupling between two dots of
the qubit in Eq. (1) decreases, Ω0 → Ω0− δΩ0, whenever
the quantum dot of the SET is occupied by an electron.
This is due to the electron’s repulsive field. In this case
the interaction term can be written as
Hint = δΩ c
0c0(a
1a2 + a
2a1) . (15)
On the other hand, in the configuration shown in Fig. 2b
where the SET is placed near one of the dots of the qubit,
the electron repulsive field displaces the qubit energy lev-
els by ∆E = U . The interaction terms in this case can
be written as
Hint = U a
0c0 . (16)
Consider the initial state where all the levels in the left
and the right reservoirs are filled with electrons up to the
Fermi levels µL,R respectively. This state will be called
the “vacuum” state |0
〉. The wave function for the entire
system can be written as
|Ψ(t)〉 =
b(1)(t)a
0l (t)a
0cl +
rl (t)a
rcl +
l<l′,r
0rll′(t)a
rclcl′ + · · ·
+b(2)(t)a
0l (t)a
0cl +
rl (t)a
rcl +
l<l′,r
0rll′(t)a
rclcl′ + . . .
〉, (17)
where b(j)(t), b
α (t) are the probability amplitudes to
find the entire system in the state described by the cor-
responding creation and annihilation operators. These
amplitudes are obtained from the Schrödinger equation
i|Ψ̇(t)〉 = H |Ψ(t)〉, supplemented with the initial condi-
tion b(1)(0) = p1, b
(2)(0) = p2, and b
α (0) = 0, where
p1,2 are the amplitudes of the initial qubit state.
Note that Eq. (17) implies a fixed electron number (N)
in the reservoirs. At the first sight it would lead to deple-
tion of the left reservoir of electrons over the time. Yet
in the limit of N →∞ (infinite reservoirs) the dynamics
of an entire system reaches its steady state before such a
depletion takes place37,38.
The behavior of the qubit and the SET is given by
the reduced density matrix, σss′ (t). It is obtained from
the entire system’s density matrix |Ψ(t)〉〈Ψ(t)| by trac-
ing out the (continuum) reservoir states. The space of
such a reduced density matrix consists of four discrete
states s, s′ = a, b, c, d, shown schematically in Fig. 3 for
the setup of Fig. 2a. The corresponding density-matrix
elements are directly related to the amplitudes b(t), for
instance,
σaa(t) = |b(1)(t)|2 +
|b(1)lr (t)|
l<l′,r<r′
|b(1)rr′ll′(t)|
2 + · · · (18a)
σdd(t) =
|b(2)0l (t)|
l<l′,r
|b(2)0rll′(t)|
l<l′<l′′,r<r′
|b(2)0rr′ll′(t)|
2 + · · · (18b)
σbd(t) =
0l (t)b
0l (t) +
l<l′,r
0rll′(t)b
0rll′(t) +
l<l′<l′′,r<r′
0rr′ll′ (t)b
0rr′ll′(t) + · · · . (18c)
In was shown in37,38 that the trace over the reservoir
states in the system’s density matrix can be performed
in the large bias limit (strong non-equilibrium limit)
V = µL − µR ≫ Γ,Ω0, U (19)
where the level (levels) of the SET carrying the current
are far away from the chemical potentials, and Γ is the
width of the level E0. In this derivation we assumed
only weak energy dependence of the transition ampli-
tudes Ωl,r ≡ ΩL,R and the density of the reservoir states,
ρ(El,r) = ρL,R. As a result we arrive at Bloch-type rate
equations for the reduced density matrix without any
additional assumptions. The general form of these equa-
tions is36,38
σ̇jj′ = i(Ej′ − Ej)σjj′ + i
σjkΩ̃k→j′ − Ω̃j→kσkj′
P2πρ(σjkΩk→k′Ωk′→j′ + σkj′Ωk→k′Ωk′→j)
P2πρ (Ωk→jΩk′→j′ +Ωk→j′Ωk′→j)σkk′ (20)
Here Ωk→k′ denotes the single-electron hopping ampli-
tude that generates the k → k′ transition. We distinguish
between the amplitudes Ω̃ describing single-electron hop-
ping between isolated states and Ω describing transitions
between isolated and continuum states. The latter can
generate transitions between the isolated states of the
system, but only indirectly, via two consecutive jumps of
an electron, into and out of the continuum reservoir states
(with the density of states ρ). These transitions are rep-
resented by the third and the fourth terms of Eq. (20).
The third term describes the transitions (k → k′ → j)
or (k → k′ → j′), which cannot change the number of
electrons in the collector. The fourth term describes the
transitions (k → j and k′ → j′) or (k → j′ and k′ → j)
which increase the number of electrons in the collector
by one. These two terms of Eq. (20) are analogues of the
“loss” (negative) and the “gain” (positive) terms in the
classical rate equations, respectively. The factor P2 = ±1
in front of these terms is due anti-commutation of the
fermions, so that P2 = −1 whenever the loss or the gain
terms in Eq. (20) proceed through a two-fermion state of
the dot. Otherwise P2 = 1.
Note that the reduction of the time-dependent
Schrödinger equation, i|Ψ̇(t)〉 = H |Ψ(t)〉, to Eqs. (20)
is performed in the limit of large bias without explicit
use of any Markov-type or weak coupling approxima-
tions. The accuracy of these equations is respectively
max(Γ,Ω0, U, T )/|µL,R−Ej |. A detailed example of this
derivation is presented in Appendix A for the case of res-
onant tunneling through a single level. The derivation
there and in Refs.37,38 were performed by assuming zero
temperature in the leads, T = 0. Yet, this assumption
is not important in the case of large bias, providing the
levels carrying the current are far away from the Fermi
energies, |µL,R − Ej | ≫ T .
IV. NO BACK-ACTION ON THE
ENVIRONMENT
A. Fluctuation of the tunneling coupling
Now we apply Eqs. (20) to investigate the qubit’s be-
havior in the configurations shown in Fig. 2. First we
consider the SET placed near the middle of the qubit,
Figs. 2a,3. In this case the electron current through the
SET will influence the coupling between two dots of the
(d)(b)(a) (c)
E E1 2
Ω0 Ω0’ ’
FIG. 3: The available discrete states of the entire system cor-
responding to the setup of Fig. 2a. ΓL,R denote the tunneling
rates to the corresponding reservoirs and Ω′0 = Ω0 − δΩ.
qubit, making it fluctuate between the values Ω0 and
Ω′0 = Ω0 − δΩ. The corresponding rate equations can be
written straightforwardly from Eqs. (20). One finds,
σ̇aa = −ΓLσaa + ΓRσbb − iΩ0(σac − σca), (21a)
σ̇bb = −ΓRσbb + ΓLσaa − iΩ′0(σbd − σdb), (21b)
σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (21c)
σ̇dd = −ΓRσdd + ΓLσcc − iΩ′0(σdb − σbd), (21d)
σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)− ΓLσac
+ ΓRσbd, (21e)
σ̇bd = −iǫ0σbd − iΩ′0(σbb − σdd)− ΓRσbd
+ ΓLσac, (21f)
where ΓL,R = 2π|ΩL,R|2ρL,R are the tunneling rates from
the reservoirs and ǫ0 = E1 − E2.
These equations display explicitly the time evolution
of the SET and the qubit. The evolution of the for-
mer is driven by the first two terms in Eqs. (21a)-(21d).
They generate charge-fluctuations inside the quantum
dot of the SET (the transitions a←→b and c←→d),
described by the “classical” Boltzmann-type dynamics.
The qubit’s evolution is described by the Bloch-type
terms (c.f. Eqs. (3)), generating the qubit transitions
(a←→c and b←→d). Thus Eqs. (21) are quite general,
since they described fluctuations of the tunneling cou-
pling driven by the Boltzmann-type dynamics.
The resulting time evolution of the qubit is given by
the qubit (reduced) density matrix:
σ11(t) = σaa(t) + σbb(t) , (22a)
σ12(t) = σac(t) + σbd(t) , (22b)
and σ22(t) = 1− σ11(t).
Similarly, the charge fluctuations of SET are deter-
mined by the probability of finding the SET occupied,
P1(t) = σbb(t) + σdd(t) . (23)
It is given by the equation
Ṗ1(t) = ΓL − ΓP1(t) , (24)
obtained straightforwardly from Eqs. (21). Here Γ =
ΓL +ΓR is the total width. The same equation for P1(t)
can be obtained if the qubit is decoupled from the SET
(δΩ = 0). Thus there is no back-action of the qubit on
the charge fluctuations inside the SET in the limit of
large bias voltage.
Consider first the stationary limit, t → ∞, where
Ṗ1(t)→ 0 and σ̇(t)→ 0. It follows from Eq. (24) that the
probability of finding the SET occupied in this limit is
P̄1 = ΓL/Γ. This implies that the fluctuations of the cou-
pling Ω0, induced by the SET, would take place around
the average value Ω = Ω0 − P̄1 δΩ.
With respect to the qubit in the stationary limit, one
easily obtains from Eqs. (21) that the qubit density ma-
trix always becomes the statistical mixture (5), when
t → ∞. This takes place for any initial conditions and
any values of the qubit and the SET parameters. There-
fore the effect of the fluctuating charge inside the SET
does not lead to relaxation of the qubit, but rather to its
decoherence.
It is important to note, however, that for the aligned
qubit, ǫ = 0, the decoherence due to fluctuations of the
tunneling coupling Ω0 is not complete. Indeed, it follows
from Eqs. (21) that d/dt[Re σ12(t)] = 0. The reason is
that the corresponding operator, a
1a2 + a
2a1 commutes
with the total Hamiltonian H = Hqb + HSET + Hint,
Eqs. (1), (14) and (15), for E1 = E2. As a result,
Re σ12(t) = Re σ12(0).
In order to determine the decoherence rate analytically,
we perform a Laplace transform on the density matrix,
σ̃(E) =
σ(t) exp(−iEt)dE. Then solving Eq. (21) we
can determine the decoherence rate from the locations
of the poles of σ̃(E) in the complex E-plane. Consider
for instance the case of ǫ0 = 0 and the symmetric SET,
ΓL = ΓR = Γ/2. One finds from Eqs. (21) and (22a) that
σ̃11(E) =
i(E − 2Ω + iΓ)
4(E − 2Ω + iΓ/2)2 + Γ2 − (2 δΩ)2
i(E + 2Ω+ iΓ)
4(E + 2Ω+ iΓ/2)2 + Γ2 − (2 δΩ)2
. (25)
Upon performing the inverse Laplace transform,
σ11(t) =
∞+i0∫
−∞+i0
σ̃11(E) e
−iEt dE
, (26)
and closing the integration contour around the poles of
the integrand, we obtain for Γ > 2δΩ and t≫ 1/Γ
σ11(t)− (1/2) ∝ e−(Γ−
Γ2−4δΩ2)t/2 sin(2Ω t). (27)
Comparing this result with Eq. (12) we find that the
decoherence rate is
Γd = 2
Γ2 − 4δΩ2
Γ≫δΩ−→ (2δΩ)2/Γ . (28)
For ǫ0 6= 0 and ǫ0,Γ ≪ Ω the decoherece rate Γd is
multiplied by an additional factor [1− (ǫ0/2Ω)2].
In a general case, ΓL 6= ΓR, we obtain in the same
limit (ΓL,R ≫ δΩ) for the decoherence rate:
(4 δΩ)2
(ΓL + ΓR)3
It is interesting to compare this result with the fluctu-
ation spectrum of the charge inside the SET, Eq. (B8),
Appendix B. We find
Γd = 2 (δωR)
2 SQ(0) , (30)
where ωR =
4Ω2 + ǫ20 is the Rabi frequency. The latter
represents the energy splitting in the diagonalized qubit
Hamiltonian. Thus δωR corresponds to the amplitude of
energy level fluctuations in a single dot.
Although Eq. (30) has been obtained for small fluc-
tuations δωR, it might be approximately correct even
if δωR is of the order of Γ. It is demonstrated in
Fig. 4, where we compare σ11(t) and σ12(t), obtained
from Eqs. (21) and (22) (solid line) with those from
Eqs. (3a) and (4) (dashed line) for the decoherence rate
Γd given by Eq. (30). The initial conditions correspond to
σ11(0) = 1 and σ12(0) = 0 (respectively, σaa(0) = ΓR/Γ
and σbb(0) = ΓR/Γ).
In the case of aligned qubit, however, Re σ12(t) =
Re σ12(0), as was explained above. On the other hand,
one always obtains from (3a) and (4) that Re [σ12(t →
∞)] = 0. Therefore the phenomenological Bloch equa-
tions are not applicable for evaluation of Re [σ12(t)],
even in the weak coupling limit (besides the case of
Re [σ12(t = 0)] = 0).
In the large coupling regime (δΩ≫ Γ) the phenomeno-
logical Bloch equations, Eqs. (3a) and (4), cannot be
used, as well. Consider for simplicity the case of ǫ = 0
and ΓL,R = Γ/2. Then one finds from Eq. (27) that
the damping oscillations between the two dots take place
at two different frequencies, 2Ω ±
(δΩ)2 − (Γ/2)2, in-
stead of the one frequency, ωR = 2Ω, given the Bloch
equations. Moreover, Eq. (30) does not reproduce the
decoherence (damping) rate in this limit. Indeed, one
obtains from Eq. (28) that the decoherence rate Γd = 2Γ
for δΩ > Γ/2, so Γd does not depend on the coupling
(δΩ) at all.
B. Fluctuation of the energy level
Consider the SET placed near one of the qubit dots,
as shown in Fig. 2b. In this case the qubit-SET inter-
action term is given by Eq. (16). As a result the energy
level E1 will fluctuate under the influence of the fluctua-
tions of the electron charge inside the SET. The available
discrete states of the entire system are shown in Fig. 5.
Using Eqs. (20) we can write the rate equations, similar
10 20 30 40 50
(a) (b)
10 20 30 40 50
Re 12
Im 12
11 σ  ( )
σ  ( )
σ  ( )
FIG. 4: The occupation probability of the first dot of the qubit
for ǫ = 2Ω, ΓL = Ω, ΓR = 2Ω and δΩ = 0.5Ω. The solid line
is the exact result, whereas the dashed line is obtained from
the Bloch-type rate equations with the decoherence rate given
by Eq. (30).
(d)(b)(a) (c)
’’’ ’ Γ
R ΓL ΓR
E0 E0
0Ω0Ω0
E2 E2
E +U1Ω
FIG. 5: The available discrete states of the entire system for
the configuration shown in Fig. 2b. Here U is the repulsion
energy between the electrons.
to Eqs. (21),
σ̇aa = −Γ′Lσaa + Γ′Rσbb − iΩ0(σac − σca), (31a)
σ̇bb = −Γ′Rσbb + Γ′Lσaa − iΩ0(σbd − σdb), (31b)
σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (31c)
σ̇dd = −ΓRσdd + ΓLσcc − iΩ0(σdb − σbd), (31d)
σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)−
ΓL + Γ
Rσbd, (31e)
σ̇bd = −i(ǫ0 + U)σbd − iΩ0(σbb − σdd)−
ΓR + Γ
Lσac , (31f)
where Γ′L,R are the tunneling rate at the energy E0+U
Let us assume that Γ′L,R = ΓL,R. Then it follows from
Eqs. (31) that the behavior of the charge inside the SET
is not affected by the qubit, the same as in the previous
case of the Rabi frequency fluctuations. Also the qubit
density matrix becomes the mixture (5) in the stationary
state for any values of the qubit and the SET parameters.
Hence, there is no qubit relaxation in this case either
(except for the static qubit, Ω0 = 0, and σ11(0) 6= σ22(0),
Eq. (11)).
Since according to Eq. (24), the probability of finding
an electron inside the SET in the stationary state is P̄1 =
ΓL/Γ, the energy level E1 of the qubit is shifted by P̄1U .
Therefore it is useful to define the “renormalized” level
displacement, ǫ = ǫ0 + P̄1U .
As in the previous case we use the Laplace transform,
σ(t)→ σ̃(E), in order to determine the decoherence rate
analytically. In the case of ΓL = ΓR = Γ/2 and ǫ = 0 we
obtain from Eqs. (31)
σ̃11(E) =
32(E + iΓ)Ω20
U2 − 4E(E + iΓ)
. (32)
The position of the pole in the second term of this expres-
sion determines the decoherence rate. In contrast with
Eq. (25), however, the exact analytical expression for the
decoherence rate (Γd) is complicated, since it is given by
a cubic equation. We therefore evaluate Γd in a different
way, by substituting E = ±2Ω0 − iγ in the second term
of Eq. (32) and then expanding the latter in powers of
γ by keeping only the first two terms of this expansion.
The decoherence rate Γd is related to γ by Γd = 4γ, as
follows from Eq. (12). Then we obtain:
2(Γ2 + 4Ω20)
for U ≪ (Ω20 + ΓΩ0)1/2
64ΓΩ20
U2 + 16Ω20
for U ≫ (Ω20 + ΓΩ0)1/2
In general, if ΓL 6= ΓR, one finds from Eqs. (31) that
Γd = 2U
2ΓLΓR/[Γ(Γ
2 + 4Ω20)] for U ≪ (Ω20 + ΓΩ0)1/2.
The same as in the previous case, Eq. (30), the deco-
herence rate in a weak coupling limit is related to the
fluctuation spectrum of the SET, SQ(ω), Eq. (B8), but
now taken at a different frequency, ω = 2Ω0. The lat-
ter corresponds to the level splitting of the diagonalized
qubit’s Hamiltonian, ωR. Thus,
Γd = U
2 SQ(ωR) , (34)
which can be applied also for ǫ 6= 0. This is illustrated
by Fig. 6 which shows σ11(t) obtained from Eqs. (31)
and (22) (solid line) with Eqs. (3a) and (4) (dashed line)
for the decoherence rate Γd given by Eq. (34). As in
the previous case, shown in Fig. 4, the initial conditions
correspond to σ11(0) = 1 and σ12(0) = 0 (respectively,
σaa(0) = ΓR/Γ and σbb(0) = ΓR/Γ). One finds from
Fig. 6 that Eq. (34) can be used for an estimation of Γd
even for U ∼ Γ,Ω0.
In contrast with the tunneling-coupling fluctuations,
Eq. (30), where the decoherence rate is given by SQ(0),
the fluctuations of the qubit’s energy level generate the
decoherence rate, determined by the fluctuation spec-
trum at Rabi frequency, SQ(ωR), Eq. (34). A similar
distinction between the decoherence rates generated by
different components of the fluctuating field, exists in a
phenomenological description of magnetic resonance27.
One can understand this distinction by diagonalizing the
qubit’s Hamiltonian. In this case the Rabi frequency,
ωR, becomes the level splitting of the qubit’s states
|±〉 = (|1〉 ± |2〉)/
2 (for ǫ = 0). So in this basis, the
tunneling-coupling fluctuations correspond to simultane-
ous fluctuations of the energy levels in the both dots.
10 20 30 40 50
(a) (b)
10 20 30 40 50
Re 12
Im 12
11 σ  ( )
σ  ( )
σ  ( )
FIG. 6: The probability of finding the electron in the first
dot of the qubit for ǫ = 2Ω0, ΓL = Ω0, ΓR = 2Ω0 and U =
0.5Ω0. The solid line is the exact result, whereas the dashed
line is obtained from the Bloch-type rate equations with the
decoherence rate given by Eq. (34).
Since these fluctuations are “in phase”, we could expect
that the corresponding dephasing rate is determined by
spectral density at zero frequency. In fact, it looks like
as fluctuations of a single dot state, considered by Levin-
son in a weak coupling limit31. On the other hand by
fluctuating the energy level in one of the dots only, one
can anticipate that the corresponding dephasing rate is
determined by the fluctuation spectrum at the Rabi fre-
quency, ωR, Eq. (34), which is a frequency of the inter-dot
transitions.
Since ωR can be controlled by the qubit’s levels dis-
placement, ǫ, the relation (34) can be implied by using
qubit for a measurement of the shot-noise spectrum of the
environment18,19,40. For instance, it can be done by at-
taching a qubit to reservoirs at different chemical poten-
tials. The corresponding resonant current which would
flow through the qubit in this case, can be evaluated via
a simple analytical expression13 that includes explicitly
the decoherence rate, Eq. (34). Thus by measuring this
current for different level displacement of the qubit (ǫ0),
one can extract the spectral density of the fluctuating
environment acting on the qubit18.
Although Eq. (34) for the decoherence rate has been
obtained by using a particular mechanism for fluctuations
of the qubit’s energy levels, we suggest that this mecha-
nism is quite general. Indeed, the rate equations (31) can
describe any fluctuating media near a qubit, driven by the
Boltzmann type of equations. Therefore it is rather nat-
ural to assume that Eq. (34) would be valid for any type
of such (classical) environment in weak coupling limit.
This implies that the decoherence rate is always deter-
mined via the spectral density of a fluctuating qubit’s
level, whereas the nature of a particular medium inducing
these fluctuations would be irrelevant. In order to sub-
stantiate this point it is important to compare Eq. (34)
with the corresponding decoherence rate induced by the
thermal environment in the framework of the spin-boson
model. In a weak damping limit this model predicts1,2
T−11 = T
2 = (q
0/2)S(ωR) , where q0 is a coupling of
the medium with the qubit levels (q0 corresponds to U in
our case) and S(ω) is a spectral density. Using Eq. (10)
one finds that this result coincides with Eq. (34).
0 5 10 15 20
0 5 10 15 20
t t1111
U/      =100ΩU/      =10
σ  ( )
σ  ( )
(a) (b)
FIG. 7: The probability of finding the electron in the first
dot of the qubit for ǫ = 0, ΓL = ΓR = Ω0 and U , as given
by Eqs. (31) (solid line) and from the Bloch-type equations
(dashed line) with the decoherence rate given by Eq. (33).
C. Strong-coupling limit and localization
Let us consider the limit of U ≫ (Ω20 + ΓΩ0)1/2. Our
rate equation (31) are perfectly valid in this region, pro-
viding only that E0 + U is deeply inside of the potential
bias, Eq. (19). We find from Eq. (33) that the deco-
herence rate is not directly related to the spectrum of
fluctuations in strong coupling limit. In addition, the ef-
fective frequency of the qubit’s Rabi oscillations (ω
decreases in this limit. Indeed, by using Eqs. (32), (26),
one finds that the main contribution to σ11(t), is coming
from a pole of σ̃11(E), which lies on the imaginary axis.
This implies that the effective frequency of Rabi oscilla-
tions strongly decreases when U ≫ (Ω20 + ΓΩ0)1/2. In
addition, the decoherence rate Γd → 0 in the same limit,
Eq. (33). As a result, the electron would localize in the
initial qubit state, Fig. 7.
The results displayed in this figure show that the solu-
tion of the Bloch-type rate equations, with the decoher-
ence rate given by Eq. (33), represents damped oscilla-
tions (dashed line). It is very far from the exact result
(solid line), obtained from Eqs. (31) and corresponding
to the electron localization in the first dot. The latter is
a result of an effective decrease of the Rabi frequency for
large U that slows down electron transitions between the
dots. Thus such an environment-induced localization is
different from the Zeno-type effect (unlike an assumption
of Ref.12). Indeed, the Zeno effect takes place whenever
the decoherence rate is much larger then the coupling be-
tween the qubit’s states13,33. However, the decoherence
rate in the strong coupling limit is much smaller then the
coupling Ω0 . In fact, the localization shown in Fig. 7
is rather similar to that in the spin-boson model1,2. It
shows that in spite of their defferences, both models trace
the same physics of the back-action of the environment
(SET) on the qubit.
V. BACK-ACTION OF THE QUBIT ON THE
ENVIRONMENT
A. Weak back-action effect
Now we investigate a weak dependence of the width’s
ΓL,R on the energy U , Fig. 5. We keep only the linear
term, Γ′L,R = ΓL,R+αL,RU , by assuming that U is small.
(A similar model has been considered in28,41). In con-
trast with the previous examples, where the widths have
not been dependent on the energy, the qubit’s oscillation
would affect the SET current and its charge correlator.
A more interesting case corresponds to αL 6= αR. Let us
take for simplicity αL = 0 and αR = α 6= 0.
Similarly to the previous case we introduce the “renor-
malized” level displacement, ǫ = ǫ0 − (ΓL/Γ)U , where
ǫ = 0 corresponds to the aligned qubit. Solving Eqs. (31)
in the steady-state limit, σ̄ = σ(t → ∞), and keeping
only the first term in expansion in powers of U , we find
for the reduced density matrix of the qubit, Eqs. (22):
− α ǫ
αΩ0(1 + c αU)
αΩ0(1 + c αU)
, (35)
where c = (αǫ − 2Γ)/(4ΓRΓ). It follows from Eqs. (35)
that the qubit’s density matrix in the steady-state is no
longer a mixture, Eq. (5) . Indeed, the probability to
occupy the lowest level is always larger than 1/2 and
σ̄12 6= 0. This implies that relaxation takes place to-
gether with decoherence. The ratio of the relaxation and
decoherence rates is given by the off-diagonal terms of
the reduced density matrix of the qubit. For ǫ = 0 one
finds from Eq. (13) that Γd/Γr = σ̄
12 − 2.
In order to find a relation between the decoherence
and relaxation rates, Γd,r, and the fluctuation spectrum
of the qubit energy level, SQ(ω), we first evaluate the
total damping rate of the qubit’s oscillations (γ). Using
Eq. (12) we find that this quantity is related to the deco-
herence and relaxation rates by γ = (Γd + 2Γr)/4. The
same as in the previous case the rate γ is determined by
poles of Laplace transformed density matrix σ(t)→ σ̃(E)
in the complex E-plane. Consider for simplicity the case
of ǫ = 0 and ΓL = ΓR = Γ/2. Performing the Laplace
transform of Eqs. (31) we look for the poles of σ11(E) at
E = ±2Ω0 − iγ by assuming that γ is small. We obtain
Γd + 2Γr =
2(Γ2 + 4Ω20)
Γ− αU Γ
2 − 4Ω20
2 (Γ2 + 4Ω20)
for U ≪ Ω0.
Now we evaluate the correlator of the charge inside the
SET, SQ(ω) which induces the energy-level fluctuations
of the qubit. Using Eqs. (31) and (B6) we find,
SQ(ω) =
2 (Γ2 + ω2)
− αU Γ
2 − ω2
4 (Γ2 + ω2)
for αU ≪ Γ. Therefore in the limit of U ≪ Ω0 and αU ≪
Γ the total damping rate of the qubit’s oscillations is
directly related to the spectral density of the fluctuations
spectrum taken at the Rabi frequency,
Γd + 2Γr = U
2SQ(2Ω0). (38)
This represents a generalization of Eq. (34) for the case
of a weak back-action of qubit oscillations on the spectral
density of the environment. As a result, the qubit dis-
plays relaxation together with decoherence. It is remark-
able that the total qubit’s damping rate is still given by
the fluctuation spectrum of the SET (environment) mod-
ulated by the qubit. Note that Eq. (38) can be applied
only if the modulation of the tunneling rate through the
SET (tunneling current) is small αU ≪ Γ, in addition to
a weak distortion of the qubit (U ≪ Ω0).
In the case of strong back-action of the qubit on the
environment the decorerence and relaxation rates of the
qubit are not directly related to the fluctuation spectrum
of the environment, even if the distortion of the qubit is
small. This point is illustrated by the following example.
B. Strong back-action
Until now we considered the case where E0 + U ≪
µL, so that the interacting electron of the SET remains
deeply inside the voltage bias. If however, the interaction
U between the qubit and the SET is such that E0 +
U ≫ µL, the qubit’s oscillation would strongly affect the
fluctuation of charge inside the SET. Indeed, the current
through the SET is blocked whenever the level E1 of the
qubit is occupied, Fig. 8. In fact, this case can be treated
with small modification of the rate equations (31), if only
µL − E0 ≫ Γ and E0 + U − µL ≫ Γ, where E0 is a level
of the SET carrying the current.
The corresponding quantum rate equations describing
the system are obtained directly from Eqs. (20). As-
suming that the widths ΓL,R are energy independent we
find16
σ̇aa = (ΓL + ΓR)σbb − iΩ0(σac − σca), (39a)
σ̇bb = −(ΓR + ΓL)σbb − iΩ0(σbd − σdb), (39b)
σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (39c)
σ̇dd = −ΓRσdd + ΓLσcc − iΩ0(σdb − σbd), (39d)
σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)−
+ ΓRσbd, (39e)
σ̇bd = −i(ǫ0 + U)σbd − iΩ0(σbb − σdd)
σbd . (39f)
Solving Eqs. (39) in the stationary limit, σ̄ = σ(t →
∞) and introducing the “renormalized” level displace-
ment, ǫ = ǫ0 −UΓL/(2Γ), we obtain for the qubit’s den-
(d)(b)(a) (c)
E +U1
FIG. 8: The available discrete states of the entire system
when the electron-electron repulsive interaction U breaks off
the current through the SET.
sity matrix, Eqs. (22) in the steady state:
σ̄11 =
− 8ǫU
16ǫ2 + 8Uǫ+ 48Ω20 + 9(U
2 + Γ2)
,(40a)
σ̄12 =
12UΩ0
16ǫ2 + 8Uǫ+ 48Ω20 + 9 (U
2 + Γ2)
, (40b)
where for simplicity we considered the symmetric case,
ΓL = ΓR = Γ/2. It follows from Eqs. (40) that similarly
to the previous example, the qubit’s density matrix is no
longer a mixture (5). The relaxation takes place together
with decoherence in this case too.
Let us consider weak distortion of the qubit by the
SET, U < Ω0. Although the values of U are restricted
from below (U ≫ Γ+µL−E0), this limit can be achieved
if the level E0 is close to the Fermi energy, providing only
that µL − E0 ≫ Γ, and Γ≪ U . Now we evaluate σ11(t)
with the rate equations (39) and then compare it with
the same quantity obtained from the Bloch equations,
Eq. (12), where Γd,r are given by Eqs. (34)and (13). The
corresponding charge-correlator, SQ(ωR), is evaluated by
Eqs. (B6) and (39). As an example, we take symmetric
qubit with aligned levels, ǫ = 0, ΓL = ΓR = 0.05Ω0 and
U = 0.5Ω0. The decoherence and relaxation rates, corre-
sponding to these parameters are respectively: Γd/Ω0 =
0.0038 and Γr/Ω0 = 0.00059.
The results are presented in Fig. 9a. The solid line
shows σ11(t), obtained from the rate equations (39),
where the dashed line is the same quantity obtained from
Eq. (12). We find that Eq. (34) (or (38)) underestimates
the actual damping rate of σ11(t) by an order of mag-
nitude). This lies in a sharp contrast with the previous
case, where the energy level of the SET is not distorted
by the qubit, Γ′L,R = ΓL,R, Fig. 5. Indeed, in this case
σ11(t) obtained Eq. (12) with Γd given by Eq. (34) and
Γr = 0, agrees very well with that obtained from the rate
equations (31), as shown in Fig. 9b.
Such an example clearly illustrates that the decoher-
ence is not related to the fluctuation spectrum of the
environment, whenever the environment is strongly af-
fected by the qubit, even if the coupling with a qubit
is small. This is a typical case of measurement, corre-
sponding to a noticeable response of the environment to
the qubit’s state (a “signal”).
20 40 60 80 100
20 40 60 80 100
t t11 11σ  ( ) σ  ( )
FIG. 9: (a) The probability of finding the electron in the first
dot of the qubit for ǫ = 0, ΓL = ΓR = 0.05Ω0 and U = 0.5Ω0.
The solid line is obtained from Eqs. (39), whereas the dashed
line corresponds to the Eq. (12) with Γd given by Eq. (34);
(b) the same for the case, shown in Fig. 5, where the solid
line corresponds to Eqs. (31).
VI. SUMMARY
In this paper we propose a simple model describing
a qubit interacting with fluctuating environment. The
latter is represented by a single electron transistor (SET)
in close proximity of the qubit. Then the fluctuations
of the charge inside the SET generate fluctuating field
acting on the qubit. In the limit of large bias voltage, the
Schrödinger equation for the entire system is reduced to
the Bloch-type rate equations. The resulting equations
are very simple, so that one can easily analyze the limits
of weak and strong coupling of the qubit with the SET.
We considered separately two different cases: (a) there
is no back-action of the qubit on the SET behavior, so
that the latter represents a “pure environment”; and (b)
the SET behavior depends on the qubit’s state. In the
latter case the SET can “measure” the qubit. The setup
corresponding to the “pure environment” is realized when
the energy level of the SET carrying the current lies
deeply inside the potential bias. The second (measure-
ment) regime of the SET is realized when the tunnel-
ing widths of the SET are energy dependent, or when
the energy level of the SET carrying the current is close
enough to the Fermi level of the corresponding reservoir.
Then the electron-electron interaction between the qubit
and the SET modulates the electron current through the
In the case of the “pure environment” (“no-
measurement” regime) we investigate separately two dif-
ferent configurations of the qubit with respect to the
SET. In the first one the SET produces fluctuations of
the off-diagonal coupling (Rabi frequency) between two
qubit’s states. In the second configuration the SET pro-
duces fluctuations of the qubit’s energy levels. In the
both cases we find no relaxation of the qubit, despite
the energy transfer between the qubit and the SET can
take place. As a result the qubit always turns asymptot-
ically to the statistical mixture. We also found that in
both cases the decoherence rate of the qubit in the weak
coupling limit is given by the spectral density of the cor-
responding fluctuating parameter. The difference is that
in the case of the off-diagonal coupling fluctuations the
spectral density is taken at zero frequency, whereas in
the case of the energy level fluctuations it is taken at the
Rabi-frequency.
In the case of the strong coupling limit, however, the
decoherence rate is not related to the fluctuation spec-
trum. Moreover we found that the electron in the qubit
is localized in this limit due to an effective decrease of
the off-diagonal coupling. This phenomenon may resem-
ble the localization in the spin-boson model in the strong
coupling limit.
If the charge correlator and the total SET current are
affected by the qubit (back-action effect), we found that
the off-diagonal density-matrix elements of the qubit sur-
vive in the steady-state limit and therefore the relax-
ation rate is not zero. We concentrated on the case of
weak coupling, when the Coulomb repulsion between the
qubit and the SET is smaller then the Rabi frequency.
The back-action of the qubit on the SET, however, can
be weak or strong. In the first case we found that the
total damping rate of the qubit due to decoherence and
relaxation is again given by the spectral density of the
SET charge fluctuations, modulated by the qubit. This
relation, however, is not working if the back-action is
strong. Indeed, we found that the damping rate of the
qubit in this case is larger by an order of magnitude than
that given by the spectral density of the corresponding
fluctuating parameter.
This looks like that in the strong back-action of the
qubit on the SET the major component of decoherence is
not coming from the fluctuation spectrum of the qubit’s
parameters only, but also from the measurement “sig-
nal” of the SET. On the first sight it could agree with
an analysis of Ref.30, suggesting that the decoherence
rate contains two components, generated by a measure-
ment and by a “pure environment” (environmental fluc-
tuations). The latter therefore represents an unavoid-
able decoherence, generated by any environment. Yet,
in a weak coupling regime such a separation seems not
working. In this case the damping (decoherence) rate is
totally determined by the environment fluctuations, even
so modulated by the qubit.
Although our model deals with a particular setup, it
bears the main physics of a fluctuating environment, act-
ing on a qubit. Indeed, the Bloch-type rate equations,
which we used in our analysis have a pronounced phys-
ical meaning: they relate the variation of qubit param-
eters with a nearby fluctuating field described by rate
equations. A particular mechanism, generated this field
should not be relevant for an evaluations of the deco-
herence and relaxation rates, but only its fluctuation
spectrum. Indeed, in the weak coupling limit our re-
sult for the decorence rate coincides with that obtained
in a framework of the spin-boson model. Thus our model
can be considered as a generic one. Its main advantage is
that it can be easily extended to multiple coupled qubits.
Such an analysis would allow to determine how decoher-
ence scales with number of qubits42, which is extremely
important for a realization of quantum computations.
In addition, our model can be extended to a more
complicated fluctuating environments, such as containing
characteristic frequencies in its spectrum. It would for-
mally correspond to a replacement of the SET in Fig. 2
by a double-dot (DD) coupled to the reservoirs43. All
these situations, however, must be a subject of a sepa-
rate investigation.
VII. ACKNOWLEDGEMENT
One of us (S.G.) thanks T. Brandes and C. Emary
for helpful discussions and important suggestions. S.G is
also grateful to the Max Planck Institute for the Physics
of Complex Systems, Dresden, Germany, and to NTT Ba-
sic Research Laboratories, Atsugi-shi, Kanagawa, Japan,
for kind hospitality.
APPENDIX A: QUANTUM-MECHANICAL
DERIVATION OF RATE EQUATIONS FOR
QUANTUM TRANSPORT
Consider the resonant tunneling through the SET,
shown schematically in Fig. 10. The entire system is
described by the Hamiltonian HSET, given by Eq. (14).
The wave function can be written in the same way as
Eq. (17), where the variables related to the qubit are
omitted,
|Ψ(t)〉 =
b(t) +
b0l(t)c
0cl +
brl(t)c
l<l′,r
b0rll′(t)c
rclcl′ + · · ·
〉. (A1)
Substituting |Ψ(t)〉 into the time-dependent Schrödinger
equation, i∂t|Ψ(t)〉 = HSET|Ψ(t)〉, and performing the
Laplace transform, b̃(E) =
exp(iEt) b(t)dt, we obtain
the following infinite set of algebraic equations for the
FIG. 10: Resonant tunneling through a single dot. µL,R are
the Fermi energies in the collector and emitter, respectively.
amplitudes b̃(E):
Eb̃(E) −
Ωlb̃0l(E) = i (A2a)
(E + El − E0)b̃0l(E)− Ωlb̃(E)
Ωr b̃lr(E) = 0 (A2b)
(E + El − Er)b̃lr(E)− Ωr b̃0l(E)
Ωl′ b̃0ll′r(E) = 0 (A2c)
(E + El + El′ − E0 − Er)b̃0ll′r(E) − Ωl′ b̃lr(E)
+ Ωlb̃l′r(E)−
Ωr′ b̃ll′rr′(E) = 0 (A2d)
· · · · · · · · ·
(The r.h.s of Eq. (A2a) reflects the initial condition.)
Let us replace the amplitude b̃ in the term
Ωb̃ of each
of the equations (A2) by its expression obtained from the
subsequent equation. For example, substituting b̃0l(E)
from Eq. (A2b) into Eq. (A2a) we obtain
E + El − E0
b̃(E)
E + El − E0
b̃lr(E) = i. (A3)
Since the states in the reservoirs are very dense (contin-
uum), one can replace the sums over l and r by integrals,
for instance
ρL(El) dEl , where ρL(El) is the
density of states in the emitter, and Ωl,r → ΩL,R(El,r).
Consider the first term
Ω2L(El)
E + El − E0
ρL(El)dEl (A4)
where Λ is the cut-off parameter. Assuming weak en-
ergy dependence of the couplings ΩL,R and the density of
states ρL,R, we find in the limit of high bias, µL = Λ→∞
S1 = −iπΩ2L(E0 − E)ρL(E0 − E) = −i
. (A5)
Consider now the second sum in Eq. (A3).
ρR(Er)dEr
ΩL(El)ΩR(Er)b̃lr(E,El, Er)
E + El − E0
ρL(El)dEl , (A6)
where we replaced b̃lr(E) by b̃(E,El, Er) and took µL =
Λ, µR = −Λ. In contrast with the first term of Eq. (A3),
the amplitude b̃ is not factorized out the integral (A6).
We refer to this type of terms as “cross-terms”. Fortu-
nately, all “cross-terms” vanish in the limit of large bias,
Λ → ∞. This greatly simplifies the problem and is very
crucial for a transformation of the Schrödinger to the rate
equations. The reason is that the poles of the integrand
in the El(Er)-variable in the “cross-terms” are on the
same side of the integration contour. One can find it by
using a perturbation series the amplitudes b̃ in powers of
Ω. For instance, from iterations of Eqs. (A2) one finds
b̃(E,El, Er) =
iΩLΩR
E(E + El − Er)(E + El − E0)
+ · · ·
The higher order powers of Ω have the same structure.
Since E → E + iǫ in the Laplace transform, all poles of
the amplitude b̃(E,El, Er) in the El-variable are below
the real axis. In this case, substituting Eq. (A7) into
Eq. (A6) we find
(E + iǫ)(E + E0 − E1 + iǫ)2(E + E0 − Er + iǫ)
+ · · ·
dEl = 0 , (A8)
Thus, S2 → 0 in the limit of µL →∞, µR → −∞.
Applying analogous considerations to the other equa-
tions of the system (A2), we finally arrive at the following
set of equations:
(E + iΓL/2)b̃(E) = i (A9a)
(E + El − E0 + iΓR/2)b̃0l(E)
− Ωlb̃(E) = 0 (A9b)
(E + El − Er + iΓL/2)b̃lr(E)
− Ωrb̃0l(E) = 0 (A9c)
(E + El + El′ − E0 − Er + iΓR/2)b̃0ll′r(E)
− Ωl′ b̃lr(E) + Ωlb̃l′r(E) = 0 (A9d)
· · · · · · · · ·
Eqs. (A9) can be transformed directly to the reduced
density matrix σ
(n,n′)
jj′ (t), where j = 0, 1 denote the state
of the SET with an unoccupied or occupied dot and n de-
notes the number of electrons which have arrived at the
collector by time t. In fact, as follows from our derivation,
the diagonal density-matrix elements, j = j′and n = n′,
form a closed system in the case of resonant tunneling
through one level, Fig. 10. The off-diagonal elements,
j 6= j′, appear in the equation of motion whenever more
than one discrete level of the system carry the transport
(see Eq. (20). Therefore we concentrate below on the di-
agonal density-matrix elements only, σ
00 (t) ≡ σ
(n,n)
00 (t)
and σ
11 (t) ≡ σ
(n,n)
11 (t). Applying the inverse Laplace
transform on finds
00 (t) =
l...,r...
dEdE′
b̃l · · ·
r · · ·
(E)b̃∗l · · ·
r · · ·
(E′)ei(E
′−E)t (A10a)
11 (t) =
l...,r...
dEdE′
0l · · ·
r · · ·
(E)b̃∗
0l · · ·
r · · ·
(E′)ei(E
′−E)t (A10b)
Consider, for instance, the term σ
11 (t) =
l |b0l(t)|2.
Multiplying Eq. (A9b) by b̃∗0l(E
′) and then subtracting
the complex conjugated equation with the interchange
E ↔ E′ we obtain
dEdE′
(E′ − E − iΓR)b̃0l(E)b̃∗0l(E′)
− 2Im
Ωlb̃0l(E)b̃
∗(E′)
′−E)t = 0 (A11)
Using Eq. (A10b) one easily finds that the first inte-
gral in Eq. (A11) equals to −i[σ̇(0)11 (t)+ΓRσ
11 (t)]. Next,
substituting
b̃0l(E) =
Ωlb̃(E)
E + El − E0 + iΓR/2
(A12)
from Eq. (A9b) into the second term of Eq. (A11), and
replacing a sum by an integral, one can perform the El-
integration in the large bias limit, µL → ∞, µR → −∞.
Then using again Eq. (A10b) one reduces the second
term of Eq. (A11) to iΓLσ
00 (t). Finally, Eq. (A11) reads
11 (t) = ΓLσ
00 (t)− ΓRσ
11 (t).
The same algebra can be applied for all other am-
plitudes b̃α(t). For instance, by using Eq. (A10a) one
easily finds that Eq. (A9c) is converted to the following
rate equation σ̇
00 (t) = −ΓLσ
00 (t) + ΓRσ
11 (t). With
respect to the states involving more than one electron
(hole) in the reservoirs (the amplitudes like b̃0ll′r(E) and
so on), the corresponding equations contain the Pauli ex-
change terms. By converting these equations into those
for the density matrix using our procedure, one finds the
“cross terms”, like
Ωlb̃l′r(E)Ωl′ b̃
′), generated by
Eq. (A9d). Yet, these terms vanish after an integration
over El(r) in the large bias limit, as the second term in
Eq. (A3). The rest of the algebra remains the same,
as described above. Finally we arrive at the following
infinite system of the chain equations for the diagonal
elements, σ
00 and σ
11 , of the density matrix,
00 (t) = −ΓLσ
00 (t) , (A13a)
11 (t) = ΓLσ
00 (t)− ΓRσ
11 (t) , (A13b)
00 (t) = −ΓLσ
00 (t) + ΓRσ
11 (t) , (A13c)
11 (t) = ΓLσ
00 (t)− ΓRσ
11 (t) , (A13d)
· · · · · · · · ·
Summing over n in Eqs. (A13) we find for the reduced
density matrix of the SET, σ(t) =
(n)(t), the fol-
lowing “classical” rate equations,
σ̇00(t) = −ΓLσ00(t) + ΓRσ11(t) (A14a)
σ̇11(t) = ΓLσ00(t)− ΓRσ11(t) (A14b)
These equations represent a particular case of our general
quantum rate equations (20), which are derived using the
above described technique37,38.
APPENDIX B: CORRELATOR OF ELECTRIC
CHARGE INSIDE THE SET.
The charge correlator inside the SET is given by
SQ(ω) = S̄Q(ω) + S̄Q(−ω), where
S̄Q(ω) =
〈δQ̂(0)δQ̂(t)〉eiωtdt . (B1)
Here δQ̂(t) = c
0(t)c0(t) − q̄ and q̄ = P̄1 = P1(t → ∞) is
the average charge inside the dot. Since the initial state,
t = 0 in Eq. (B1) corresponds to the steady state, one
can represent the time-correlator as
〈δQ̂(0)δQ̂(t)〉 =
q=0,1
Pq(0)(q − q̄)(〈Qq(t)〉 − q̄) , (B2)
where Pq(0) is the probability of finding the charge q =
0, 1 inside the quantum dot in the steady state, such that
P1(0) = q̄ and P0(0) = 1 − q̄, and 〈Qq(t)〉 = P (q)1 (t) is
the average charge in the dot at time t, starting with the
initial condition P
1 (0) = q. Substituting Eq. (B2) into
Eq. (B1) we finally obtain
S̄Q(ω) = q̄(1 − q̄)[P̃ (1)1 (ω)− P̃
1 (ω)] , (B3)
where P̃
1 (ω) is a Laplace transform of P
1 (t). These
quantities are obtained directly from the rate equations,
such that q̄ = σ̄bb + σ̄dd and P̃
1 (ω) = σ̃
bb (ω) +
dd (ω), where σ̄ = σ(t → ∞) and σ̃(q)(ω) is the
Laplace transform σ(q)(t) with the initial conditions
corresponding to the occupied (q = 1) or unoccupied
(q = 0) SET. In order to find these quantities it is use-
ful to rewrite the rate equations in the matrix form,
σ̇(t) = Mσ(t), representing σ(t) as the eight-vector,
σ = {σaa, σbb, σcc, σdd, σac, σca, σbd, σdb} and M as the
corresponding 8× 8-matrix. Applying the Laplace trans-
form we find the following matrix equation,
(i ω I +M)σ̃(q)(ω) = −σ(q)(0) , (B4)
where I is the unit matrix and σ(q)(0) is the initial con-
dition for the density-matrix obtained by projecting the
total wave function (17) on occupied (q = 1) and unoc-
cupied (q = 0) states of the SET in the limit of t → ∞,
σ(1)(0) = N1{0, σ̄bb, 0, σ̄dd, 0, 0, σ̄bd, σ̄db} , (B5a)
σ(0)(0) = N0{σ̄aa, 0, σ̄cc, 0, σ̄ac, σ̄ca, 0, 0} , (B5b)
and N1 = 1/q̄ and N0 = 1/(1− q̄) are the corresponding
normalization factors. Finally one obtains:
SQ(ω) = 2q̄(1− q̄)Re [σ̃(1)bb (ω) + σ̃
dd (ω)
− σ̃(0)bb (ω)− σ̃
dd (ω)]. (B6)
In the case shown in Fig. 2 one finds from Eqs. (21) or
Eqs. (31) for Γ′L,R = ΓL,R that σ̄ac = σbd = 0, q̄ = ΓL/Γ
and σ̃
bb (ω) + σ̃
dd (ω) = P̃
1 (ω). The latter equation is
given by
(iω − Γ)P̃ (q)1 (ω) = −q +
. (B7)
Substituting Eq. (B7) into Eq. (B3) one obtains:
SQ(ω) =
2ΓLΓR
Γ(ω2 + Γ2)
. (B8)
Obviously, for a more general case when Γ′L,R 6= ΓL,R,
or when the electron-electron interaction excites the elec-
tron inside the SET above the Fermi level, Fig. 8, the ex-
pressions for SQ(ω), obtained from Eq. (B6) have a more
complicated than Eq. (B8).
∗ Electronic address: shmuel.gurvitz@weizmann.ac.il
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mailto:shmuel.gurvitz@weizmann.ac.il

  We consider an electrostatic qubit, interacting with a fluctuating charge of
single electron transistor (SET) in the framework of exactly solvable model.
The SET plays a role of the fluctuating environment affecting the qubit's
parameters in a controllable way. We derive the rate equations describing
dynamics of the entire system for both weak and strong qubit-SET coupling.
Solving these equation we obtain decoherence and relaxation rates of the qubit,
as well as the spectral density of the fluctuating qubit's parameters. We found
that in the weak coupling regime the decoherence and relaxation rates are
directly related to the spectral density taken at Rabi or at zero frequency,
depending on what a particular qubit's parameters is fluctuating. This relation
holds also in the presence of weak back-action of the qubit on the fluctuating
environment. In the case of strong back-action, such simple relationship no
longer holds, even if the qubit-SET coupling is small. It does not hold either
in the strong-coupling regime, even in the absence of the back-action. In
addition, we found that our model predicts localization of the qubit in the
strong-coupling regime, resembling that of the spin-boson model.

Quantum mechanical approach to decoherence and relaxation generated by
fluctuating environment
S.A. Gurvitz∗
Department of Particle Physics, Weizmann Institute of Science,
Rehovot 76100, Israel and Theoretical Division and CNLS,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
D. Mozyrsky
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: November 4, 2018)
We consider an electrostatic qubit, interacting with fluctuating charge of a single electron tran-
sistor (SET) in the framework of an exactly solvable model. The SET plays role of an environment
affecting the qubits’ parameters in a controllable way. We derive the rate equations describing the
dynamics of the entire system for an arbitrary qubit-SET coupling. Solving these equations we
obtain decoherence and relaxation rates of the qubit, as well as the spectral density of qubit param-
eters’ fluctuations. We found that in a weak coupling regime decoherence and relaxation rates are
directly related to the spectral density taken at either zero or Rabi frequency, depending on which
qubit parameter is fluctuating. In the latter case our result coincides with that of the spin-boson
model in the weak coupling limit, despite different origin of the fluctuations. We show that this
relation holds also in the presence of weak back-action of the qubit on the environment. In case of
strong back-action such a simple relationship no longer holds, even if qubit-SET coupling is small.
It does not hold also in the strong coupling regime, even in the absence of the back-action. In
addition, we found that our model predicts localization of the qubit in the strong-coupling regime,
resembling that in the spin-boson model.
PACS numbers: 03.65.Yz, 05.60.Gg, 73.23.-b, 73.23.Hk
I. INTRODUCTION
The influence of environment on a single quantum sys-
tem is the issue of crucial importance in quantum in-
formation science. It is mainly associated with decoher-
ence, or dephasing, which transforms any pure state of
a quantum system into a statistical mixture. Despite a
large body of theoretical work devoted to decoherence, its
mechanism has not been clarified enough. For instance,
how decoherence is related to environmental noise, in
particular in the presence of back-action of the system
on the environment (quantum measurements). More-
over, decoherence is often intermixed with relaxation. Al-
though each of them represents an irreversible process,
decoherence and relaxation affect quantum systems in
quite different ways.
In order to establish a relation between the fluctua-
tion spectrum and decoherence and relaxation rates one
needs a model that describes the effects of decoherence
and relaxation in a consistent quantum mechanical way.
An obvious candidate is the spin-boson model1,2 which
represents the environment as a bath of harmonic oscil-
lators at equilibrium, where the fluctuations obey Gaus-
sian statistics3. Despite its apparent simplicity, the spin-
boson model cannot be solved exactly2. Also, it is hard
to manipulate the fluctuation spectrum in the framework
of this model. In addition, mesoscopic structures may
couple only to a few isolated fluctuators, like spins, lo-
cal currents, background charge fluctuations, etc. This
would require models of the environment, different from
Electrodes
FIG. 1: Electrostatic qubit, realized by an electron trapped
in a coupled-dot system (a), and its schematic representation
by a double-well (b). Ω0 denotes the coupling between the
two dots.
the spin-boson model (see for instace4,5,6,7,8,9,10,11,12). In
general, the environment can be out of equilibrium, like
a steady-state fluctuating current, interacting with the
qubit13,14,15,16. This for instance, takes place in the con-
tinuous measurement (monitoring) of quantum systems17
and in the “control dephasing” experiments18,19,20. All
these types on non-Gaussian and non-equilibrium envi-
ronments attracted recently a great deal of attention21.
In this paper we consider an electrostatic qubit, which
can be viewed as a generic example of two-state systems.
It is realized by an electron trapped in coupled quantum
dots22,23,24, Fig. 1. Here E1 and E2 denote energies of
the electron states in each of the dots and Ω0 is a cou-
pling between these states. It is reasonable to assume
that the decoherence of a qubit is associated with fluctu-
http://arxiv.org/abs/0704.0194v3
Electrostatic Qubit
Single Electron Transistor
(a) (b)
FIG. 2: Qubit near Single Electron Transistor. Here El,r and
E0 denote the energy levels in the left (right) reservoirs and in
the quantum dot, respectively, and µL,R are the correspond-
ing chemical potentials. The electric current I generates fluc-
tuations of the electrostatic opening between two dots (a), or
it fluctuates the energy level of the nearest dot (b).
ations of the qubit parameters, E1,2 and Ω0, generated
by the environment. Indeed, a stochastic averaging of
the Schrödinger equation over these fluctuations param-
eters results in the qubit’s decoherence, which transfers
any qubit state into a statistical mixture25,26. In general,
one can expect that the fluctuating environment should
result in the qubit’s relaxation, as well, as for instance in
the phenomenological Redfield’s description of relaxation
in the magnetic resonance27.
As a quantum mechanical model of the environment
we consider a Single Electron Transistor (SET) capac-
itively coupled to the qubit, e.g., Fig. 2. Such setup
has been contemplated in numerous solid state quantum
computing architectures where SET plays role of a read-
out device16,17,28,29 and contains most of the generic fea-
tures of a fluctuating non-equilibrium environment. The
discreteness of the electron charge creates fluctuations
in the electrostatic field near the SET. If the electro-
static qubit is placed near the SET, this fluctuating field
should affect the qubit behavior as shown in Fig. 2. It
can produce fluctuations of the tunneling coupling be-
tween the dots (off-diagonal coupling) by narrowing the
electrostatic opening connecting these dots, as in Fig. 2a,
or make the energy levels of the dots fluctuate, as shown
schematically in Fig. 2b. Note that while in some regimes
the SET operates as a measuring device16,17, in other
regimes it corresponds purely to a source of noise. In-
deed, if the energy level E0, Fig. 2, is deeply inside
the voltage bias – the case we consider in the begin-
ning, the SET current is not modulated by the qubit
electron. In this case the SET represents only the fluc-
tuating environment affecting the qubit behavior (“pure
environment”30).
A similar model of the fluctuating environment has
been studied mostly for small bias (linear response) or
for the environment in an equilibrium. Here, however,
we consider strongly non-equilibrium case where the bias
voltage applied on the SET (V = µL − µR) is much
larger than the levels widths and the coupling between
the SET and the qubit. In this limit our model can be
solved exactly for both weak and strong coupling (but
is still smaller than the bias voltage). This constitutes
an essential advantage with regard to perturbative treat-
ments of similar models. For instance, the results of our
model can be compared in different regimes with phe-
nomenological descriptions used in the literature. Such
a comparison would allow us to determine the regions
where these phenomenological models are valid.
Since our model is very simple in treatment, the deco-
herence and relaxation rates can be extracted from the
exact solution analytically, as well as the time-correlator
of the electric charge inside the SET. This would make
it possible to establish a relation between the frequency-
dependent fluctuation spectrum of the environment and
the decoherence and the relaxation rates of the qubit,
and to determine how far this relation can be extended.
We expect that such a relation should not depend on
a source of fluctuations. This point can be verified by
a comparison with a similar results obtained for equi-
librium environment in the framework of the spin-boson
model1,2.
It is also important to understand how the decoher-
ence and relaxation rates depend on the frequency of
the environmental fluctuations. This problem has been
investigated in many phenomenological approaches for
“classical” environments at equilibrium. Yet, there still
exists an ambiguity in the literature related to this point
for non-equilibrium environment. For instance, it was
found by Levinson that the decoherence rate, generated
by fluctuations of the energy level in a single quantum dot
is proportional to the spectral density of fluctuations at
zero frequency31. The same result, but for a double-dot
system has been obtained by Rabenstein et al.32. On
the other hand, it follows from the Redfied’s approach
that the corresponding decoherence rate is proportional
to the spectral density at the frequency of the qubit’s
oscillations (the Rabi frequency)27. Since our model is
the exactly solvable one, we can resolve this ambiguity
and establish the appropriate physical conditions that
can result in different relations of decoherence rate to
the environmental fluctuations.
The most important results of our study are related to
the situation when back-action of the qubit on the envi-
ronment takes place. This problem did not receive such
a considerable amount of attention in the literature as,
for example, the case of “inert” environment. This is in
spite of a fact that the back-action always takes place in
the presence of measurement. There are many questions
related to the effects of a back-action. For instance, what
would be a relation between decoherence (relaxation) of
the qubit and the noise spectrum of the environment?
Or, how decoherence is affected by a strong response of
environment? We believe that our model appears to be
more suitable for studying these and other problems re-
lated to the back-action than most of the other existing
approaches.
The plan of this paper is as follows: Sect. II presents a
phenomenological description of decoherence and relax-
ation in the framework of Bloch equations, applied to the
electrostatic qubit. Sect. III contains description of the
model and the quantum rate-equation formalism, used
for its solution. Detailed quantum-mechanical derivation
of these equations for a specific example is presented in
Appendix A. Sect. IV deals with a configuration where
the SET can generate only decoherence of the qubit. We
consider separately the situations when SET produces
fluctuations of the tunneling coupling (Rabi frequency)
or of the energy levels. The results are compared with
the SET fluctuation spectrum, evaluated in Appendix B.
Sect. V deals with a configuration where the SET gener-
ates both decoherence and relaxation of the qubit. Sect.
VI is summary.
II. DECOHERENCE AND RELAXATION OF A
QUBIT
In this section we describe in a general phenomeno-
logical framework the effect of decoherence and relax-
ation on the qubit behavior. Although the results are
known, there still exists some confusion in the literature
in this issue. We therefore need to define precisely these
quantities and demonstrate how the corresponding deco-
herenece and relaxation rates can be extracted from the
qubit density matrix.
Let us consider an electrostatic qubit, realized by an
electron trapped in coupled quantum dots, Fig. 1. This
system is described by the following tunneling Hamilto-
Hqb = E1a
1a1 + E2a
2a2 − Ω0(a
2a1 + a
1a2) (1)
where a
1,2, a1,2 are the creation and annihilation opera-
tors of the electron in the first or in the second dot. For
simplicity we consider electrons as spinless fermions. In
addition, we assume that a
1a1 + a
2a2 = 1, so that only
one electron is present in the double-dot. The electron
wave function can be written as
|Ψ(t)〉 =
b(1)(t)a
1 + b
(2)(t)a
〉 (2)
where b(1,2)(t) are the probability amplitudes for find-
ing the electron in the first or second well, obtained
from the Schrödinger equation i∂t|Ψ(t)〉 = Hqb|Ψ(t)〉 (we
adopt the units where ~ = 1 and the electron charge
e = 1). The corresponding density matrix, σjj′ (t) =
b(j)(t)b(j
′)∗(t), with j, j′ = {1, 2}, is obtained from the
equation i∂t σ = [H,σ]. This can be written explicitly as
σ̇11 = iΩ0(σ21 − σ12) (3a)
σ̇12 = −iǫσ12 + iΩ0(1− 2σ11) , (3b)
where σ22(t) = 1 − σ11(t), σ21(t) = σ∗12(t) and ǫ =
E1 − E2. Solving these equations one easily finds that
the electron oscillates between the two dots (Rabi oscil-
lations) with frequency ωR =
4Ω20 + ǫ
2. For instance,
for the initial conditions σ11(0) = 1 and σ12(0) = 1, the
probability of finding the electron in the second dot is
σ22(t) = 2(Ω0/ωR)
2(1− cosωRt). This result shows that
for ǫ≫ Ω0 the amplitude of the Rabi oscillations is small,
so the electron remains localized in its initial state.
The situation is different when the qubit interacts with
the environment. In this case the (reduced) density ma-
trix of the qubit σ(t) is obtained by tracing out the en-
vironment variables from the total density matrix. The
question is how to modify Eqs. (3), written for an isolated
qubit, in order to obtain the reduced density matrix of
the qubit, σ(t). In general one expects that the environ-
ment could affect the qubit in two different ways. First, it
can destroy the off-diagonal elements of the qubit density
matrix. This process is usually referred to as decoherence
(or dephasing). It can be accounted for phenomenolog-
ically by introducing an additional (damping) term in
Eq. (3b),
σ̇12 = −iǫσ12 + iΩ0(1− 2σ11)−
σ12 (4)
where Γd is the decoherence rate. As a result the qubit
density-matrix σ(t) becomes a statistical mixture in the
stationary limit,
t→∞−→
1/2 0
0 1/2
. (5)
This happens for any initial conditions and even for large
level displacement, ǫ ≫ Ω0,Γd (provided that Ω0 6= 0).
Note that the statistical mixture (5) is proportional to
the unity matrix and therefore it remains the same in
any basis.
Secondly, the environment can put the qubit in its
ground state, for instance via photon or phonon emis-
sion. This process is usually referred to as relaxation.
For a symmetric qubit we would have
t→∞−→
1/2 1/2
1/2 1/2
. (6)
In contrast with decoherence, Eq. (5), the relaxation pro-
cess puts the qubit into a pure state. That implies that
the corresponding density matrix can be always written
as δ1iδ1j in a certain basis (the basis of the qubit eigen-
states). This is in fact the essential difference between
decoherence and relaxation. With respect to elimination
of the off-diagonal density matrix elements, note that re-
laxation would eliminate these terms only in the qubit’s
eigenstates basis. In contrast, decoherence eliminates the
off-diagonal density matrix element in any basis (Eq. (5)).
In fact, if the environment has some energy, it can put
the qubit into an exited state. However, if the qubit is
finally in a pure state, such excitation process generated
by the environment affects the qubit in the same way as
relaxation: it eliminates the off-diagonal density matrix
elements only in a certain qubit’s basis. Therefore exci-
tation of the qubit can be described phenomenologically
on the same footing as relaxation.
It is often claimed that decoherence is associated with
an absence of energy transfer between the system and
the environment, in contrast with relaxation (excitation).
This distinction is not generally valid. For instance, if the
initial qubit state corresponds to the electron in the state
|E2〉, Fig. 1, the final state after decoherence corresponds
to an equal distribution between the two dots, 〈E〉 =
(E1 +E2)/2. In the case of E1 ≫ E2, this process would
require a large energy transfer between the qubit and the
environment. Therefore decoherence can be consistently
defines as a process leading to a statistical mixture, where
all states of the system have equal probabilities (as in
Eq. (5)).
The relaxation (excitation) process can be described
most simply by diagonalizing the qubit Hamiltonian,
Eqs. (1), to obtain Hqb = E+a
+a+ + E−a
−a−, where
the operators a± are obtained by the corresponding ro-
tation of the operators a1,2
30. Here E+ and E− are the
ground (symmetric) and excited (antisymmetric) state
energies. Then the relaxation process can be described
phenomenologically in the new qubit basis |±〉 = a†±|0¯〉
σ̇−−(t) = −Γrσ−−(t) (7a)
σ̇+−(t) = i(E− − E+)σ+−(t)−
σ+−(t) , (7b)
where σ++(t) = 1 − σ−−(t), σ−+(t) = σ∗+−(t) and Γr is
the relaxation rate.
In order to add decoherence, we return to the orig-
inal qubit basis |1, 2〉 = a†1,2|0¯〉 and add the damping
term to the equation for the off-diagonal matrix elements,
Eq. (4). We arrive at the quantum rate equation describ-
ing the qubit’s behavior in the presence of both decoher-
ence and relaxation30,33,
σ̇11 = iΩ0(σ21 − σ12)− Γr
(σ12 + σ21)−
(2σ11 − 1) + Γr
σ̇12 = −iǫσ12 +
iΩ0 + Γr
(1− 2σ11) + Γr
σ12 − κ2(σ12 + σ21)
σ12 , (8b)
where ǫ̃ = (ǫ2 + 4Ω20)
1/2 and κ = Ω0/ǫ̃. In fact, these
equations can be derived in the framework of a particu-
lar model, representing an electrostatic qubit interacting
with the point-contact detector and the environment, de-
scribed by the Lee model Hamiltonian33.
Equations (8) can be rewritten in a simpler form by
mapping the qubit density matrix σ = {σ11, σ12, σ21} to
a “polarization” vector S(t) via σ(t) = [1 + τ · S(t)]/2,
where τx,y,z are the Pauli matrices. For instance, one
obtains for the symmetric case, ǫ = 0,
Ṡz = −
Sz − 2Ω0 Sy (9a)
Ṡy = 2Ω0 Sz −
Γd + Γr
Sy (9b)
Ṡx = −
Γd + 2Γr
(Sx − S̄x) (9c)
where S̄x = Sx(t→∞) = 2Γr/(Γd+2Γr). One finds that
Eqs. (9) have a form of the Bloch equations for spin-
precession in the magnetic field27, where the effect of
environment is accounted for by two relaxation times for
the different spin components: the longitudinal T1 and
the transverse T2, related to Γd and 2Γr as
T−11 =
Γd + 2Γr
, and T−12 =
Γd + Γr
, (10)
The corresponding damping rates, the so-called “depolar-
ization” (Γ1 = 1/T1) and the “dephasing” (Γ2 = 1/T2)
are used for phenomenological description of two-level
systems34. However, neither Γ1 nor Γ2 taken alone would
drive the qubit density matrix into a statistical mixture
Eq. (5) or into a pure state Eq. (6).
In contrast, our definition of decoherence and relax-
ation (excitation) is associated with two opposite effects
of the environment on the qubit: the first drives it into
a statistical mixture, whereas the second drives it into
a pure state. We expect therefore that such a natural
distinction between decoherence and relaxation would be
more useful for finding a relation between these quantities
and the environmental behavior than other alternative
definitions of these quantities existing in the literature.
In general, the two rates, Γd,r, introduced in phe-
nomenological equations (8), (9), are consistent with our
definitions of decoherence and relaxation. The only ex-
ception is the case of Γr = 0 and Ω0 = 0, where are no
transitions between the qubit’s states even in the pres-
ence of the environment (“static” qubit). One easily finds
from Eqs. (3a), (4) that σ12(t) → 0 for t → ∞, whereas
the diagonal density-matrix elements of the qubit remain
unchanged (so-called “pure dephasing”5,34):
t→∞−→
σ11(0) 0
0 σ22(0)
. (11)
Thus, if the initial probabilities of finding the qubit in
each of its states are not equal, σ11(0) 6= σ22(0), then the
final qubit state is neither a mixture nor a pure state, but
a combination of the both. It implies that Γd in Eqs. (8)
would also generate relaxation (excitation) of the qubit.
Note that in this case the off-diagonal density-matrix el-
ements, absent in Eq.(11), would reappear in a different
basis. This implies that the “pure dephasing”5,34 occurs
only in a particular basis.
Let us evaluate the probability of finding the electron
in the first dot, σ11(t). Solving Eqs. (9) for the initial
conditions σ11(0) = 1, σ12(0) = 0, we find
σ11(t) =
e−Γrt/2
−e−t + C2e
where e± =
(Γd ± Ω̃), Ω̃ =
Γ2d − 64Ω20 and C1,2 =
1±(Γd/Ω̃). Solving the same equations in the limit of t→
∞, we find that the steady-state qubit density matrix is
t→∞−→
1/2 Γr/(Γd + 2Γr)
Γr/(Γd + 2Γr) 1/2
. (13)
Thus the off-diagonal elements of the density matrix
can provide us with a ratio of relaxation to decoherence
rates33.
III. DESCRIPTION OF THE MODEL
Consider the setup shown in Fig. 2. The entire system
can be described by the following tunneling Hamiltonian,
represented by a sum of the qubit and SET Hamiltonians
and the interaction term, H = Hqb +HSET +Hint. Here
Hqb is given by Eq. (1) and describes the qubit. The sec-
ond term, HSET, describes the single-electron transistor.
It can be written as
HSET =
l cl +
rcr + E0c
l c0 +Ωrc
rc0 +H.c.) , (14)
where c
l,r and cl,r are the creation and annihilation elec-
tron operators in the state El,r of the right or left reser-
voir; c
0 and c0 are those for the level E0 inside the quan-
tum dot; and Ωl,r are the couplings between the level E0
and the level El,r in the left (right) reservoir. In order to
avoid too lengthy formulaes, our summation indices l, r
indicate simultaneously the left and the right leads of the
SET, where the corresponding summation is carried out.
As follows from the Hamiltonian (14), the quantum dot
of the SET contains only one level (E0). This assumption
has been implied only for the sake of simplicity for our
presentation, although our approach is well suited for a
case of n levels inside the SET, E0c
0c0 →
n Enc
and even when the interaction between these levels is in-
cluded (providing that the latter is much less or much
larger than the bias V )35,36. We also assumed a weak
energy dependence of the couplings Ωl,r ≃ ΩL,R.
The interaction between the qubit and the SET, Hint,
depends on a position of the SET with respect to the
qubit. If the SET is placed near the middle of the qubit,
Fig. 2a, then the tunneling coupling between two dots of
the qubit in Eq. (1) decreases, Ω0 → Ω0− δΩ0, whenever
the quantum dot of the SET is occupied by an electron.
This is due to the electron’s repulsive field. In this case
the interaction term can be written as
Hint = δΩ c
0c0(a
1a2 + a
2a1) . (15)
On the other hand, in the configuration shown in Fig. 2b
where the SET is placed near one of the dots of the qubit,
the electron repulsive field displaces the qubit energy lev-
els by ∆E = U . The interaction terms in this case can
be written as
Hint = U a
0c0 . (16)
Consider the initial state where all the levels in the left
and the right reservoirs are filled with electrons up to the
Fermi levels µL,R respectively. This state will be called
the “vacuum” state |0
〉. The wave function for the entire
system can be written as
|Ψ(t)〉 =
b(1)(t)a
0l (t)a
0cl +
rl (t)a
rcl +
l<l′,r
0rll′(t)a
rclcl′ + · · ·
+b(2)(t)a
0l (t)a
0cl +
rl (t)a
rcl +
l<l′,r
0rll′(t)a
rclcl′ + . . .
〉, (17)
where b(j)(t), b
α (t) are the probability amplitudes to
find the entire system in the state described by the cor-
responding creation and annihilation operators. These
amplitudes are obtained from the Schrödinger equation
i|Ψ̇(t)〉 = H |Ψ(t)〉, supplemented with the initial condi-
tion b(1)(0) = p1, b
(2)(0) = p2, and b
α (0) = 0, where
p1,2 are the amplitudes of the initial qubit state.
Note that Eq. (17) implies a fixed electron number (N)
in the reservoirs. At the first sight it would lead to deple-
tion of the left reservoir of electrons over the time. Yet
in the limit of N →∞ (infinite reservoirs) the dynamics
of an entire system reaches its steady state before such a
depletion takes place37,38.
The behavior of the qubit and the SET is given by
the reduced density matrix, σss′ (t). It is obtained from
the entire system’s density matrix |Ψ(t)〉〈Ψ(t)| by trac-
ing out the (continuum) reservoir states. The space of
such a reduced density matrix consists of four discrete
states s, s′ = a, b, c, d, shown schematically in Fig. 3 for
the setup of Fig. 2a. The corresponding density-matrix
elements are directly related to the amplitudes b(t), for
instance,
σaa(t) = |b(1)(t)|2 +
|b(1)lr (t)|
l<l′,r<r′
|b(1)rr′ll′(t)|
2 + · · · (18a)
σdd(t) =
|b(2)0l (t)|
l<l′,r
|b(2)0rll′(t)|
l<l′<l′′,r<r′
|b(2)0rr′ll′(t)|
2 + · · · (18b)
σbd(t) =
0l (t)b
0l (t) +
l<l′,r
0rll′(t)b
0rll′(t) +
l<l′<l′′,r<r′
0rr′ll′ (t)b
0rr′ll′(t) + · · · . (18c)
In was shown in37,38 that the trace over the reservoir
states in the system’s density matrix can be performed
in the large bias limit (strong non-equilibrium limit)
V = µL − µR ≫ Γ,Ω0, U (19)
where the level (levels) of the SET carrying the current
are far away from the chemical potentials, and Γ is the
width of the level E0. In this derivation we assumed
only weak energy dependence of the transition ampli-
tudes Ωl,r ≡ ΩL,R and the density of the reservoir states,
ρ(El,r) = ρL,R. As a result we arrive at Bloch-type rate
equations for the reduced density matrix without any
additional assumptions. The general form of these equa-
tions is36,38
σ̇jj′ = i(Ej′ − Ej)σjj′ + i
σjkΩ̃k→j′ − Ω̃j→kσkj′
P2πρ(σjkΩk→k′Ωk′→j′ + σkj′Ωk→k′Ωk′→j)
P2πρ (Ωk→jΩk′→j′ +Ωk→j′Ωk′→j)σkk′ (20)
Here Ωk→k′ denotes the single-electron hopping ampli-
tude that generates the k → k′ transition. We distinguish
between the amplitudes Ω̃ describing single-electron hop-
ping between isolated states and Ω describing transitions
between isolated and continuum states. The latter can
generate transitions between the isolated states of the
system, but only indirectly, via two consecutive jumps of
an electron, into and out of the continuum reservoir states
(with the density of states ρ). These transitions are rep-
resented by the third and the fourth terms of Eq. (20).
The third term describes the transitions (k → k′ → j)
or (k → k′ → j′), which cannot change the number of
electrons in the collector. The fourth term describes the
transitions (k → j and k′ → j′) or (k → j′ and k′ → j)
which increase the number of electrons in the collector
by one. These two terms of Eq. (20) are analogues of the
“loss” (negative) and the “gain” (positive) terms in the
classical rate equations, respectively. The factor P2 = ±1
in front of these terms is due anti-commutation of the
fermions, so that P2 = −1 whenever the loss or the gain
terms in Eq. (20) proceed through a two-fermion state of
the dot. Otherwise P2 = 1.
Note that the reduction of the time-dependent
Schrödinger equation, i|Ψ̇(t)〉 = H |Ψ(t)〉, to Eqs. (20)
is performed in the limit of large bias without explicit
use of any Markov-type or weak coupling approxima-
tions. The accuracy of these equations is respectively
max(Γ,Ω0, U, T )/|µL,R−Ej |. A detailed example of this
derivation is presented in Appendix A for the case of res-
onant tunneling through a single level. The derivation
there and in Refs.37,38 were performed by assuming zero
temperature in the leads, T = 0. Yet, this assumption
is not important in the case of large bias, providing the
levels carrying the current are far away from the Fermi
energies, |µL,R − Ej | ≫ T .
IV. NO BACK-ACTION ON THE
ENVIRONMENT
A. Fluctuation of the tunneling coupling
Now we apply Eqs. (20) to investigate the qubit’s be-
havior in the configurations shown in Fig. 2. First we
consider the SET placed near the middle of the qubit,
Figs. 2a,3. In this case the electron current through the
SET will influence the coupling between two dots of the
(d)(b)(a) (c)
E E1 2
Ω0 Ω0’ ’
FIG. 3: The available discrete states of the entire system cor-
responding to the setup of Fig. 2a. ΓL,R denote the tunneling
rates to the corresponding reservoirs and Ω′0 = Ω0 − δΩ.
qubit, making it fluctuate between the values Ω0 and
Ω′0 = Ω0 − δΩ. The corresponding rate equations can be
written straightforwardly from Eqs. (20). One finds,
σ̇aa = −ΓLσaa + ΓRσbb − iΩ0(σac − σca), (21a)
σ̇bb = −ΓRσbb + ΓLσaa − iΩ′0(σbd − σdb), (21b)
σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (21c)
σ̇dd = −ΓRσdd + ΓLσcc − iΩ′0(σdb − σbd), (21d)
σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)− ΓLσac
+ ΓRσbd, (21e)
σ̇bd = −iǫ0σbd − iΩ′0(σbb − σdd)− ΓRσbd
+ ΓLσac, (21f)
where ΓL,R = 2π|ΩL,R|2ρL,R are the tunneling rates from
the reservoirs and ǫ0 = E1 − E2.
These equations display explicitly the time evolution
of the SET and the qubit. The evolution of the for-
mer is driven by the first two terms in Eqs. (21a)-(21d).
They generate charge-fluctuations inside the quantum
dot of the SET (the transitions a←→b and c←→d),
described by the “classical” Boltzmann-type dynamics.
The qubit’s evolution is described by the Bloch-type
terms (c.f. Eqs. (3)), generating the qubit transitions
(a←→c and b←→d). Thus Eqs. (21) are quite general,
since they described fluctuations of the tunneling cou-
pling driven by the Boltzmann-type dynamics.
The resulting time evolution of the qubit is given by
the qubit (reduced) density matrix:
σ11(t) = σaa(t) + σbb(t) , (22a)
σ12(t) = σac(t) + σbd(t) , (22b)
and σ22(t) = 1− σ11(t).
Similarly, the charge fluctuations of SET are deter-
mined by the probability of finding the SET occupied,
P1(t) = σbb(t) + σdd(t) . (23)
It is given by the equation
Ṗ1(t) = ΓL − ΓP1(t) , (24)
obtained straightforwardly from Eqs. (21). Here Γ =
ΓL +ΓR is the total width. The same equation for P1(t)
can be obtained if the qubit is decoupled from the SET
(δΩ = 0). Thus there is no back-action of the qubit on
the charge fluctuations inside the SET in the limit of
large bias voltage.
Consider first the stationary limit, t → ∞, where
Ṗ1(t)→ 0 and σ̇(t)→ 0. It follows from Eq. (24) that the
probability of finding the SET occupied in this limit is
P̄1 = ΓL/Γ. This implies that the fluctuations of the cou-
pling Ω0, induced by the SET, would take place around
the average value Ω = Ω0 − P̄1 δΩ.
With respect to the qubit in the stationary limit, one
easily obtains from Eqs. (21) that the qubit density ma-
trix always becomes the statistical mixture (5), when
t → ∞. This takes place for any initial conditions and
any values of the qubit and the SET parameters. There-
fore the effect of the fluctuating charge inside the SET
does not lead to relaxation of the qubit, but rather to its
decoherence.
It is important to note, however, that for the aligned
qubit, ǫ = 0, the decoherence due to fluctuations of the
tunneling coupling Ω0 is not complete. Indeed, it follows
from Eqs. (21) that d/dt[Re σ12(t)] = 0. The reason is
that the corresponding operator, a
1a2 + a
2a1 commutes
with the total Hamiltonian H = Hqb + HSET + Hint,
Eqs. (1), (14) and (15), for E1 = E2. As a result,
Re σ12(t) = Re σ12(0).
In order to determine the decoherence rate analytically,
we perform a Laplace transform on the density matrix,
σ̃(E) =
σ(t) exp(−iEt)dE. Then solving Eq. (21) we
can determine the decoherence rate from the locations
of the poles of σ̃(E) in the complex E-plane. Consider
for instance the case of ǫ0 = 0 and the symmetric SET,
ΓL = ΓR = Γ/2. One finds from Eqs. (21) and (22a) that
σ̃11(E) =
i(E − 2Ω + iΓ)
4(E − 2Ω + iΓ/2)2 + Γ2 − (2 δΩ)2
i(E + 2Ω+ iΓ)
4(E + 2Ω+ iΓ/2)2 + Γ2 − (2 δΩ)2
. (25)
Upon performing the inverse Laplace transform,
σ11(t) =
∞+i0∫
−∞+i0
σ̃11(E) e
−iEt dE
, (26)
and closing the integration contour around the poles of
the integrand, we obtain for Γ > 2δΩ and t≫ 1/Γ
σ11(t)− (1/2) ∝ e−(Γ−
Γ2−4δΩ2)t/2 sin(2Ω t). (27)
Comparing this result with Eq. (12) we find that the
decoherence rate is
Γd = 2
Γ2 − 4δΩ2
Γ≫δΩ−→ (2δΩ)2/Γ . (28)
For ǫ0 6= 0 and ǫ0,Γ ≪ Ω the decoherece rate Γd is
multiplied by an additional factor [1− (ǫ0/2Ω)2].
In a general case, ΓL 6= ΓR, we obtain in the same
limit (ΓL,R ≫ δΩ) for the decoherence rate:
(4 δΩ)2
(ΓL + ΓR)3
It is interesting to compare this result with the fluctu-
ation spectrum of the charge inside the SET, Eq. (B8),
Appendix B. We find
Γd = 2 (δωR)
2 SQ(0) , (30)
where ωR =
4Ω2 + ǫ20 is the Rabi frequency. The latter
represents the energy splitting in the diagonalized qubit
Hamiltonian. Thus δωR corresponds to the amplitude of
energy level fluctuations in a single dot.
Although Eq. (30) has been obtained for small fluc-
tuations δωR, it might be approximately correct even
if δωR is of the order of Γ. It is demonstrated in
Fig. 4, where we compare σ11(t) and σ12(t), obtained
from Eqs. (21) and (22) (solid line) with those from
Eqs. (3a) and (4) (dashed line) for the decoherence rate
Γd given by Eq. (30). The initial conditions correspond to
σ11(0) = 1 and σ12(0) = 0 (respectively, σaa(0) = ΓR/Γ
and σbb(0) = ΓR/Γ).
In the case of aligned qubit, however, Re σ12(t) =
Re σ12(0), as was explained above. On the other hand,
one always obtains from (3a) and (4) that Re [σ12(t →
∞)] = 0. Therefore the phenomenological Bloch equa-
tions are not applicable for evaluation of Re [σ12(t)],
even in the weak coupling limit (besides the case of
Re [σ12(t = 0)] = 0).
In the large coupling regime (δΩ≫ Γ) the phenomeno-
logical Bloch equations, Eqs. (3a) and (4), cannot be
used, as well. Consider for simplicity the case of ǫ = 0
and ΓL,R = Γ/2. Then one finds from Eq. (27) that
the damping oscillations between the two dots take place
at two different frequencies, 2Ω ±
(δΩ)2 − (Γ/2)2, in-
stead of the one frequency, ωR = 2Ω, given the Bloch
equations. Moreover, Eq. (30) does not reproduce the
decoherence (damping) rate in this limit. Indeed, one
obtains from Eq. (28) that the decoherence rate Γd = 2Γ
for δΩ > Γ/2, so Γd does not depend on the coupling
(δΩ) at all.
B. Fluctuation of the energy level
Consider the SET placed near one of the qubit dots,
as shown in Fig. 2b. In this case the qubit-SET inter-
action term is given by Eq. (16). As a result the energy
level E1 will fluctuate under the influence of the fluctua-
tions of the electron charge inside the SET. The available
discrete states of the entire system are shown in Fig. 5.
Using Eqs. (20) we can write the rate equations, similar
10 20 30 40 50
(a) (b)
10 20 30 40 50
Re 12
Im 12
11 σ  ( )
σ  ( )
σ  ( )
FIG. 4: The occupation probability of the first dot of the qubit
for ǫ = 2Ω, ΓL = Ω, ΓR = 2Ω and δΩ = 0.5Ω. The solid line
is the exact result, whereas the dashed line is obtained from
the Bloch-type rate equations with the decoherence rate given
by Eq. (30).
(d)(b)(a) (c)
’’’ ’ Γ
R ΓL ΓR
E0 E0
0Ω0Ω0
E2 E2
E +U1Ω
FIG. 5: The available discrete states of the entire system for
the configuration shown in Fig. 2b. Here U is the repulsion
energy between the electrons.
to Eqs. (21),
σ̇aa = −Γ′Lσaa + Γ′Rσbb − iΩ0(σac − σca), (31a)
σ̇bb = −Γ′Rσbb + Γ′Lσaa − iΩ0(σbd − σdb), (31b)
σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (31c)
σ̇dd = −ΓRσdd + ΓLσcc − iΩ0(σdb − σbd), (31d)
σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)−
ΓL + Γ
Rσbd, (31e)
σ̇bd = −i(ǫ0 + U)σbd − iΩ0(σbb − σdd)−
ΓR + Γ
Lσac , (31f)
where Γ′L,R are the tunneling rate at the energy E0+U
Let us assume that Γ′L,R = ΓL,R. Then it follows from
Eqs. (31) that the behavior of the charge inside the SET
is not affected by the qubit, the same as in the previous
case of the Rabi frequency fluctuations. Also the qubit
density matrix becomes the mixture (5) in the stationary
state for any values of the qubit and the SET parameters.
Hence, there is no qubit relaxation in this case either
(except for the static qubit, Ω0 = 0, and σ11(0) 6= σ22(0),
Eq. (11)).
Since according to Eq. (24), the probability of finding
an electron inside the SET in the stationary state is P̄1 =
ΓL/Γ, the energy level E1 of the qubit is shifted by P̄1U .
Therefore it is useful to define the “renormalized” level
displacement, ǫ = ǫ0 + P̄1U .
As in the previous case we use the Laplace transform,
σ(t)→ σ̃(E), in order to determine the decoherence rate
analytically. In the case of ΓL = ΓR = Γ/2 and ǫ = 0 we
obtain from Eqs. (31)
σ̃11(E) =
32(E + iΓ)Ω20
U2 − 4E(E + iΓ)
. (32)
The position of the pole in the second term of this expres-
sion determines the decoherence rate. In contrast with
Eq. (25), however, the exact analytical expression for the
decoherence rate (Γd) is complicated, since it is given by
a cubic equation. We therefore evaluate Γd in a different
way, by substituting E = ±2Ω0 − iγ in the second term
of Eq. (32) and then expanding the latter in powers of
γ by keeping only the first two terms of this expansion.
The decoherence rate Γd is related to γ by Γd = 4γ, as
follows from Eq. (12). Then we obtain:
2(Γ2 + 4Ω20)
for U ≪ (Ω20 + ΓΩ0)1/2
64ΓΩ20
U2 + 16Ω20
for U ≫ (Ω20 + ΓΩ0)1/2
In general, if ΓL 6= ΓR, one finds from Eqs. (31) that
Γd = 2U
2ΓLΓR/[Γ(Γ
2 + 4Ω20)] for U ≪ (Ω20 + ΓΩ0)1/2.
The same as in the previous case, Eq. (30), the deco-
herence rate in a weak coupling limit is related to the
fluctuation spectrum of the SET, SQ(ω), Eq. (B8), but
now taken at a different frequency, ω = 2Ω0. The lat-
ter corresponds to the level splitting of the diagonalized
qubit’s Hamiltonian, ωR. Thus,
Γd = U
2 SQ(ωR) , (34)
which can be applied also for ǫ 6= 0. This is illustrated
by Fig. 6 which shows σ11(t) obtained from Eqs. (31)
and (22) (solid line) with Eqs. (3a) and (4) (dashed line)
for the decoherence rate Γd given by Eq. (34). As in
the previous case, shown in Fig. 4, the initial conditions
correspond to σ11(0) = 1 and σ12(0) = 0 (respectively,
σaa(0) = ΓR/Γ and σbb(0) = ΓR/Γ). One finds from
Fig. 6 that Eq. (34) can be used for an estimation of Γd
even for U ∼ Γ,Ω0.
In contrast with the tunneling-coupling fluctuations,
Eq. (30), where the decoherence rate is given by SQ(0),
the fluctuations of the qubit’s energy level generate the
decoherence rate, determined by the fluctuation spec-
trum at Rabi frequency, SQ(ωR), Eq. (34). A similar
distinction between the decoherence rates generated by
different components of the fluctuating field, exists in a
phenomenological description of magnetic resonance27.
One can understand this distinction by diagonalizing the
qubit’s Hamiltonian. In this case the Rabi frequency,
ωR, becomes the level splitting of the qubit’s states
|±〉 = (|1〉 ± |2〉)/
2 (for ǫ = 0). So in this basis, the
tunneling-coupling fluctuations correspond to simultane-
ous fluctuations of the energy levels in the both dots.
10 20 30 40 50
(a) (b)
10 20 30 40 50
Re 12
Im 12
11 σ  ( )
σ  ( )
σ  ( )
FIG. 6: The probability of finding the electron in the first
dot of the qubit for ǫ = 2Ω0, ΓL = Ω0, ΓR = 2Ω0 and U =
0.5Ω0. The solid line is the exact result, whereas the dashed
line is obtained from the Bloch-type rate equations with the
decoherence rate given by Eq. (34).
Since these fluctuations are “in phase”, we could expect
that the corresponding dephasing rate is determined by
spectral density at zero frequency. In fact, it looks like
as fluctuations of a single dot state, considered by Levin-
son in a weak coupling limit31. On the other hand by
fluctuating the energy level in one of the dots only, one
can anticipate that the corresponding dephasing rate is
determined by the fluctuation spectrum at the Rabi fre-
quency, ωR, Eq. (34), which is a frequency of the inter-dot
transitions.
Since ωR can be controlled by the qubit’s levels dis-
placement, ǫ, the relation (34) can be implied by using
qubit for a measurement of the shot-noise spectrum of the
environment18,19,40. For instance, it can be done by at-
taching a qubit to reservoirs at different chemical poten-
tials. The corresponding resonant current which would
flow through the qubit in this case, can be evaluated via
a simple analytical expression13 that includes explicitly
the decoherence rate, Eq. (34). Thus by measuring this
current for different level displacement of the qubit (ǫ0),
one can extract the spectral density of the fluctuating
environment acting on the qubit18.
Although Eq. (34) for the decoherence rate has been
obtained by using a particular mechanism for fluctuations
of the qubit’s energy levels, we suggest that this mecha-
nism is quite general. Indeed, the rate equations (31) can
describe any fluctuating media near a qubit, driven by the
Boltzmann type of equations. Therefore it is rather nat-
ural to assume that Eq. (34) would be valid for any type
of such (classical) environment in weak coupling limit.
This implies that the decoherence rate is always deter-
mined via the spectral density of a fluctuating qubit’s
level, whereas the nature of a particular medium inducing
these fluctuations would be irrelevant. In order to sub-
stantiate this point it is important to compare Eq. (34)
with the corresponding decoherence rate induced by the
thermal environment in the framework of the spin-boson
model. In a weak damping limit this model predicts1,2
T−11 = T
2 = (q
0/2)S(ωR) , where q0 is a coupling of
the medium with the qubit levels (q0 corresponds to U in
our case) and S(ω) is a spectral density. Using Eq. (10)
one finds that this result coincides with Eq. (34).
0 5 10 15 20
0 5 10 15 20
t t1111
U/      =100ΩU/      =10
σ  ( )
σ  ( )
(a) (b)
FIG. 7: The probability of finding the electron in the first
dot of the qubit for ǫ = 0, ΓL = ΓR = Ω0 and U , as given
by Eqs. (31) (solid line) and from the Bloch-type equations
(dashed line) with the decoherence rate given by Eq. (33).
C. Strong-coupling limit and localization
Let us consider the limit of U ≫ (Ω20 + ΓΩ0)1/2. Our
rate equation (31) are perfectly valid in this region, pro-
viding only that E0 + U is deeply inside of the potential
bias, Eq. (19). We find from Eq. (33) that the deco-
herence rate is not directly related to the spectrum of
fluctuations in strong coupling limit. In addition, the ef-
fective frequency of the qubit’s Rabi oscillations (ω
decreases in this limit. Indeed, by using Eqs. (32), (26),
one finds that the main contribution to σ11(t), is coming
from a pole of σ̃11(E), which lies on the imaginary axis.
This implies that the effective frequency of Rabi oscilla-
tions strongly decreases when U ≫ (Ω20 + ΓΩ0)1/2. In
addition, the decoherence rate Γd → 0 in the same limit,
Eq. (33). As a result, the electron would localize in the
initial qubit state, Fig. 7.
The results displayed in this figure show that the solu-
tion of the Bloch-type rate equations, with the decoher-
ence rate given by Eq. (33), represents damped oscilla-
tions (dashed line). It is very far from the exact result
(solid line), obtained from Eqs. (31) and corresponding
to the electron localization in the first dot. The latter is
a result of an effective decrease of the Rabi frequency for
large U that slows down electron transitions between the
dots. Thus such an environment-induced localization is
different from the Zeno-type effect (unlike an assumption
of Ref.12). Indeed, the Zeno effect takes place whenever
the decoherence rate is much larger then the coupling be-
tween the qubit’s states13,33. However, the decoherence
rate in the strong coupling limit is much smaller then the
coupling Ω0 . In fact, the localization shown in Fig. 7
is rather similar to that in the spin-boson model1,2. It
shows that in spite of their defferences, both models trace
the same physics of the back-action of the environment
(SET) on the qubit.
V. BACK-ACTION OF THE QUBIT ON THE
ENVIRONMENT
A. Weak back-action effect
Now we investigate a weak dependence of the width’s
ΓL,R on the energy U , Fig. 5. We keep only the linear
term, Γ′L,R = ΓL,R+αL,RU , by assuming that U is small.
(A similar model has been considered in28,41). In con-
trast with the previous examples, where the widths have
not been dependent on the energy, the qubit’s oscillation
would affect the SET current and its charge correlator.
A more interesting case corresponds to αL 6= αR. Let us
take for simplicity αL = 0 and αR = α 6= 0.
Similarly to the previous case we introduce the “renor-
malized” level displacement, ǫ = ǫ0 − (ΓL/Γ)U , where
ǫ = 0 corresponds to the aligned qubit. Solving Eqs. (31)
in the steady-state limit, σ̄ = σ(t → ∞), and keeping
only the first term in expansion in powers of U , we find
for the reduced density matrix of the qubit, Eqs. (22):
− α ǫ
αΩ0(1 + c αU)
αΩ0(1 + c αU)
, (35)
where c = (αǫ − 2Γ)/(4ΓRΓ). It follows from Eqs. (35)
that the qubit’s density matrix in the steady-state is no
longer a mixture, Eq. (5) . Indeed, the probability to
occupy the lowest level is always larger than 1/2 and
σ̄12 6= 0. This implies that relaxation takes place to-
gether with decoherence. The ratio of the relaxation and
decoherence rates is given by the off-diagonal terms of
the reduced density matrix of the qubit. For ǫ = 0 one
finds from Eq. (13) that Γd/Γr = σ̄
12 − 2.
In order to find a relation between the decoherence
and relaxation rates, Γd,r, and the fluctuation spectrum
of the qubit energy level, SQ(ω), we first evaluate the
total damping rate of the qubit’s oscillations (γ). Using
Eq. (12) we find that this quantity is related to the deco-
herence and relaxation rates by γ = (Γd + 2Γr)/4. The
same as in the previous case the rate γ is determined by
poles of Laplace transformed density matrix σ(t)→ σ̃(E)
in the complex E-plane. Consider for simplicity the case
of ǫ = 0 and ΓL = ΓR = Γ/2. Performing the Laplace
transform of Eqs. (31) we look for the poles of σ11(E) at
E = ±2Ω0 − iγ by assuming that γ is small. We obtain
Γd + 2Γr =
2(Γ2 + 4Ω20)
Γ− αU Γ
2 − 4Ω20
2 (Γ2 + 4Ω20)
for U ≪ Ω0.
Now we evaluate the correlator of the charge inside the
SET, SQ(ω) which induces the energy-level fluctuations
of the qubit. Using Eqs. (31) and (B6) we find,
SQ(ω) =
2 (Γ2 + ω2)
− αU Γ
2 − ω2
4 (Γ2 + ω2)
for αU ≪ Γ. Therefore in the limit of U ≪ Ω0 and αU ≪
Γ the total damping rate of the qubit’s oscillations is
directly related to the spectral density of the fluctuations
spectrum taken at the Rabi frequency,
Γd + 2Γr = U
2SQ(2Ω0). (38)
This represents a generalization of Eq. (34) for the case
of a weak back-action of qubit oscillations on the spectral
density of the environment. As a result, the qubit dis-
plays relaxation together with decoherence. It is remark-
able that the total qubit’s damping rate is still given by
the fluctuation spectrum of the SET (environment) mod-
ulated by the qubit. Note that Eq. (38) can be applied
only if the modulation of the tunneling rate through the
SET (tunneling current) is small αU ≪ Γ, in addition to
a weak distortion of the qubit (U ≪ Ω0).
In the case of strong back-action of the qubit on the
environment the decorerence and relaxation rates of the
qubit are not directly related to the fluctuation spectrum
of the environment, even if the distortion of the qubit is
small. This point is illustrated by the following example.
B. Strong back-action
Until now we considered the case where E0 + U ≪
µL, so that the interacting electron of the SET remains
deeply inside the voltage bias. If however, the interaction
U between the qubit and the SET is such that E0 +
U ≫ µL, the qubit’s oscillation would strongly affect the
fluctuation of charge inside the SET. Indeed, the current
through the SET is blocked whenever the level E1 of the
qubit is occupied, Fig. 8. In fact, this case can be treated
with small modification of the rate equations (31), if only
µL − E0 ≫ Γ and E0 + U − µL ≫ Γ, where E0 is a level
of the SET carrying the current.
The corresponding quantum rate equations describing
the system are obtained directly from Eqs. (20). As-
suming that the widths ΓL,R are energy independent we
find16
σ̇aa = (ΓL + ΓR)σbb − iΩ0(σac − σca), (39a)
σ̇bb = −(ΓR + ΓL)σbb − iΩ0(σbd − σdb), (39b)
σ̇cc = −ΓLσcc + ΓRσdd − iΩ0(σca − σac), (39c)
σ̇dd = −ΓRσdd + ΓLσcc − iΩ0(σdb − σbd), (39d)
σ̇ac = −iǫ0σac − iΩ0(σaa − σcc)−
+ ΓRσbd, (39e)
σ̇bd = −i(ǫ0 + U)σbd − iΩ0(σbb − σdd)
σbd . (39f)
Solving Eqs. (39) in the stationary limit, σ̄ = σ(t →
∞) and introducing the “renormalized” level displace-
ment, ǫ = ǫ0 −UΓL/(2Γ), we obtain for the qubit’s den-
(d)(b)(a) (c)
E +U1
FIG. 8: The available discrete states of the entire system
when the electron-electron repulsive interaction U breaks off
the current through the SET.
sity matrix, Eqs. (22) in the steady state:
σ̄11 =
− 8ǫU
16ǫ2 + 8Uǫ+ 48Ω20 + 9(U
2 + Γ2)
,(40a)
σ̄12 =
12UΩ0
16ǫ2 + 8Uǫ+ 48Ω20 + 9 (U
2 + Γ2)
, (40b)
where for simplicity we considered the symmetric case,
ΓL = ΓR = Γ/2. It follows from Eqs. (40) that similarly
to the previous example, the qubit’s density matrix is no
longer a mixture (5). The relaxation takes place together
with decoherence in this case too.
Let us consider weak distortion of the qubit by the
SET, U < Ω0. Although the values of U are restricted
from below (U ≫ Γ+µL−E0), this limit can be achieved
if the level E0 is close to the Fermi energy, providing only
that µL − E0 ≫ Γ, and Γ≪ U . Now we evaluate σ11(t)
with the rate equations (39) and then compare it with
the same quantity obtained from the Bloch equations,
Eq. (12), where Γd,r are given by Eqs. (34)and (13). The
corresponding charge-correlator, SQ(ωR), is evaluated by
Eqs. (B6) and (39). As an example, we take symmetric
qubit with aligned levels, ǫ = 0, ΓL = ΓR = 0.05Ω0 and
U = 0.5Ω0. The decoherence and relaxation rates, corre-
sponding to these parameters are respectively: Γd/Ω0 =
0.0038 and Γr/Ω0 = 0.00059.
The results are presented in Fig. 9a. The solid line
shows σ11(t), obtained from the rate equations (39),
where the dashed line is the same quantity obtained from
Eq. (12). We find that Eq. (34) (or (38)) underestimates
the actual damping rate of σ11(t) by an order of mag-
nitude). This lies in a sharp contrast with the previous
case, where the energy level of the SET is not distorted
by the qubit, Γ′L,R = ΓL,R, Fig. 5. Indeed, in this case
σ11(t) obtained Eq. (12) with Γd given by Eq. (34) and
Γr = 0, agrees very well with that obtained from the rate
equations (31), as shown in Fig. 9b.
Such an example clearly illustrates that the decoher-
ence is not related to the fluctuation spectrum of the
environment, whenever the environment is strongly af-
fected by the qubit, even if the coupling with a qubit
is small. This is a typical case of measurement, corre-
sponding to a noticeable response of the environment to
the qubit’s state (a “signal”).
20 40 60 80 100
20 40 60 80 100
t t11 11σ  ( ) σ  ( )
FIG. 9: (a) The probability of finding the electron in the first
dot of the qubit for ǫ = 0, ΓL = ΓR = 0.05Ω0 and U = 0.5Ω0.
The solid line is obtained from Eqs. (39), whereas the dashed
line corresponds to the Eq. (12) with Γd given by Eq. (34);
(b) the same for the case, shown in Fig. 5, where the solid
line corresponds to Eqs. (31).
VI. SUMMARY
In this paper we propose a simple model describing
a qubit interacting with fluctuating environment. The
latter is represented by a single electron transistor (SET)
in close proximity of the qubit. Then the fluctuations
of the charge inside the SET generate fluctuating field
acting on the qubit. In the limit of large bias voltage, the
Schrödinger equation for the entire system is reduced to
the Bloch-type rate equations. The resulting equations
are very simple, so that one can easily analyze the limits
of weak and strong coupling of the qubit with the SET.
We considered separately two different cases: (a) there
is no back-action of the qubit on the SET behavior, so
that the latter represents a “pure environment”; and (b)
the SET behavior depends on the qubit’s state. In the
latter case the SET can “measure” the qubit. The setup
corresponding to the “pure environment” is realized when
the energy level of the SET carrying the current lies
deeply inside the potential bias. The second (measure-
ment) regime of the SET is realized when the tunnel-
ing widths of the SET are energy dependent, or when
the energy level of the SET carrying the current is close
enough to the Fermi level of the corresponding reservoir.
Then the electron-electron interaction between the qubit
and the SET modulates the electron current through the
In the case of the “pure environment” (“no-
measurement” regime) we investigate separately two dif-
ferent configurations of the qubit with respect to the
SET. In the first one the SET produces fluctuations of
the off-diagonal coupling (Rabi frequency) between two
qubit’s states. In the second configuration the SET pro-
duces fluctuations of the qubit’s energy levels. In the
both cases we find no relaxation of the qubit, despite
the energy transfer between the qubit and the SET can
take place. As a result the qubit always turns asymptot-
ically to the statistical mixture. We also found that in
both cases the decoherence rate of the qubit in the weak
coupling limit is given by the spectral density of the cor-
responding fluctuating parameter. The difference is that
in the case of the off-diagonal coupling fluctuations the
spectral density is taken at zero frequency, whereas in
the case of the energy level fluctuations it is taken at the
Rabi-frequency.
In the case of the strong coupling limit, however, the
decoherence rate is not related to the fluctuation spec-
trum. Moreover we found that the electron in the qubit
is localized in this limit due to an effective decrease of
the off-diagonal coupling. This phenomenon may resem-
ble the localization in the spin-boson model in the strong
coupling limit.
If the charge correlator and the total SET current are
affected by the qubit (back-action effect), we found that
the off-diagonal density-matrix elements of the qubit sur-
vive in the steady-state limit and therefore the relax-
ation rate is not zero. We concentrated on the case of
weak coupling, when the Coulomb repulsion between the
qubit and the SET is smaller then the Rabi frequency.
The back-action of the qubit on the SET, however, can
be weak or strong. In the first case we found that the
total damping rate of the qubit due to decoherence and
relaxation is again given by the spectral density of the
SET charge fluctuations, modulated by the qubit. This
relation, however, is not working if the back-action is
strong. Indeed, we found that the damping rate of the
qubit in this case is larger by an order of magnitude than
that given by the spectral density of the corresponding
fluctuating parameter.
This looks like that in the strong back-action of the
qubit on the SET the major component of decoherence is
not coming from the fluctuation spectrum of the qubit’s
parameters only, but also from the measurement “sig-
nal” of the SET. On the first sight it could agree with
an analysis of Ref.30, suggesting that the decoherence
rate contains two components, generated by a measure-
ment and by a “pure environment” (environmental fluc-
tuations). The latter therefore represents an unavoid-
able decoherence, generated by any environment. Yet,
in a weak coupling regime such a separation seems not
working. In this case the damping (decoherence) rate is
totally determined by the environment fluctuations, even
so modulated by the qubit.
Although our model deals with a particular setup, it
bears the main physics of a fluctuating environment, act-
ing on a qubit. Indeed, the Bloch-type rate equations,
which we used in our analysis have a pronounced phys-
ical meaning: they relate the variation of qubit param-
eters with a nearby fluctuating field described by rate
equations. A particular mechanism, generated this field
should not be relevant for an evaluations of the deco-
herence and relaxation rates, but only its fluctuation
spectrum. Indeed, in the weak coupling limit our re-
sult for the decorence rate coincides with that obtained
in a framework of the spin-boson model. Thus our model
can be considered as a generic one. Its main advantage is
that it can be easily extended to multiple coupled qubits.
Such an analysis would allow to determine how decoher-
ence scales with number of qubits42, which is extremely
important for a realization of quantum computations.
In addition, our model can be extended to a more
complicated fluctuating environments, such as containing
characteristic frequencies in its spectrum. It would for-
mally correspond to a replacement of the SET in Fig. 2
by a double-dot (DD) coupled to the reservoirs43. All
these situations, however, must be a subject of a sepa-
rate investigation.
VII. ACKNOWLEDGEMENT
One of us (S.G.) thanks T. Brandes and C. Emary
for helpful discussions and important suggestions. S.G is
also grateful to the Max Planck Institute for the Physics
of Complex Systems, Dresden, Germany, and to NTT Ba-
sic Research Laboratories, Atsugi-shi, Kanagawa, Japan,
for kind hospitality.
APPENDIX A: QUANTUM-MECHANICAL
DERIVATION OF RATE EQUATIONS FOR
QUANTUM TRANSPORT
Consider the resonant tunneling through the SET,
shown schematically in Fig. 10. The entire system is
described by the Hamiltonian HSET, given by Eq. (14).
The wave function can be written in the same way as
Eq. (17), where the variables related to the qubit are
omitted,
|Ψ(t)〉 =
b(t) +
b0l(t)c
0cl +
brl(t)c
l<l′,r
b0rll′(t)c
rclcl′ + · · ·
〉. (A1)
Substituting |Ψ(t)〉 into the time-dependent Schrödinger
equation, i∂t|Ψ(t)〉 = HSET|Ψ(t)〉, and performing the
Laplace transform, b̃(E) =
exp(iEt) b(t)dt, we obtain
the following infinite set of algebraic equations for the
FIG. 10: Resonant tunneling through a single dot. µL,R are
the Fermi energies in the collector and emitter, respectively.
amplitudes b̃(E):
Eb̃(E) −
Ωlb̃0l(E) = i (A2a)
(E + El − E0)b̃0l(E)− Ωlb̃(E)
Ωr b̃lr(E) = 0 (A2b)
(E + El − Er)b̃lr(E)− Ωr b̃0l(E)
Ωl′ b̃0ll′r(E) = 0 (A2c)
(E + El + El′ − E0 − Er)b̃0ll′r(E) − Ωl′ b̃lr(E)
+ Ωlb̃l′r(E)−
Ωr′ b̃ll′rr′(E) = 0 (A2d)
· · · · · · · · ·
(The r.h.s of Eq. (A2a) reflects the initial condition.)
Let us replace the amplitude b̃ in the term
Ωb̃ of each
of the equations (A2) by its expression obtained from the
subsequent equation. For example, substituting b̃0l(E)
from Eq. (A2b) into Eq. (A2a) we obtain
E + El − E0
b̃(E)
E + El − E0
b̃lr(E) = i. (A3)
Since the states in the reservoirs are very dense (contin-
uum), one can replace the sums over l and r by integrals,
for instance
ρL(El) dEl , where ρL(El) is the
density of states in the emitter, and Ωl,r → ΩL,R(El,r).
Consider the first term
Ω2L(El)
E + El − E0
ρL(El)dEl (A4)
where Λ is the cut-off parameter. Assuming weak en-
ergy dependence of the couplings ΩL,R and the density of
states ρL,R, we find in the limit of high bias, µL = Λ→∞
S1 = −iπΩ2L(E0 − E)ρL(E0 − E) = −i
. (A5)
Consider now the second sum in Eq. (A3).
ρR(Er)dEr
ΩL(El)ΩR(Er)b̃lr(E,El, Er)
E + El − E0
ρL(El)dEl , (A6)
where we replaced b̃lr(E) by b̃(E,El, Er) and took µL =
Λ, µR = −Λ. In contrast with the first term of Eq. (A3),
the amplitude b̃ is not factorized out the integral (A6).
We refer to this type of terms as “cross-terms”. Fortu-
nately, all “cross-terms” vanish in the limit of large bias,
Λ → ∞. This greatly simplifies the problem and is very
crucial for a transformation of the Schrödinger to the rate
equations. The reason is that the poles of the integrand
in the El(Er)-variable in the “cross-terms” are on the
same side of the integration contour. One can find it by
using a perturbation series the amplitudes b̃ in powers of
Ω. For instance, from iterations of Eqs. (A2) one finds
b̃(E,El, Er) =
iΩLΩR
E(E + El − Er)(E + El − E0)
+ · · ·
The higher order powers of Ω have the same structure.
Since E → E + iǫ in the Laplace transform, all poles of
the amplitude b̃(E,El, Er) in the El-variable are below
the real axis. In this case, substituting Eq. (A7) into
Eq. (A6) we find
(E + iǫ)(E + E0 − E1 + iǫ)2(E + E0 − Er + iǫ)
+ · · ·
dEl = 0 , (A8)
Thus, S2 → 0 in the limit of µL →∞, µR → −∞.
Applying analogous considerations to the other equa-
tions of the system (A2), we finally arrive at the following
set of equations:
(E + iΓL/2)b̃(E) = i (A9a)
(E + El − E0 + iΓR/2)b̃0l(E)
− Ωlb̃(E) = 0 (A9b)
(E + El − Er + iΓL/2)b̃lr(E)
− Ωrb̃0l(E) = 0 (A9c)
(E + El + El′ − E0 − Er + iΓR/2)b̃0ll′r(E)
− Ωl′ b̃lr(E) + Ωlb̃l′r(E) = 0 (A9d)
· · · · · · · · ·
Eqs. (A9) can be transformed directly to the reduced
density matrix σ
(n,n′)
jj′ (t), where j = 0, 1 denote the state
of the SET with an unoccupied or occupied dot and n de-
notes the number of electrons which have arrived at the
collector by time t. In fact, as follows from our derivation,
the diagonal density-matrix elements, j = j′and n = n′,
form a closed system in the case of resonant tunneling
through one level, Fig. 10. The off-diagonal elements,
j 6= j′, appear in the equation of motion whenever more
than one discrete level of the system carry the transport
(see Eq. (20). Therefore we concentrate below on the di-
agonal density-matrix elements only, σ
00 (t) ≡ σ
(n,n)
00 (t)
and σ
11 (t) ≡ σ
(n,n)
11 (t). Applying the inverse Laplace
transform on finds
00 (t) =
l...,r...
dEdE′
b̃l · · ·
r · · ·
(E)b̃∗l · · ·
r · · ·
(E′)ei(E
′−E)t (A10a)
11 (t) =
l...,r...
dEdE′
0l · · ·
r · · ·
(E)b̃∗
0l · · ·
r · · ·
(E′)ei(E
′−E)t (A10b)
Consider, for instance, the term σ
11 (t) =
l |b0l(t)|2.
Multiplying Eq. (A9b) by b̃∗0l(E
′) and then subtracting
the complex conjugated equation with the interchange
E ↔ E′ we obtain
dEdE′
(E′ − E − iΓR)b̃0l(E)b̃∗0l(E′)
− 2Im
Ωlb̃0l(E)b̃
∗(E′)
′−E)t = 0 (A11)
Using Eq. (A10b) one easily finds that the first inte-
gral in Eq. (A11) equals to −i[σ̇(0)11 (t)+ΓRσ
11 (t)]. Next,
substituting
b̃0l(E) =
Ωlb̃(E)
E + El − E0 + iΓR/2
(A12)
from Eq. (A9b) into the second term of Eq. (A11), and
replacing a sum by an integral, one can perform the El-
integration in the large bias limit, µL → ∞, µR → −∞.
Then using again Eq. (A10b) one reduces the second
term of Eq. (A11) to iΓLσ
00 (t). Finally, Eq. (A11) reads
11 (t) = ΓLσ
00 (t)− ΓRσ
11 (t).
The same algebra can be applied for all other am-
plitudes b̃α(t). For instance, by using Eq. (A10a) one
easily finds that Eq. (A9c) is converted to the following
rate equation σ̇
00 (t) = −ΓLσ
00 (t) + ΓRσ
11 (t). With
respect to the states involving more than one electron
(hole) in the reservoirs (the amplitudes like b̃0ll′r(E) and
so on), the corresponding equations contain the Pauli ex-
change terms. By converting these equations into those
for the density matrix using our procedure, one finds the
“cross terms”, like
Ωlb̃l′r(E)Ωl′ b̃
′), generated by
Eq. (A9d). Yet, these terms vanish after an integration
over El(r) in the large bias limit, as the second term in
Eq. (A3). The rest of the algebra remains the same,
as described above. Finally we arrive at the following
infinite system of the chain equations for the diagonal
elements, σ
00 and σ
11 , of the density matrix,
00 (t) = −ΓLσ
00 (t) , (A13a)
11 (t) = ΓLσ
00 (t)− ΓRσ
11 (t) , (A13b)
00 (t) = −ΓLσ
00 (t) + ΓRσ
11 (t) , (A13c)
11 (t) = ΓLσ
00 (t)− ΓRσ
11 (t) , (A13d)
· · · · · · · · ·
Summing over n in Eqs. (A13) we find for the reduced
density matrix of the SET, σ(t) =
(n)(t), the fol-
lowing “classical” rate equations,
σ̇00(t) = −ΓLσ00(t) + ΓRσ11(t) (A14a)
σ̇11(t) = ΓLσ00(t)− ΓRσ11(t) (A14b)
These equations represent a particular case of our general
quantum rate equations (20), which are derived using the
above described technique37,38.
APPENDIX B: CORRELATOR OF ELECTRIC
CHARGE INSIDE THE SET.
The charge correlator inside the SET is given by
SQ(ω) = S̄Q(ω) + S̄Q(−ω), where
S̄Q(ω) =
〈δQ̂(0)δQ̂(t)〉eiωtdt . (B1)
Here δQ̂(t) = c
0(t)c0(t) − q̄ and q̄ = P̄1 = P1(t → ∞) is
the average charge inside the dot. Since the initial state,
t = 0 in Eq. (B1) corresponds to the steady state, one
can represent the time-correlator as
〈δQ̂(0)δQ̂(t)〉 =
q=0,1
Pq(0)(q − q̄)(〈Qq(t)〉 − q̄) , (B2)
where Pq(0) is the probability of finding the charge q =
0, 1 inside the quantum dot in the steady state, such that
P1(0) = q̄ and P0(0) = 1 − q̄, and 〈Qq(t)〉 = P (q)1 (t) is
the average charge in the dot at time t, starting with the
initial condition P
1 (0) = q. Substituting Eq. (B2) into
Eq. (B1) we finally obtain
S̄Q(ω) = q̄(1 − q̄)[P̃ (1)1 (ω)− P̃
1 (ω)] , (B3)
where P̃
1 (ω) is a Laplace transform of P
1 (t). These
quantities are obtained directly from the rate equations,
such that q̄ = σ̄bb + σ̄dd and P̃
1 (ω) = σ̃
bb (ω) +
dd (ω), where σ̄ = σ(t → ∞) and σ̃(q)(ω) is the
Laplace transform σ(q)(t) with the initial conditions
corresponding to the occupied (q = 1) or unoccupied
(q = 0) SET. In order to find these quantities it is use-
ful to rewrite the rate equations in the matrix form,
σ̇(t) = Mσ(t), representing σ(t) as the eight-vector,
σ = {σaa, σbb, σcc, σdd, σac, σca, σbd, σdb} and M as the
corresponding 8× 8-matrix. Applying the Laplace trans-
form we find the following matrix equation,
(i ω I +M)σ̃(q)(ω) = −σ(q)(0) , (B4)
where I is the unit matrix and σ(q)(0) is the initial con-
dition for the density-matrix obtained by projecting the
total wave function (17) on occupied (q = 1) and unoc-
cupied (q = 0) states of the SET in the limit of t → ∞,
σ(1)(0) = N1{0, σ̄bb, 0, σ̄dd, 0, 0, σ̄bd, σ̄db} , (B5a)
σ(0)(0) = N0{σ̄aa, 0, σ̄cc, 0, σ̄ac, σ̄ca, 0, 0} , (B5b)
and N1 = 1/q̄ and N0 = 1/(1− q̄) are the corresponding
normalization factors. Finally one obtains:
SQ(ω) = 2q̄(1− q̄)Re [σ̃(1)bb (ω) + σ̃
dd (ω)
− σ̃(0)bb (ω)− σ̃
dd (ω)]. (B6)
In the case shown in Fig. 2 one finds from Eqs. (21) or
Eqs. (31) for Γ′L,R = ΓL,R that σ̄ac = σbd = 0, q̄ = ΓL/Γ
and σ̃
bb (ω) + σ̃
dd (ω) = P̃
1 (ω). The latter equation is
given by
(iω − Γ)P̃ (q)1 (ω) = −q +
. (B7)
Substituting Eq. (B7) into Eq. (B3) one obtains:
SQ(ω) =
2ΓLΓR
Γ(ω2 + Γ2)
. (B8)
Obviously, for a more general case when Γ′L,R 6= ΓL,R,
or when the electron-electron interaction excites the elec-
tron inside the SET above the Fermi level, Fig. 8, the ex-
pressions for SQ(ω), obtained from Eq. (B6) have a more
complicated than Eq. (B8).
∗ Electronic address: shmuel.gurvitz@weizmann.ac.il
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mailto:shmuel.gurvitz@weizmann.ac.il
