PHYSICAL REVIEW B 67, 125323 共2003兲
Steering of a bosonic mode with a double quantum dot
T. Brandes and N. Lambert
Department of Physics, University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88,
Manchester M60 1QD, United Kingdom
共Received 1 October 2002; published 31 March 2003兲
We investigate the transport and coherence properties of a double quantum dot coupled to a single damped
boson mode. The numerical results reveal how the properties of the boson distribution can be steered by
altering parameters of the electronic system such as the energy difference between the dots. Quadrature
amplitude variances and the Wigner function are employed to illustrate how the state of the boson mode can be
controlled by a stationary electron current through the dots.
DOI: 10.1103/PhysRevB.67.125323
PACS number共s兲: 73.21.La, 71.38.⫺k, 62.25.⫹g, 42.50.⫺p
I. INTRODUCTION
The two-level system coupled to a single bosonic mode
共Rabi Hamiltonian1兲 is probably one of the best-studied models for the interaction of matter with light.2 Cavity quantum
electrodynamics is an example where the coupling between
atoms and photons can be studied in detail and used in order
to, e.g., transfer quantum coherence from light to matter
共control of tunneling by electromagnetic fields3,4兲 and vice
versa.5–7
Quantum optics usually deals with closed two-level systems where the total electron number on individual atoms
remains constant and does not fluctuate during the interaction with the photon. This restriction can be lifted in semiconductor quantum dots 共artificial atoms兲 by tunnel coupling
to electron reservoirs. For example, in semiconductor cavity
quantum electrodynamics,8 the interaction of light with excitons can be steered by ‘‘pumping’’ the cavity by resonant
tunneling of electrons and holes.
In this paper, we examine a single bosonic mode interacting with a system of two 共bound兲 electronic states which are
themselves coupled to a continuum of free electrons. As a
concrete realization, we investigate the stationary electron
transport through a double quantum dot coupled to electron
reservoirs and a single photon or phonon mode. The main
idea is to control the density matrix of the coupled dot-boson
system by external parameters such as the reservoir chemical
potentials and the tunnel couplings. These parameters 共which
in experiments can be controlled via gate voltages兲 then determine the reduced density operators of the dot and the boson and allow one, e.g., to modify the state of the boson by
driving a stationary electron current through the dots.
One of the motivations for our work are experiments in
double quantum dots where the coherent coupling between
classical light 共microwaves兲 and electrons can be detected in
electronic transport.9–12 The open two-level-boson model describes the full quantum version of these systems with the
light replaced by a photon field with its own quantum dynamics.
On the other hand, quantum effects 共such as spontaneous
emission兲 relevant for transport through double dots have
been found to be due to phonons rather than photons. In fact,
the importance of electron-phonon coupling for transport in
coupled dots is well-established by now.13–17 Suggestions
0163-1829/2003/67共12兲/125323共10兲/$20.00
have been made14,18,19 to explore ‘‘semiconductor phonon
cavity QED’’ in nanostructures where phonons become experimentally controllable. In contrast to standard semiconductor cavity QED,8 boundary conditions for vibration
modes lead to extremely nonlinear phonon-dispersion relations. For example, Van Hove singularities in the phonon
density of states show up at certain frequencies,18 signatures
of which seem to be relevant for transport through quantum
dots embedded into free-standing structures.20 Such a situation would then be described 共within a strongly idealized
model兲 by the coupling of a single frequency boson mode to
a few-level quantum dot.
The influence of nanomechanical vibrational properties on
single-electron tunneling in fact has emerged as a whole new
area of mesoscopic transport, triggered by the possibility to
explore electron transport through individual molecules21–25
or in free-standing26 –29 and movable30–33 nanostructures.
Phonons then are no longer a mere source of dissipation but
become experimentally controllable with the possibility, e.g.,
to create phonon confinement18,34 –36 or the analogon of
quantum optical phenomena such as coherent and squeezed
phonon states.37– 40
The paper is organized as follows. In Sec. II, we introduce
the model and derive a master equation for the density operator. We discuss transport properties in Sec. III where a
comparison to analytical solutions is made. In Sec. IV, the
reduced boson density operator, its Wigner function, and the
fluctuation properties of the boson mode under a stationary
electron current are discussed. Finally, we conclude with a
discussion in Sec. V.
II. MODEL AND METHOD
We consider an idealized situation of a single-cavity boson mode 共photon or phonon兲 coupled to a two-level electronic system 共dot兲 which itself is connected to external electron reservoirs. Our aim is to determine the stationary
reduced density operator ␳ (t) of the dot-boson system for
large times t, treating the dot and the boson on equal footing.
The single bosonic mode 共photon or phonon兲 interacts with
the electrons within the dots. We set ប⫽1 in the following.
A. The Hamiltonian
An artificial ‘‘open’’ two-level system can be realized by
two quantum dots 共which for simplicity we call ‘‘left’’ and
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PHYSICAL REVIEW B 67, 125323 共2003兲
T. BRANDES AND N. LAMBERT
‘‘right’’兲 coupled to each other by a tunnel barrier and to a
left source and right drain electron reservoir with chemical
potentials ␮ L and ␮ R . In the following, we are only interested in the regime of high bias voltage V⫽ ␮ L ⫺ ␮ R ⬎0. We
assume that the charging energy U required to add an additional electron to the double dot is much greater than V
共strong Coulomb blockade regime兲. Therefore, electrons tunnel through the structure only from the left to the right. Only
one additional electron can tunnel into the double dot at a
time, and the effective Hilbert space of the electronic system
can be defined by three states10,16 empty, left, and right, 兩 0 典
⫽ 兩 N L ,N R 典 , 兩 L 典 ⫽ 兩 N L ⫹1,N R 典 , and 兩 R 典 ⫽ 兩 N L ,N R ⫹1 典 . The
energies of the two states with an additional electron in the
left 共right兲 dot are denoted as ␧ L (␧ R ). Higher excited states
for both dots are assumed to lie outside the energy window
␮ L Ⰷ␧ L ,␧ R Ⰷ ␮ R ,
共1兲
which defines the regime where nonlinear transport does not
significantly depend on ␮ L and ␮ R at low temperatures. This
corresponds to the standard situation10,14,16 where only the
two lowest hybridized electronic states in the double dot contribute to transport.
The Hamiltonian H D of the double dot is given in terms
of the dot operators n L ª 兩 L 典具 L 兩 , n R ª 兩 R 典具 R 兩 , and a term H T
describing the tunneling between the left and the right dot,
H D ⫽H A ⫹H T ⬅␧ L n L ⫹␧ R n R ⫹T c 共 P⫹ P † 兲 ,
P⫽ 兩 L 典具 R 兩 , P † ⫽ 兩 R 典具 L 兩 .
and the tunnel Hamiltonian
H V⫽
H⫽H 0 ⫹H T ⫹H V ⫹H ep ,
H ep ⫽ 共 ␣ n L ⫹ ␤ n R ⫹ ␥ * P⫹ ␥ P † 兲共 a⫹a † 兲 .
B. Equations of motion
The density matrix of the total system 共dot, boson, and
reservoirs兲 is given by the Liouville–von Neumann equation
and formulated in the interaction picture with respect to H 0 .
For the nonlinear transport window, Eq. 共1兲 considered here,
the chemical potentials in the equilibrium reservoirs are such
that the Fermi distributions in the left and the right leads can
well be approximated by f L (␧)⫽1 and f R (␧)⫽0. We treat
the reservoir coupling term H V in second-order perturbation
theory, neglecting Kondo physics41 throughout, so that transport can be described by a master equation.10,16,42,43 Transforming back into the Schrödinger picture will produce the
following master equation for the reduced density operator
of the system 共dot ⫹ boson兲,
d
␳ 共 t 兲 ⫽⫺i 关 H A ⫹H p ⫹H T ⫹H ep , ␳ 共 t 兲兴
dt
H res ⫽
兺k ␧ Lk c †k c k ⫹ 兺k ␧ Rk d †k d k ,
⌫L
关 s s † ␳ 共 t 兲 ⫺2s L† ␳ 共 t 兲 s L ⫹ ␳ 共 t 兲 s L s L† 兴
2 L L
⫺
⌫R †
关 s s ␳ 共 t 兲 ⫺2s R ␳ 共 t 兲 s R† ⫹ ␳ 共 t 兲 s R† s R 兴
2 R R
⫺
␥b
共 2a ␳ a † ⫺a † a ␳ ⫺ ␳ a † a 兲 .
2
共5兲
共6兲
共9兲
The perturbation from the reservoir is given by the tunnel
rates
共4兲
where the sum is over all wave vectors k in both the left 共L兲
and right 共R兲 reservoirs and spin polarization is assumed for
simplicity. The coupling between dot and reservoirs is described with two dot operators
s L ⫽ 兩 0 典具 L 兩 , s R ⫽ 兩 0 典具 R 兩 ;
⫺
共3兲
Here, we assume the coupling constants ␣ , ␤ , and ␥ as given
parameters. Their precise form can be calculated from microscopic details such as the many-body wave functions of the
dot electrons. It should be noted that the interaction between
the bosonic system and the reservoirs is not considered here.
The two electron reservoirs are described by
共8兲
where H 0 ⫽H A ⫹H p ⫹H res is the free Hamiltonian in the
interaction picture introduced below.
共2兲
with frequency ␻ and creation/annihilation operators a and
a † fulfilling 关 a,a † 兴 ⫽1. The coupling between the electronic
and the bosonic system is described by four microscopic
coupling constants,
共7兲
with tunnel matrix elements V Lk and V Rk .
The total Hamiltonian is written as
The tunnel matrix element T c is used to describe the strength
of the tunneling process.
The Hilbert space of the bosonic system is spanned by the
usual number or Fock eigenstates 兩 n 典 ,n⫽0,1,2,3, . . . of the
harmonic oscillator
H p⫽ ␻ a †a
兺k 共 V Lk c †k s L ⫹V Rk d †k s R ⫹c.c兲
⌫ L/R ⫽2 ␲
L/R
2
兺k 兩 V L/R
k 兩 ␦ 共 ␧ L/R ⫺␧ k 兲 .
共10兲
In the last line in Eq. 共9兲, we introduced a term that describes
damping of the bosonic system at a rate ␥ b 共Ref. 44兲 corresponding to photon or phonon cavity losses.
Taking matrix elements of the master equation, Eq. 共9兲,
one obtains a system of linear equations for the matrix elements of the density operator which are given explicitly in
Appendix A. In the stationary state of these equations, the
time derivatives are zero, and a coupled set of linear equations is found. In order to numerically solve these equations,
the bosonic Hilbert space has to be truncated at a finite number N of boson states. The total number of equations then is
5N 2 ⫹10N⫹5, remembering that there is always an equation for n⫽m⫽0. The numerical solution becomes a standard matrix inversion, though book-keeping of the matrix
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STEERING OF A BOSONIC MODE WITH A DOUBLE . . .
elements has to be done carefully. The trace over electron
and boson degrees of freedom has to obey the normalization
condition
兺n 关 具 n,0兩 ␳ stat兩 n,0典 ⫹ 具 n,R 兩 ␳ stat兩 n,R 典 ⫹ 具 n,L 兩 ␳ stat兩 n,L 典 兴 ⫽1.
共11兲
We mention that without this condition, the matrix becomes
singular, and cannot be solved.
III. TRANSPORT PROPERTIES
The first quantity of interest that one can obtain from the
matrix elements is the stationary electron current through the
double dot.
A. Stationary current:
numerical results
In the stationary state, the current operator is defined via
the rate of electrons tunneling from the left dot to the right
dot,
Î⫽
⳵
n ⫽iT C 共 P † ⫺ P 兲 ⫹i 共 ␥ P † ⫺ ␥ * P 兲共 a⫹a † 兲 . 共12兲
⳵t L
Here and in the following, we set the electron charge ⫺e
⫽1. For a boson mode with wave vector Q, the electronboson coupling constants can be expressed in terms of matrix
elements of the left and right dot states, ␣ ⫽␭ 具 L 兩 e iQr兩 L 典 ,
␤ ⫽␭ 具 R 兩 e iQr兩 R 典 , ␥ ⫽␭ 具 L 兩 e iQr兩 R 典 , where ␭ is the microscopic constant for the interaction of the boson with the electron in two dimensions.
Our formalism works for arbitrary choices of coupling
parameters ␣ , ␤ , ␥ , but for simplicity we restrict ourselves
to simple, nontrivial cases for the numerical calculations. We
set ␥ ⫽0, which corresponds to neglecting the nondiagonal
terms which for relatively sharp peaked electron densities in
the dots only weakly contribute.45 For a sharp electron density profile, one further has ␤ ⫽ ␣ e iQd, where d is the vector
connecting the left and right dot centers. Identical energy
shifts in both dots, corresponding to ␤ ⫽ ␣ , have no effect on
transport. In the following, we choose Qd⫽ ␲ and therefore
electron-boson coupling constants are
␣ ⫽⫺ ␤ ⬅2g, ␥ ⫽0.
as first derived by Stoof and Nazarov.10
For finite electron-boson coupling and positive ␧, resonances appear when
␧⬅␧ L ⫺␧ R ⫽n ␻ ,
共15兲
i.e., the energy gap of the dots becomes equal to multiple
integers of the boson energy. At these resonances, electrons
tunneling from the left to the right dot can excite the boson
system.
The current profiles with differing damping rates, Fig. 2,
illustrate that the system is extremely sensitive even to very
small ‘‘damping.’’ For small but finite boson damping, we
found that two profiles for N⫽5 and N⫽10 match in an
extremely close manner. This illustrates that a finite boson
damping removes numerical problems due to the truncation
of the boson Hilbert space at a finite N. As expected, there
are no peaks at all on the absorption side of the profile (␧
⬍0): the damped boson relaxes to its ground state.
共13兲
Furthermore, the tunneling rates between the reservoirs and
the dots are generally kept smaller than the energies of the
dots and boson states. This produces clear and sharply defined transport characteristics. We use the boson frequency as
an energy scale and set ␻ ⫽1 in the following. Using the
typical phonon cavity energy ប ␻ ⬇100 ␮ eV from a recent
experiment with a free-standing phonon cavity20 will yield a
typical electron current scale of e ␻ ⫽24 nA, as shown in
Figs. 1 and 2.
Figure 1 illustrates the stationary current as a function of
the energy difference ␧. For zero boson coupling g⫽0, we
reproduce the Lorentzian resonant tunneling profile
I stat ⫽T 2c ⌫ R / 关 T 2c 共 2⫹⌫ R /⌫ L 兲 ⫹⌫ R2 /4⫹␧ 2 兴
FIG. 1. Stationary current through double quantum dot coupled
to single boson mode with ⌫ L ⫽⌫ R ⫽0.1, T c ⫽0.01, ␥ b ⫽0.05, and
varying 2g⫽ ␣ ⫽⫺ ␤ . 关Electron-boson coupling, cf. Eq. 共4兲兴. N:
number of boson states.
共14兲
FIG. 2. Current as in Fig. 1 for coupling 2g⫽0.1 and different
boson damping ␥ b , alongside corresponding polaron transformation result, Eq. 共19兲.
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PHYSICAL REVIEW B 67, 125323 共2003兲
T. BRANDES AND N. LAMBERT
B. Perturbation theory in T c : Polaron transformation
method
An analytical expression for the stationary current can be
obtained from a polaron transformation and a polarontransformed master equation that is valid for small coupling
T c between the two dots.16 After tracing out the bosonic
degrees of freedom, one obtains an equation of motion for
the time-dependent expectation values of the operators n L/R
and P (†) which can be transformed into the Laplace space,
具 n L 典 共 z 兲 ⫽⫺i
Tc
⌫L
兵 具 P 典 共 z 兲 ⫺ 具 P † 典 共 z 兲 其 ⫹ 兵 1/z⫺ 具 n L 典 共 z 兲
z
z
should at least determine ␳ B self-consistently. For small couplings g, however, the assumption of a decoupled time evolution of the boson should work well, and we proceed with a
master equation for ␳ B ,
␥b
d
␳ B共 t 兲 ⫽⫺i 关 ␻ a † a, ␳ B兴 ⫺ 共 2a ␳ Ba † ⫺a † a ␳ B⫺ ␳ Ba † a 兲
dt
2
共20兲
that describes the free time evolution of the boson in presence of boson damping with damping constant ␥ b ⬎0. Assuming the boson initial condition to be its ground state, one
obtains (t⬎0)
⫺ 具 n R典共 z 兲其 ,
C 共 t 兲 ⫽exp兵 ⫺ 兩 ␰ 兩 2 共 1⫺e ⫺[( ␥ b /2)⫹i ␻ ]t 兲 其 , ␰ ⫽
Tc
⌫R
具 n R 典 共 z 兲 ⫽i 兵 具 P 典 共 z 兲 ⫺ 具 P † 典 共 z 兲 其 ⫺ 具 n R 典 共 z 兲 ,
z
z
* 共 z 兲其
具 P 典 共 z 兲 ⫽⫺iT c 兵 具 n L 典 共 z 兲 C ␧ 共 z 兲 ⫺ 具 n R 典 共 z 兲 C ⫺␧
⫺
⌫R
具 P 典 共 z 兲 C ␧共 z 兲 .
2
共16兲
Here, the coupling to the boson system enters through the
correlation function
C ␧共 z 兲 ⫽
冕
⬁
0
dte ⫺zt⫹i␧t C 共 t 兲 ,
4g
.
␻
共21兲
Figure 2 shows that the analytical expression for the current,
Eq. 共19兲, compares quite well with the numerical results. In
particular, this demonstrates that the formulation in terms of
a general boson correlation function C(t) works reasonably
well not only for boson systems with continous mode
spectrum,16,45 but also for the single mode case discussed
here, at least for small-coupling constants g. We point out
that the existence of a finite boson damping is crucial here
and that for larger g, the comparison becomes worse, as expected.
IV. BOSON DISTRIBUTION
冉 冊
4g
C 共 t 兲 ⫽ 具 X 共 t 兲 X † 共 t⫽0 兲 典 , X⫽D
␻
共17兲
of the displacement operator of the bosonic mode,
D 共 ␰ 兲 ⫽exp共 ␰ a † ⫺ ␰ * a 兲 .
共18兲
Turning now to properties of the bosonic system, our primary interest is in how the boson mode can be controlled via
parameters of the electron subsystem such as the bias ␧ and
the tunnel couplings ⌫ R/L . In the stationary state, the reduced density operator of the boson is
Equations 共16兲 can be solved algebraically. One then obtains
the expectation value 具 I 典 t→⬁ of the stationary current from
the 1/z coefficient of the I(z) expansion into a Laurent
series46 for z→0,
␳ b ⬅ lim Trdot␳ 共 t 兲 ⫽ lim
t→⬁
兺
t→⬁ i⫽0,L,R
␳ ii 共 t 兲 .
共22兲
A. Boson state detection
2Re共 C ␧ 兲 ⫹⌫ R 兩 C ␧ 兩 2
2
I
⫽T
,
具 典 t→⬁ c
2
⌫R
2
C ⫹2T c B ␧
1⫹
2 ␧
B ␧ :⫽Re
再冉
冏
1⫹
⌫R
C
2 ␧
冊冋
冏
冉
C ⫺␧ C *
⌫L
␧
⫹
1⫹
⌫R
⌫L
⌫R
冊 册冎
,
共19兲
where C ␧ ⬅C ␧ (z→0).
The correlation function C(t) enters via a factorization
assumption in the polaron-transformed master equation and
has to be calculated from an effective density operator ␳ B of
the bosonic mode. For dissipative pseudospin-boson systems
with a continuous spectrum of modes and a thermal equilibrium for ␳ B , reasonable results can be obtained within this
method for small coupling, in particular, in comparison with
perturbation theory.45,47 However, in the present case with
just one single mode it is not clear whether or not a factorization into an effective dot and an effective boson density
operator yields reasonable results, and in principle, one
Before discussing the boson state ␳ b , we describe a possible experimental scheme to directly detect its properties.
The basic idea is to use another nearby double quantum dot
as detector of the boson, similar to the scheme for detecting
quantum noise in mesoscopic conductors as proposed by
Aguado and Kouwenhoven.48
The detector consists of a double quantum dot very similar to the one discussed above. The boson state gives a contribution to the inelastic, stationary current
I d 共 E 兲 ⬇T 2d P 共 E 兲 ⫽2T 2d Re
冕
⬁
0
dte iEt C d 共 t 兲 ;
C d 共 t 兲 ⫽Tr关 ␳ b X d 共 t 兲 X †d 兴 , X d ⫽D
冉 冊
4g d
␻
共23兲
through the detector in the lowest order in the detector tunnel
coupling T d . Here, E denotes the detector dot energy difference, D again denotes the unitary displacement operator, Eq.
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共18兲, g d the coupling constant in the detector double dot, and
␳ b the boson state. Equation 共23兲 describes the detector current I(E) resulting from a given boson state ␳ b , as calculated in the following section.
Before discussing our numerical results for ␳ b , it is illustrative to look at a few special cases. For example, if the
boson state ␳ b describes a thermal equilibrium at temperature
T, one obtains the inelastic current spectrum with absorption
E⬍0 and emission E⬎0 branches16, as has been observed
by Fujisawa et al.14 for the case of equilibrium 共multimode兲
phonons.
If ␳ b ⫽ 兩 n 典具 n 兩 describes a pure, n-boson number state, the
function C d (t) in the detector current, Eq. 共23兲, becomes
( ␰ ⫽4g d / ␻ )
C d共 t 兲 ⫽ 具 n 兩 D 共 ␰ e i␻t 兲 D 共 ⫺ ␰ 兲兩 n 典
⫽e ⫺i 兩 ␰ 兩
2 sin ␻ t ⫺(1/2) 兩 ␰ (1⫺e i ␻ t ) 兩 2
e
FIG. 3. Boson profiles n⫽0,2,4, . . . ,10 for ␥ b ⫽0.0001, N
⫽10, 2g⫽ ␣ ⫽⫺ ␤ ⫽0.3, 10T C ⫽⌫ L ⫽⌫ R ⫽0.1.
L 0n 共 兩 ␰ 共 1⫺e i ␻ t 兲 兩 2 兲 ,
共24兲
where L 0n is a Laguerre polynomial, cf. Appendix B. Here we
assume that the boson time evolution is undamped in the
detector, i.e., governed by the boson Hamiltonian H b
⫽ ␻ a † a. Expanding up to the lowest order in ␰ ⫽4g d / ␻ and
using L 0n (x)⫽1⫺nx⫹O(x 2 ), one obtains from Eqs. 共23兲
and 共24兲,
I d 共 E 兲 ⬇T 2d 2 ␲ 关 兵 1⫺ 兩 ␰ 兩 2 共 2n⫹1 兲 其 ␦ 共 E 兲 ⫹ 兩 ␰ 兩 2 兵 n ␦ 共 E⫹ ␻ 兲
⫹ 共 n⫹1 兲 ␦ 共 E⫺ ␻ 兲 其 兴 .
共25兲
Clearly, the height of the inelastic current peaks is determined by the quantum number n, i.e., the absorption peak at
E⫽⫺ ␻ ⬍0 scales with n, and the 共stimulated兲 emission
peak at E⫽ ␻ ⬎0 scales with n⫹1.
The above example can be even generalized. In Appendix
C we show how to reconstruct an arbitrary boson state ␳ b
⫽ 兺 nm ␳ nm 兩 n 典具 m 兩 from the function P(E) 关or equivalently
the stationary detector current spectrum I d (E), Eq. 共23兲兴.
This demonstrates that the properties of the boson state discussed in the following can be directly related to an experimentally accessible quantity.
B. Numerical results
We first analyze the boson mode via moments of its distribution function such as occupation number and Fano factor before discussing the full distribution function in the
Wigner function representation.
The boson occupation probability p n ⫽( ␳ b ) nn is the probability for the boson to be in one particular number state n.
Furthermore, the boson Fano factor F Q is calculated from the
boson occupancy as the variance of boson number over the
mean number of bosons,
F Q⫽
具 n̂ 2 典 ⫺ 具 n̂ 典 2
具 n̂ 典
.
x⫽
冑2
, p⫽
i 共 ⫺a⫹a † 兲
冑2
共27兲
can be obtained from ␳ b , cf. Eq. 共A7兲 in Appendix A. They
provide an indication of the quantum fluctuations in the boson states, and prove useful in comparison with the Fano
factor profile. It should also be apparent that ⌬x 2 ⌬ p 2 ⭓ 41 .
We have checked that the probability p n fulfills the normalization 兺 n p n ⫽1. For accurate results, the occupation
probability should tend to 0 as n→N, where N is the dimension of the truncated boson Hilbert space. This condition is
easily achieved in presence of a finite boson damping rate,
␥ b ⬎0. In this case, it is possible to obtain a boson distribution which is centered around the lower boson number states,
and is excited when a resonant interaction with the electron
occurs.
It is more illustrative to first discuss a weakly damped
case, as shown in Fig. 3. We can see quite clearly how at the
positive resonance energies, the occupation probabilities of
the states spread into the higher number states.
The Fano factor F Q illustrates fluctuation properties of the
boson mode with F Q ⫽1 corresponding to a Poissonian boson number distribution and F Q ⬍1 (F Q ⬎1) to a sub
共super兲-Poissonian distribution. Figure 4 illustrates the Fano
factor vs ␧ for the strongly damped case.
This profile suggests the reduced boson states are coherent 共Poissonian兲 for ␧⬍0, and super-Poissonian at the resonance energies, ␧⫽ ␻ ,2␻ , . . . .
C. Coherent state and x-p variances
For ␧Ⰶ0, nearly no current is flowing, the electron is
predominatly localized in the left dot, and one can approximate the operator ␴ z ⫽ 兩 L 典具 L 兩 ⫺ 兩 R 典具 R 兩 by its expectation
value 具 ␴ z 典 ⫽1. Then, the boson system is effectively described by
共26兲
Similarly, the variance ⌬x 2 ⫽ 具 x 2 典 ⫺ 具 x 典 2 and ⌬p 2 ⫽ 具 p 2 典
⫺ 具 p 典 2 of the position and a momentum coordinate
共 a⫹a † 兲
H eff⫽2g 共 a⫹a † 兲 ⫹ ␻ a † a,
共28兲
which is a shifted oscillator, the ground state 兩 GS 典 of which
is the coherent state 兩 GS 典 ⫽ 兩 ⫺2g/ ␻ 典 with a 兩 z 典 ⫽z 兩 z 典 . This
125323-5
PHYSICAL REVIEW B 67, 125323 共2003兲
T. BRANDES AND N. LAMBERT
D. Wigner function
The Wigner function of the density operator ␳ b for a
bosonic mode a † is a representation of ␳ b in x-p space, cf.
Eq. 共27兲. It is defined as49
1
x⫹i p
,
W 共 x,p 兲 ª Tr共 ␳ b D 共 2 ␣ 兲 U 0 兲 , ␣ ⫽
␲
冑2
FIG. 4. Fano vs ␧ profile for 2g⫽ ␣ ⫽⫺ ␤ ⫽0.1, N⫽10, ␥ b
⫽0.05, 10T C ⫽⌫ L ⫽⌫ R ⫽0.1. Inset: Comparison between coherent
state, Eq. 共29兲, and numerical result for boson distribution p n at ␧
⫽⫺4.
can be seen by introducing new operators bªa⫹2g/ ␻
whence H eff⫽ ␻ b † b⫺4g 2 / ␻ and b 兩 GS 典 ⫽0. In this regime,
we have
␳ b ⬇ 兩 z 典具 z 兩 , z⫽⫺2g/ ␻ ,
where D( ␣ )ªexp关␣a†⫺␣*a兴, Eq. 共18兲, is a unitary displacement operator and U 0 ªexp关i␲a†a兴 is a parity operator
for the boson.50 W(x,p) is a symmetric Gaussian for a pure
coherent boson state and a symmetric Gaussian multiplied
with a polynomial for a pure number state.44 In our model,
the shape of W(x,p) therefore indicates how close to these
limiting cases the actual stationary state ␳ b of the boson, Eq.
共22兲, is. In particular, this is a convenient way to represent
the ‘‘steering’’ of the boson mode when external parameters
共such as ␧ or ⌫ R/L ) are changed.51
Using the Fock state basis 兵 兩 n 典 ,n⫽0,1,2, . . . 其 and U 0
⬁
⫽ 兺 n⫽0
(⫺1) n 兩 n 典具 n 兩 , we find
共29兲
and p n ⫽ 兩 具 n 兩 GS 典 兩 2 is given by a Poisson distribution, p n
2
⫽ 兩 z 兩 2n e ⫺ 兩 z 兩 /n!. One can plot this against the numerically
obtained p n and check that the distribution is indeed Poissonian for small ␧Ⰶ0. This is represented in the inset of Fig. 4
in detail for the first four-boson Fock 共number兲 states.
The quantum fluctuations of the boson system are described by the variance of the position and momentum operators. Figure 5 presents a direct comparison of the position
and momentum variance of a strongly damped system. The
uncertainty principle holds, ⌬x⌬p⭓ 21 , and thus ⌬x 2 ⌬p 2
⭓ 14 . The state with minimum uncertainty occurs as expected, for ␧ L ⬍␧ R , with ⌬x 2 ⌬ p 2 ⫽ 41 . This is a reassuring
result, since it suggests both that the boson is coherent in this
area, and that the Fano factor’s Poissonian distribution for
this coherent state is correct. We mention that since both
⌬x⬎0.5 and ⌬p⬎0.5, we find no squeezing of the boson
mode for the parameters checked here.
1
W 共 x,p 兲 ⫽
␲
⬁
兺
n,m⫽0
共 ⫺1 兲 n 具 n 兩 ␳ b 兩 m 典具 m 兩 D 共 2 ␣ 兲 兩 n 典 .
共31兲
It is useful to split the sum into diagonal and nondiagonal
parts, to use 具 m 兩 ␳ b 兩 n 典 ⫽ 具 n 兩 ␳ b 兩 m 典 * ; and (m⭓n)
具 m 兩 D共 ␣ 兲兩 n 典 ⫽
冑
n! m⫺n ⫺(1/2) 兩 ␣ 兩 2 m⫺n
␣
e
Ln 共兩␣兩2兲
m!
⫽ 具 m 兩 D † 共 ␣ 兲 兩 n 典 * ⫽ 共 ⫺1 兲 m⫺n 具 m 兩 D 共 ␣ 兲 兩 n 典 * ,
共32兲
where again ␣ ⫽(x⫹ip)/ 冑2 and L m⫺n
is a Laguerre polyn
nomial, cf. Appendix B. This leads to
1
W 共 x,p 兲 ⫽
␲
⬁
兺
n⫽0
1
⫹
␲
共 ⫺1 兲 n 具 n 兩 ␳ b 兩 n 典具 n 兩 D 共 2 ␣ 兲 兩 n 典
⬁
⬁
兺 兺
n⫽0 m⫽n⫹1
共 ⫺1 兲 n 2 Re关 具 n 兩 ␳ b 兩 m 典
⫻具 m 兩 D 共 2 ␣ 兲 兩 n 典 兴 .
FIG. 5. Position variance, momentum variance, and an estimation of the combined uncertainty. 2g⫽ ␣ ⫽⫺ ␤ ⫽0.1, N⫽10, ␥ b
⫽0.05, 10T C ⫽⌫ L ⫽⌫ R ⫽0.1.
共30兲
共33兲
Using our numerical results for the stationary state ␳ of
the coupled system, we easily obtain Wigner functions from
Eqs. 共22兲 and 共33兲 for different values of the energy difference ␧ between the two dots as shown in Fig. 6. Between
two resonance energies ␧⫽n ␻ , W(x,p) closely resembles
a Gaussian. At and close to the resonance energies, the
distribution spreads out in rings around the origin, which is
consistent with the increased Fock state occupation numbers
and the increased position and momentum variances at these
energies.
125323-6
PHYSICAL REVIEW B 67, 125323 共2003兲
STEERING OF A BOSONIC MODE WITH A DOUBLE . . .
FIG. 6. Wigner distribution functions for the
bosonic mode. Parameters are ⌫ L ⫽⌫ R ⫽T c
⫽0.1, ␥ b ⫽0.005, g⫽0.2, N⫽20. Stationary
current resonances occur at ␧⫽0.0,1.0,2.0, . . . .
V. DISCUSSION
Our investigation is based on a numerical method which
allows the density matrix of the boson-dot system to be
solved for arbitrary dot parameters (␧, T c , ⌫ L/R ) and boson
damping rate ␥ b . Since we have to truncate the boson Hilbert space, the electron-boson coupling constant g has to be
restricted to small values gⱗ1. In contrast with the determination of the 共pure兲 eigenstates of an isolated dot-boson system 共Rabi Hamiltonian兲, the numerical effort for our mixed
state 共density operator兲 is much bigger here. Although not
discussed in this paper, we suggest that the strong-coupling
regime could be reached numerically by a polaron transformation of the master equation16 without the factorization assumption employed in Sec. III B. Fortunately, at present the
small-coupling regime seems to be valid for experimental
situations with quantum dots.13–16
Our results suggest that there is no resonant interaction on
the ␧⬍0 side of the current peak as long as the boson system
is damped and any many-body excited electron states can be
ignored. For ␧⬎0, we have found strong excitations of the
boson mode occurring at resonances given by multiples n ␻
of the boson frequency ␻ . These correspond to the photosatellite peaks in resonant tunneling in electromagnetic ac
fields9,10 which in the classical case, however, appear on both
sides of the main resonance n⫽0 in contrast with the quantum case considered here. The resonances at ␧⫽n ␻ can thus
be interpreted as the emission of phonons 共photons兲 by the
electron as it tunnels through the dot. It should be possible to
detect them in the stationary current through double quantum
dots in phonon resonators. With increasing electron-boson
coupling, the visibility of these side peaks increases whereas
the main resonance at ␧⫽0 is reduced.
We have discussed that an additional double dot can serve
as a detector of the stationary boson state ␳ b , if its energy
difference E is tuned independently. In the single mode case
considered here, the stationary detector current spectrum
I d (E) is uniquely related to the components of ␳ b in the
number state basis. In order to characterize the stationary
boson state itself, we have investigated the Fano factor and
the quantum fluctuations in the quadrature amplitudes of the
boson. For ␧Ⰶ0, the boson state is perfectly described as a
coherent state whereas for ␧ⲏ0, the interaction with the
quantum dot produces occupied excited states in the boson
mode that could best be visualized by a broadening of the
phase-space 共Wigner兲 distribution function of the boson.
Although in this paper we have only calculated stationary
properties, our method can be extended to calculate the noise
spectrum via the quantum regression theorem within the
master-equation framework as well.44 It remains a task for
the future to analyze the relation between the boson quantum
fluctuations and the current noise in detail.
ACKNOWLEDGMENTS
We acknowledge inspiring discussions with A. Vourdas
共who suggested the analysis of the x and p variances兲 and C.
Emary. This work was supported by Projects Nos. EPSRC
GR44690/01 and DFG Br1528/4-1, the WE Heraeus Foundation, and the UK Quantum Circuits Network.
APPENDIX A:
MATRIX ELEMENTS OF THE DENSITY
OPERATOR
The matrix elements of the dot-boson density operator ␳
are defined as
i
␳ nm
ª 具 n,i 兩 ␳ 兩 i,m 典 , i⫽0,L,R;
ij
␳ nm
ª 具 n,i 兩 ␳ 兩 j,m 典 , i, j⫽0,L,R,
共A1兲
where n and m refer to boson Fock states.
The matrix elements for the empty dot state obey
d 0
0
0
R
␳ ⫽ 关 i ␻ 共 m⫺n 兲 ␳ nm
兴 ⫺⌫ L 关 ␳ nm
兴 ⫹⌫ R 关 ␳ nm
兴
dt nm
⫹
␥b
0
0
冑n⫹1 冑m⫹1⫺ ␳ nm
关 2 ␳ n⫹1,m⫹1
共 n⫹m 兲兴 ,
2
共A2兲
those for the right dot state
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PHYSICAL REVIEW B 67, 125323 共2003兲
T. BRANDES AND N. LAMBERT
d R
R
RL
LR
R
␳ ⫽i 关 ␻ 共 m⫺n 兲 ␳ nm
⫹T c 共 ␳ nm
⫺ ␳ nm
兲 ⫹ ␤ 共 冑m ␳ nm⫺1
dt nm
1
⌬x 2 ⫽ Trdot
2
R
R
R
⫹ 冑m⫹1 ␳ nm⫹1
⫺ 冑n ␳ n⫺1m
⫺ 冑n⫹1 ␳ n⫹1m
兲
⫹ 冑n⫹1 冑n⫹2 ␳ n,n⫹2 兴 ⫺
RL
RL
LR
⫹ ␥ * 共 冑m ␳ nm⫺1
⫹ 冑m⫹1 ␳ nm⫹1
兲 ⫺ ␥ 共 冑n ␳ n⫺1m
册
LR
R
⫹ 冑n⫹1 ␳ n⫹1m
兲兴 ⫺⌫ R 关 ␳ nm
兴
⫹
兺n 关 冑n 冑n⫺1 ␳ n,n⫺2 ⫹ 共 2n⫹1 兲 ␳ n,n
冋
1
Trdot
2
兺n 共 冑n ␳ n,n⫺1
2
⫹ 冑n⫹1 ␳ n,n⫹1 兲 ,
␥b
R
R
冑n⫹1 冑m⫹1⫺ ␳ nm
关 2 ␳ n⫹1,m⫹1
共 n⫹m 兲兴 ,
2
共A3兲
1
⌬ p 2 ⫽ Trdot
2
共A7兲
兺n 关 ⫺ 冑n 冑n⫺1 ␳ n,n⫺2 ⫹ 共 2n⫹1 兲 ␳ n,n
and for the left dot state
⫺ 冑n⫹1 冑n⫹2 ␳ n,n⫹2 兴 ⫺
d L
L
RL
LR
␳ ⫽i 关 ␻ 共 m⫺n 兲 ␳ nm
⫹T c 共 ⫺ ␳ nm
⫹ ␳ nm
兲
dt nm
⫹i 冑n⫹1 ␳ n,n⫹1 兲 .
册
冋
1
Trdot
2
兺n 共 ⫺i 冑n ␳ n,n⫺1
2
共A8兲
L
L
L
⫹ ␣ 共 ⫺ 冑n⫹1 ␳ n⫹1m
⫺ 冑n ␳ n⫺1m
⫹ 冑m ␳ nm⫺1
APPENDIX B: DISPLACEMENT OPERATOR,
COHERENT STATES
L
R
RL
⫹ 冑m⫹1 ␳ nm⫹1
⫺ 冑n ␳ n⫺1m
兲 ⫹ ␥ * 共 ⫺ 冑n⫹1 ␳ n⫹1m
兲
LR
LR
0
⫹ ␥ 共 冑m ␳ nm⫺1
⫹ 冑m⫹1 ␳ nm⫹1
兲兴 ⫹⌫ L 关 ␳ nm
兴
⫹
Here, we summarize some useful properties of the unitary
displacement operator
␥b
L
L
冑n⫹1 冑m⫹1⫺ ␳ nm
关 2 ␳ n⫹1,m⫹1
共 n⫹m 兲兴 .
2
D 共 z 兲 ⬅e za
共A4兲
RL
is
The equation of motion for the off-diagonal elements ␳ nm
d RL
RL
RL
␳ ⫽i 关 ␳ nm
关 ␻ 共 m⫺n 兲 ⫹ 共 ␧ L ⫺␧ R 兲兴 ⫹ ␤ 共 ⫺ 冑n⫹1 ␳ n⫹1m
dt nm
RL
RL
RL
⫺ 冑n ␳ n⫺1m
⫹ 冑m ␳ nm⫺1
兲 ⫹ ␣ 共 冑m⫹1 ␳ nm⫹1
兲
⫽„D † 共 z 兲 …⫺1 ⫽D † 共 ⫺z 兲
2
†
2
†
⫽e ⫺(1/2) 兩 z 兩 e za e ⫺z * a ⫽e (1/2) 兩 z 兩 e ⫺z * a e za ,
兩 z 典 ⫽D 共 z 兲 兩 0 典 .
⌫ R RL
关 ␳ nm 兴
2
共B2兲
Extremely useful is the relation
D 共 ␣ ⫹ ␤ 兲 ⫽D 共 ␣ 兲 D 共 ␤ 兲 e ⫺i Im( ␣␤ * )
␥b
RL
RL
冑n⫹1 冑m⫹1⫺ ␳ nm
⫹ 关 2 ␳ n⫹1,m⫹1
共 n⫹m 兲兴 .
2
共A5兲
Taking the trace over all the boson and electron states, Eq.
共12兲 for the electron current operator reads
具 ␤ 兩 D 共 ␣ 兲 兩 ␤ 典 ⫽e ⫺(1/2) 兩 ␣ 兩 e 2i Im( ␣␤ * ) .
具 m 兩 D 共 ␣ 兲 兩 n 典 ⫽e (1/2) 兩 ␣ 兩
n
冑
RL
n⫹1 兲 ⫺ ␥ * 共 ␳ n,n⫺1
冑
RL
n⫹ ␳ n,n⫹1
共B4兲
Number state matrix elements can be obtained using 兩 n 典
⬅(1/冑n!)(a † ) n 兩 0 典 ,
LR
RL
LR
冑n
⫺ ␳ n,n
兴 ⫹ 兺 i 关 ␥ 共 ␳ n,n⫺1
具 Î 典 ⫽ 兺 iT C 关 ␳ n,n
LR
⫹ ␳ n,n⫹1
共B3兲
for arbitrary complex numbers ␣ , ␤ . Coherent-state matrix
elements of D( ␣ ) follow as
2
n
共B1兲
where z is a complex number and we used the operator exfor
†关 A,B 兴 ,A‡
ponential
e A⫹B ⫽e A e B e ⫺(1/2)[A,B]
⫽†关 A,B 兴 ,B‡⫽0, cf. also Ref. 44. A coherent boson state 兩 z 典
is defined as an eigenstate of the annihilation operator, a 兩 z 典
⫽z 兩 z 典 , where z is a complex number. It can be generated
from the boson vacuum 兩 0 典 as
LL
LL
RR
⫹ ␥ 共 ⫺ 冑n⫹1 ␳ n⫹1m
⫺ 冑n ␳ n⫺1m
⫹ 冑m⫹1 ␳ nm⫹1
RR
RR
LL
⫹ 冑m ␳ nm⫺1
⫺ ␳ nm
兲 ⫹T c 共 ␳ nm
兲兴 ⫺
† ⫺z a
*
冑n⫹1 兲兴 .
2
1
⳵ n⫹m
冑n!m! ⳵ z m1 ⳵ z n2
†
⫻ 具 0 兩 e (z 1 ⫺ ␣ * )a e (z 2 ⫹ ␣ )a 兩 0 典 兩 z 1 ⫽z 2 ⫽0 .
共A6兲
共B5兲
The variances of the boson position and momentum coordinate, Eq. 共27兲, are obtained by performing the trace over the
dot variables 0,L,R,
†
With 具 0 兩 e (z 1 ⫺ ␣ * )a e (z 2 ⫹ ␣ )a 兩 0 典 ⫽e (z 1 ⫺ ␣ * )(z 2 ⫹ ␣ ) for m⭓n
the differentiation yields
125323-8
PHYSICAL REVIEW B 67, 125323 共2003兲
STEERING OF A BOSONIC MODE WITH A DOUBLE . . .
具 m 兩 D 共 ␣ 兲 兩 n 典 ⫽e (1/2) 兩 ␣ 兩
2
1
⳵n
共z2
n
冑n!m! ⳵ z 2
⫹ ␣ 兲 m e ⫺(z 2 ⫹ ␣ ) ␣ * 兩 z 2 ⫽0 .
共B6兲
The functions c nm (t) can be calculated analytically using Eq.
共B4兲 and the matrix elements 具 m 兩 D(z) 兩 n 典 of the unitary displacement operator, Eq. 共32兲. They are periodic in 2 ␲ / ␻ ,
and from integrating Eq. 共C1兲 over one period, one obtains a
k
,
linear relation between the Fourier coefficients P̃ k and c nm
Comparison with the generating function of the Laguerre
polynomials51 yields Eq. 共32兲. A corresponding expression
can be derived for n⭓m.
P̃ k ⫽
APPENDIX C: RELATION BETWEEN P„E… AND
STATIONARY BOSON STATE ␳ b
P̃ k ⬅
Again we assume that the boson time evolution is undamped in the detector, i.e., governed by the boson Hamiltonian H b ⫽ ␻ a † a. Expressing an arbitrary boson state ␳ b in
the number state basis, ␳ b ⫽ 兺 nm ␳ nm 兩 n 典具 m 兩 , and Fourier
transforming the function P(E), Eq. 共23兲, yields ( ␰
⫽4g d / ␻ )
P̃ 共 t 兲 ⬅
⫽
1
2␲
冕
⬁
⫺⬁
dEe ⫺iEt P 共 E 兲 ⫽C d 共 t 兲 ⫹C d* 共 ⫺t 兲
␳ nm c nm 共 t 兲 ,
兺
nm
c nm 共 t 兲 ⬅ 具 m 兩 D 共 ␰ e i ␻ t 兲 D 共 ⫺ ␰ 兲 兩 n 典
⫹ 具 m 兩 D 共 ␰ 兲 D 共 ⫺ ␰ e ⫺i ␻ t 兲 兩 n 典 .
共C1兲
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k
⬅
c nm
␻
2␲
␻
2␲
k
c nm
␳ nm ,
兺
nm
冕
2␲/␻
0
冕
2␲/␻
0
P̃ 共 t 兲 e ik ␻ t dt,
c nm 共 t 兲 e ik ␻ t dt.
共C2兲
The P̃ k have to be determined by numerical integration from
the 共experimentally given兲 P(E). Regarding (nm) as a
single index, Eq. 共C2兲 is a linear equation that can then be
solved for the coefficients ␳ nm of the boson state by inverting
k
, the coefficients of which are given as Fouthe matrix c nm
rier coefficients of known expressions. In practical terms, the
number of boson states as well as the number of Fourier
coefficients have to be restricted in order to make this inversion feasible.
Equation 共C2兲 establishes the relation between P(E) 关or,
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