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Quantum transport in a two-level quantum dot driven by coherent and stochastic fields
Author’s Accepted Manuscript Quantum transport in a two-level quantum dot driven by coherent and stochastic fields Sha-Sha Ke, Ling-E Miao, Zhen Guo, Yong Guo, Huai-Wu Zhang, Hai-Feng Lü www.elsevier.com/locate/ssc
PII: DOI: Reference:
S0038-1098(16)30257-5 http://dx.doi.org/10.1016/j.ssc.2016.09.017 SSC13045
To appear in: Solid State Communications Received date: 14 June 2016 Revised date: 1 September 2016 Accepted date: 21 September 2016 Cite this article as: Sha-Sha Ke, Ling-E Miao, Zhen Guo, Yong Guo, Huai-Wu Zhang and Hai-Feng Lü, Quantum transport in a two-level quantum dot driven by coherent and stochastic fields, Solid State Communications, http://dx.doi.org/10.1016/j.ssc.2016.09.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Quantum transport in a two-level quantum dot driven by coherent and stochastic fields Sha-Sha Ke,1 Ling-E Miao,1 Zhen Guo,1 Yong Guo,2 Huai-Wu Zhang,1 and Hai-Feng L¨ u1 1
State Key Laboratory of Electronic Thin Films and Integrated Devices and School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China 2 Department of Physics, Tsinghua University, Beijing 100084, China We study theoretically the current and shot noise properties flowing through a two-level quantum dot driven by a strong coherent field and a weak stochastic field. The interaction x(t) between the quantum dot and the stochastic field is assumed to be a Gaussian-Markovian random process with zero mean value and correlation function x(t)x(t) = Dκe−κ|t−t | , where D and κ are the strength and bandwidth of the stochastic field, respectively. It is found that the stochastic field could enhance the resonant effect between the quantum dot and the coherent field, and generate new resonant points. At the resonant points, the state population difference between two levels is suppressed and the current is considerably enhanced. The zero-frequency shot noise of the current varies dramatically between sub- and super-Poissonian characteristics by tuning the stochastic field appropriately. PACS numbers: 72.25.Dc, 73.63.Nm, 72.70.+m
I.
INTRODUCTION
There have been extensive investigations focused on the effect of time-varying external fields on quantum transport in mesoscopic systems.1–5 All-optically controlled single electron transport in quantum dots are particularly attractive.6–13 The interaction between the electronic transport and the light-matter interaction leads to a rich phenomenology. For instance, in twolevel quantum dot10,11 and Λ-type double-dot12 systems driven by an optical field, it has been shown that the zero-frequency shot noise can be tuned between suband super-Poissonian distributions due to the dynamical channel blockade induced by the optical field. One of the most influential recent advances in the field of superconducting quantum circuits is the ability to coherently couple microwave photons in cavity to mesoscopic electronic conductors.14–28 The interesting resonance phenomena have been observed, as well as a modification of the cavity behavior due to finite bias transport in a quantum dot device.14–18 Very recently, the photon mediated interaction between distant quantum dot circuits has been demonstrated by using the cavity quantum electrodynamics architectures, where the two dots are separated by 80μm.19 Theoretically, it has been analyzed how the coupling to a common photon mode generates entanglement between distant charge qubits realized in double quantum dots and how this entanglement manifests in the transport properties of the system.22–24 The study of the photon-assisted tunneling through a quantum conductor is also extended to the case of nonclassical light, such as squeezed states and Fock states.28 It is found that the transport property of the conductor could serve as a nontrivial probe of the nonclassical microwave states. In the field of quantum optics, it has been a long standing issue to investigate the effect of laser noise on the resonance fluorescence and absorption spectra in the
atom-light interaction.29–37 Over the last two decades, it was revealed that laser noise can induce the relatively large intensity fluctuations of the resonance fluorescence of a macroscopic sample of atoms.29,30 The stochastic field was also proposed to achieve phase-dependent spectra in resonance fluorescence in a two-level atom to avoid preparing a squeezed vacuum.34 Another typical effect induced by a stochastic perturbation in atom systems is that collisional noise can unexpectedly create phase correlation between the neighbouring atomic dressed states.35 This phase correlation is responsible for quantum interference between dressed state transition channels, resulting in anomalous modifications of the resonance fluorescence spectra. When an external light field is applied to a quantum circuit, it is important to take account of the amplitude fluctuations of the interaction between the coherent field and quantum conductor. However, it remains unknown how the input of a stochastic field affects quantum transport in a mesoscopic conductor. In this paper we investigate the quantum transport through a two-level quantum dot driven by a strong coherent field and a weak stochastic field with a wide bandwidth. By adopting the Born-Markov and rotating-wave approximations, we eliminate the stochastic variable x(t) to obtain a master equation for the reduced density matrix of the quantum dot. It is shown that the stochastic field effectively forms a reservoir. We have a reservoir with tunable amplitude and frequency and it strongly affects such physical properties as the state population, the tunneling current, and the shot noise. The paper is organized as follows. In Sec. II we introduce the model Hamiltonian of the system. An effective master equation for the reduced density operator of the quantum dot is derived when the coherent driving field is much stronger than the stochastic part. In Sec. III we discuss the effect of the stochastic field on the transport property of the device. We find that the presence of the additionally
stochastic field may modulate electronic bunching and antibunching in the transport, leading to the shot noise varying between sub- and super-Poissonian distributions. Finally, a summary is presented in Sec. IV.
II.
MODEL AND FORMULA
The schematic setup studied here is shown in Fig. 1. The system consists of a two-level quantum dot connected to two electronic reservoirs by tunnel barriers. The Coulomb repulsion inside the dot is assumed to be so large that only single occupation is allowed at one time. The inelastic transitions from the upper to the lower state could be induced by the lattice vibrations at low temperatures. In analogy to resonance fluorescence in quantum optics, the two-level dot is bichromatically driven by a coherent field with a frequency ωL and a constant amplitude Ec as well as by a stochastic field with a central frequency ωs and a stochastically fluctuating amplitude Es (t). Thus, the electron in the dot is coherently delocalized between both levels performing photon-assisted Rabi oscillations. For simplicity, the spin-resolved physics is not taken into account here and we consider spinless electrons. Before introducing the model Hamiltonian, we briefly discuss the effect of other noises in nanodevices, such as the thermal noise, the background charge (1/f ) noise in the sample, and electrical noise in the gate voltages.38 Such noises could induce the fluctuations of the device parameters and give rise to decoherence of the system. Up to now, the fluctuations with 1/f spectral density of different variables and physical origin have been widely observed in various quantum nanodevices. The 1/f noise destroys quantum coherent dynamics and is considered as the main source of decoherence. Experimentally, it has been pointed out that the amplitude of low-frequency fluctuation in energy spacing is estimated to be the order of 1μeV,38,39 which is much smaller than the strength of the stochastic field considered in the following discussion. For the thermal noise (Johnson-Nyquist noise), it is generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. When limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution. For an interacting nanodevice, the thermal noise is dominant at relatively low bias voltages.40
FIG. 1: A two-level quantum dot coupled to two electronic leads. Electrons in the dot oscillate between two levels driven by strong coherent field and a weak stochastic field.
We describe our system by the Hamiltonian H(t) =
εi d†i di +
i
Ω −iωL t † (e d2 d1 + eiωL t d†1 d2 ) 2
1 + x(t)[e−iωs t d†2 d1 + eiωs t d†1 d2 ] 2 ωQ a†Q aQ + λQ (d†2 d1 aQ + H.c.) + Q
+
kη
Q
εkη c†kη ckη
+
Vη (c†kη di + H.c.), (1)
kηi
where di (i = 1, 2) and ckη (η = L, R) are annihilation operators of electrons in the dot and in the leads, respectively. The fourth and fifth terms of Eq. (1) describe the phonon field induced by lattice vibrations and its interaction with the two-level dot respectively, where aQ is the annihilation operator for phonon.10 Here Ω = 2|d · eEc |/ is the Rabi frequency of the coherent field, and x(t) = 2|d · eEs (t)|/ represents the stochastic amplitude of the dot-stochastic-field interaction, which is assumed to be a Guassian-Markovian random process, with zero mean value and correlation functions29–33 x(t) = 0,
(2)
x(t)x(t ) = Dκe−κ|t−t | ,
(3)
where D is the strength of the stochastic process and κ can be associated with the bandwidth of the stochastic field. The last two terms of Eq. (1) describe the electrodes and the dot-electrode tunneling, where c†kη (ckη ) is the electron creation (annihilation) operator in the lead η with an energy εkη and Vη is the lead-dot coupling strength. In a frame rotating at the frequency ωL the master equation for the system is of the form ρ˙ = −i[Hd−c + Hd−s , ρ] + Ld (ρ) + Lp (ρ),
(4)
in which Δ Ω σz + (σ+ + σ− ), 2 2
(5)
1 x(t)[e−iδt d†2 d1 + eiδt d†1 d2 ], 2
(6)
γ (2σ− ρσ+ − σ+ σ− ρ − ρσ+ σ− ), 2
(7)
Hd−c =
Hd−s =
Lp (ρ) = and Ld (ρ) =
1 + Γ (2d† ρdi − di d†i ρ − ρdi d†i ) 2 iη ηi i +
1 − Γ (2di ρd†i − d†i di ρ − ρd†i di ). 2 iη ηi
(8)
Hd−c and Hd−s respectively describe the interaction of the quantum dot with the coherent and the stochastic fields. σz = d†2 d2 −d†1 d1 , σ+ = d†2 d1 and σ− = d†1 d2 are respectively Pauli matrice, raising and lowering operators. Here we assume ε2 = −ε1 = ε, then Δ = ε2 − ε1 − ωL = 2ε − ωL denotes the detuning of the dot’s resonance frequency and the frequency of coherent part of the driving field, δ = ωs − ωL is the frequency difference between the coherent and stochastic components of the driving field. The operator Lp (ρ) represents inelastic transitions from the upper to the lower state where γ = 2π|λε2 −ε1 |2 is the phonon spontaneous emission rate. The operator Ld (ρ) represents the one-way tunneling process of injection of electrons into quantum dot and the damping of the dot by the one-way electron tunneling out to the electrode. − Here, Γ+ ηi (Γηi )(η = L, R and i = 1, 2) describes the tunneling rate of electron of the state |i (i = 1, 2) into (out of) the dot from (into) the η lead. The temperature± dependent tunneling rates are defined as Γ± ηi = Γη fη (εi ), where Γη is the tunneling strength. It is assumed that the dot is symmetrically coupled to the external leads, i.e., ΓL = ΓR = Γ0 . Here, fη+ (ε) = {1 + e(ε−μη )/kB T }−1 is the Fermi distribution function of the lead η (η = L, R), and fη− (ε) = 1 − fη+ (ε). In the following discussion, the Fermi energy of the left lead is considered infinite, so no electrons can tunnel from the quantum dot to the left lead.
For simplicity, we assume the intensity of the coherent part is much greater than that of the stochastic field, and the bandwidth κ of the stochastic field is much greater than the tunneling rate Γ0 and the phonon spontaneous emission rate γ, i.e., Ω
√ Dκ,
κ Γ0 , γ.
(9)
One can then invoke standard perturbative techniques to eliminate the stochastic variable x(t). To do this we at first disregard the electronic tunneling and relaxation, respectively represented by Ld (ρ) and Lp (ρ), since these quantities undergo no change in the elimination procedure. Firstly, we perform a canonical transformation on the master Eq. (4) by ρ˜(t) = eiHd−c t ρe−iHd−c t . Then the master equation in this interaction picture takes the form ˜ d−s (t), ρ˜(t)], ρ˜˙ (t) = −i[H
(10)
˜ d−s (t) = 1 x(t)[e−iδt σ H ˜+ (t) + eiδt σ ˜− (t)]. 2
(11)
where
We now integrate Eq. (10) formally to give ρ˜(t) = ρ˜(0) −
i 2
+x(t )e
iδt
t
0
[x(t )e−iδt σ ˜+ (t )
σ ˜− (t ), ρ(t )]dt .
(12)
Substituting for ρ˜(t) inside the commutator in Eq. (10) leads to ρ˜˙ (t)
i = − [x(t)e−iδt σ ˜+ (t) + x(t)eiδt σ ˜− (t), ρ˜(0)] 2 t 1 [x(t)e−iδt σ ˜+ (t) + x(t)eiδt σ ˜− (t), (13) − 4 0
˜+ (t ) + x(t )eiδt σ ˜− (t ), ρ˜(t )]]dt . [x(t )e−iδt σ Because the the correlation time κ−1 of the stochastic field is very short compared to the radiative lifetime γ −1 of the quantum dot, one can eliminate the stochastic variable x(t).33 Substituting Eqs. (2) and (3) in, we have
1 i ˜+ (t) + x(t)eiδt σ ˜− (t), ρ˜(0)] − ρ˜˙ (t) = − [x(t)e−iδt σ 2 4
t
0
[x(t)e−iδt σ ˜+ (t) + x(t)eiδt σ ˜− (t),
˜+ (t ) + x(t )eiδt σ ˜− (t ), ρ˜(t )]]dt [x(t )e−iδt σ t 1 = − x(t)x(t ){[e−iδ(t−t ) σ ˜+ (t), [˜ σ− (t ), ρ˜(t )]] + [eiδ(t−t ) σ ˜− (t), [˜ σ+ (t ), ρ˜(t )]]}dt 4 0 Dκ t −κ(t−t ) −iδ(t−t ) = − e {[e σ ˜+ (t), [˜ σ− (t ), ρ˜(t )]] + [eiδ(t−t ) σ ˜− (t), [˜ σ+ (t ), ρ˜(t )]]}dt 4 0 Dκ t −(κ+iδ)τ = − {e [˜ σ+ (t), [˜ σ− (t − τ ), ρ(t ˜ − τ )]] + e−(κ−iδ)τ [˜ σ− (t), [˜ σ+ (t − τ ), ρ(t ˜ − τ )]]}dτ. 4 0 √ The assumption that Ω Dκ, κ γ, Γ0 validates the Markovian approximation, i.e., ρ˜(t − τ ) ≈ ρ˜(t).
(15)
We can also extend the upper limit of the integral to infinity by transformation ρ˜(t) back to the original picture via ρ = exp(−iHd−ct)˜ ρ(t) exp(iHd−c t). The resultant master equation for the reduced density operator ρ takes the form ρ˙ = −i[
where S+
with
= Dκ
0 †
∞
¯ + Δ)Ω ¯ − Δ)Ω (Ω (Ω Dκ 2ΔΩ − + [ 2 ¯ ¯ ¯ ], 4Ω κ − iδ κ − i(δ − Ω) κ − i(δ + Ω) Ω2 Ω2 Dκ 2Ω2 − − = ¯2 [ ¯ ¯ ], 4Ω κ − iδ κ − i(δ − Ω) κ − i(δ + Ω) ¯ + Δ)2 ¯ − Δ)2 (Ω (Ω Dκ 2Δ + + = ¯2 [ ¯ ¯ ], 4Ω κ − iδ κ − i(δ − Ω) κ − i(δ + Ω) (18)
B0 = B1 B2
† Ω (d d2 − d†1 d1 ) + (d†2 d1 + d†1 d2 ), ρ] 2 2 2
1 − ([d†1 d2 , [S+ , ρ]] + [d†2 d1 , [S− , ρ]]) 4 +Lp (ρ) + Ld (ρ),
¯= where Ω (16)
(14)
√ Ω2 + Δ2 .
The reduced density matrices in vector form could be chosen as ρ = (ρ00 , ρ11 , ρ22 , ρ12 , ρ21 ) and its time evaluation is governed by the master equations
dτ e−(k−iδ)τ e−iHd−c τ σ+ eiHd−c τ
ρ˙ = Wρ,
(19)
= (S− )
= B0 (d†2 d2 − d†1 d1 ) + B1 d†1 d2 + B2 d†2 d1 ,
(17)
where the matrix W are given by
⎞ + −(2ΓL + Γ+ Γ− Γ− 0 0 R1 + ΓR2 ) R1 R2 ⎜ ΓL + Γ+ −Γ− 2γ − 2Re(B2 ) 2B0∗ + iΩ/2 2B0 − iΩ/2 ⎟ R1 R1 + 2Re(B2 ) ⎟ ⎜ + − ∗ ⎟ W=⎜ Γ + Γ −2Re(B ) −Γ − 2γ + 2Re(B ) −2B − iΩ/2 −2B L 2 2 0 + iΩ/2 ⎟ 0 ⎜ R2 R2 ⎠ ⎝ 0 iΩ/2 −iΩ/2 −ΓN − γ + iΔ −2B1 0 −iΩ/2 iΩ/2 −2B1∗ −ΓN − γ − iΔ ⎛
+ − − and ΓN = (2ΓL + Γ+ R1 + ΓR2 + ΓR1 + ΓR2 )/2. Here we have considered that μL → ∞ and no electron tunnels from the dot to the left reservoir.
(20)
case is Iη = e
ˆ η ρ(0) ]k , [Γ
(21)
k
We can solve the master equation at the stationary case and obtain the transport property subsequently. In the sequential regime, the current Iη in the steady-state
ˆ η ρ(0) elements where the summation goes over all vector Γ (0) (k = 1, 2, ..., 5), ρ is the steady state solution of Eq. ˆ η are the matrixes of the current operator and (20), Γ
they are given by ⎛ ⎜ ˆL = ⎜ Γ ⎜ ⎝ ⎛ ⎜ ⎜ ˆR = ⎜ Γ ⎜ ⎝
0 ΓL ΓL 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎞ ⎟ ⎟ ⎟, ⎠
− 0 −Γ− R1 −ΓR2 + ΓR1 0 0 Γ+ 0 0 R2 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎞ ⎟ ⎟ ⎟. ⎟ ⎠
(22)
We are interested in the zero-frequency shot noise. It is well known that the noise power spectra can be expressed as the Fourier transform of the current-current correlation function9,10,12,13,40–44 ∞ SIi Ij (ω) = 2 dteiωt [Ii (t)Ij (0) − 2Ii Ij ] −∞
= 2Ii (t)Ij (0)ω − 2Ii ω Ij ω .
(23)
Here, Ii and Ij are the electrical currents across the i and j junctions and t is the time. Furthermore, the currentcurrent correlation function of the currents Ii and Ij in the ω-space can be expressed as
ˆ i Tˆ (ω)Γ ˆ j Tˆ (−ω)Γ ˆ j ρ(0) + Γ ˆ i ρ(0) , Ii (t)Ij (0)ω = Γ k
k
(24)
−1 where Tˆ(±ω) = ∓iωˆI − W and I is the unit matrix. For electron transport in the mesoscopic device, the current fluctuation is sensitively affected by different interactions, thus the shot noise can be enhanced or suppressed with respect to the Poissonian value. Due to current conservation, for the two terminal system, there is SLL = SRR = −SLR = −SRL . In the following, we use the component −SLR ≡ S to represent the noise of charge current and then the Fano factor F = S(ω = 0)/2eI is introduced to represent the deviation from Poissonian shot noise for which F = 1. III.
RESULTS AND DISCUSSION
In the following discussion, we focus on the quantum transport through the quantum dot driven by a strong coherent field, e.g., Ω kB T Γ0 . The restriction of the low-order perturbation theory requires that all the energy scales are much larger than the dot-lead coupling strength. In the calculation, the symmetric dot-lead coupling strengths ΓL,R = Γ0 are considered and the temperature kB T is taken as 10Γ0 . The energy unit is taken as 1meV. The Fermi energy of the left lead is considered infinite, so no electrons can tunnel from the dot to the left lead.
FIG. 2: The diagonal elements of the density matrix in the energy basis as a function of the detuning Δ for different frequency difference between the coherent field and stochastic field (a) δ = 0 and (b) δ = 2. Other parameters are taken as D = 0.1, κ = 0.05, Ω = 1, Γ0 = ΓL,R = 1×10−3 , γ = 1×10−4 , ε1 = −0.2, ε2 = 0.2, μR = 0, and kB T = 0.01. The energy unit is taken as 1meV.
In Fig. 2 we demonstrate the dependence of the populations in the quantum dot on the detuning Δ between the coherent field frequency and the dot energy difference, where the effect of frequency difference δ between the coherent field and the stochastic field is considered. In the resonant case that Δ ≈ 0, the electron in the dot performs rapid Rabi oscillation under strong coherent field. The steady-state population difference ρ11 −ρ22 approaches to zero. In the presence of the detuning Δ, the population difference appears and the empty state ρ00 is also suppressed. In the absence of the frequency difference δ = 0, the population difference is almost independent on the stochastic field strength D and it is given by ρ11 − ρ22 ≈
4(3γ + Γ0 )Δ2 . 4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2
(25)
The population difference ρ11 − ρ22 increases monotonically as a function of Δ and saturates in the large detuning case. The appearance of the detuning could suppress the Rabi resonance between two dot energy levels, and their competition determined the state population and coherence in the dot. It is shown in Fig. 2b that the frequency difference δ between the coherent field and the stochastic field could modulate the state populations in the dot considerably. Besides the resonant point, the population difference ρ11 − ρ22 reduces to a very small value at a certain detuning point in the presence of a frequency difference δ. This implies that a new resonant √ point is generated and its position is nearly about Δ = δ 2 − Ω2 . At this point, the population difference
can be simplified as ρ11 − ρ22
Δ2 16(3γ + Γ0 ) ≈ √ 2 . 7D Δ2 + Ω2 + Δ
factor in the strong driving field is given by
(26)
It is clearly shown that there is no population inversion in this case and ρ11 − ρ22 reduces to a small value for D γ, Γ0 . The transport property of the device sensitively relies on the state populations in the quantum dot. In the absence of a stochastic field, the quantum transport driven by a normal pumping field has been widely discussed in the previous literatures in detail.10 In the presence of a strong coherent field, we mainly concentrate on the transport in the large detuning case to investigate the effect of the stochastic field.
FIG. 3: (A) the current I and (b) noise Fano factor F as a function of the chemical potential μR in the right lead for different stochastic field strength D = 0, 0.005, 0.02, and 0.1. Other parameters are taken as those in Fig. 2 and Δ = 3, δ = 3.
In Fig. 3 we present the current and noise Fano factor as a function of the chemical potential μR in the right lead for different stochastic field strength D. By tuning the chemical potential μR in the right lead, we control the tunneling of electrons between the dot and the two leads. When μR lies between two dot energy levels, the transport is in the dynamical channel blockade regime, which is of special interest. For the nondriven case (Ω = 0), the occupation of the lower level blocks the transport through the upper level, leading to super-Poissonian noise. In the presence of a driving field, the dynamical channel blockade could be considerably suppressed by the process of pumping electrons from the lower to the upper level by means of the absorption of one photon. Correspondingly, the electronic noise turns out to be sub-Poissonian. For D = 0 and ε1 μR ε2 , the current and noise Fano
I ≈ F ≈
2Γ0 (2Γ0 + γ)Ω2 , 4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2 16(2Γ0 + 3γ)(Γ0 + 3γ)Δ4 + 41(2Γ0 + γ)2 Ω4
+
[4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2 ]2 4(30Γ20 + 101Γ0 γ + 38γ 2 )(Γ0 + 3γ)Δ2 Ω2 [4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2 ]
2
.(27)
In the large detuning case Δ Ω and resonant case Δ = 0, we have FΔΩ ≈ 1 + FΔ=0 ≈
Γ0 , Γ0 + 3γ
41
0.837, 49
(28)
which gives the bound of the noise Fano factor. It is found that the shot noise varies between sub- and superPoissonian type as a function of the detuning Δ. It is illustrated in Fig. 3 that with the increase of μR , the transport windows are closed one by one as the chemical potential of the right lead scans the energy levels in the dot. The introduction of a stochastic field affects the character of the electron current and noise only in the regime that μR lies between two dot energy levels, i.e., ε1 μR ε2 . In this case, the stochastic field could enhance the resonant effect between the quantum dot and the coherent field, while it suppresses the dephasing effect induced by large detuning. Correspondingly, the current flowing through the dot is enhanced, and it is interesting that the shot noise is weakened as a noise field is introduced to the device. With the increase of the stochastic field strength D, the super-Poissonian shot noise becomes a sub-Poissonian type. The frequency difference δ between the driving field and the stochastic field is another tunable parameter in a real experiment. Fig. 4 exhibits the dependence of the current and noise Fano factor on the frequency difference δ in the regime of ε1 μR ε2 . It is shown in Fig. 4a that the current flowing through the dot is strongly enhanced for appropriate δ. Two resonant peaks appear in the current and they √ are located around the points ¯ where Ω ¯ = Ω2 + Δ2 . From Eq. (18), it is δ = ±Ω, obvious that the coefficients B0 , B1 and B2 are resonant when the central frequency of the stochastic field is tuned ¯ At these points, the frequency difference to δ = 0, ±Ω. δ compensates the detuning of the interaction between the driving field and the dot energy levels, where the resonant Rabi oscillation of the electron in the dot is re¯ it is indicated covered. At the resonance points δ = ±Ω, in Fig. 4 that a huge difference in the amplitudes of the resonance peaks appears. At these two points, the values of B1 are the same and they affects the state populations in a same way. However, the values of B0 and B2 at these two points are quite different, which results the difference in the amplitudes of the resonance peaks. At ¯ the state populations of two states are the point δ = Ω,
favorable to equally occupied. The suppression of population difference relieves the dynamical channel blockade effect, leading to a sub-Poissonian shot noise. Especially, ¯ could be simplified as the current at δ = ±Ω
−1 16(3γ + Γ0 )Δ2 Iδ=−Ω¯ ≈ 2Γ0 7 + , ¯ − Δ)2 4(γ + 2Γ0 )Ω2 + D(Ω −1 16(3γ + Γ0 )Δ2 Iδ=Ω¯ ≈ 2Γ0 7 + .(29) ¯ + Δ)2 4(γ + 2Γ0 )Ω2 + D(Ω It can be deduced that Iδ=Ω¯ > Iδ=−Ω¯ . The upper bound of Iδ=Ω¯ is 2Γ0 /7, while for Δ Ω, the lower bound of Iδ=−Ω¯ is 2Γ0 (γ + 2Γ0 )Ω2 /(3γ + Γ0 )Δ2 .
FIG. 4: (a) The current I and (b) noise Fano factor F as a function of the frequency difference δ between the driving field and the stochastic field for different stochastic field strength D = 0, 0.005, 0.02, and 0.1. Other parameters are taken as those in Fig. 3 while μR = 0.
It is demonstrated in Fig. 4a that a rather small stochastic field (D/Ω = 1/200) could induce the dramat-
1 2 3
4
5 6 7 8
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ically enhancement of the current. The further increase of the stochastic field hardly enhances the peak values of the current, while the resonant peaks in the current are broadened as the increase of D. Besides the stochastic field strength, the frequency difference δ could also modulate the shot noise ranged between sub- and superPoissonian distributions, as shown in Fig. 4b. Near the ¯ the current flowing through the device points δ = ±Ω, gets maximum enhancement and the dynamical channel blockade is thus weakened, leading the suppression of the noise Fano factor.
IV.
SUMMARY
In conclusion, we investigate the effect of a stochastic field on the quantum transport flowing through a twolevel quantum dot irradiated by a strong coherent field. It is found that the import of a stochastic field affects the character of the electron current and noise only in the dynamical channel blockade case, in which the Fermi energy of the drain reservoir lies between two dot energy levels. The stochastic field could enhance the resonant effect between the quantum dot and the coherent field, and suppress the population difference between two dot levels induced by large detuning. New resonant points could be produced when the central frequency of the stochastic field equals to the detuning approximately. At the resonant points, the population difference in the dot reduces to a small value. The current is considerably enhanced by the stochastic field. By tuning the strength and frequency of the stochastic field, the zero-frequency shot noise of the current varies dramatically between sub- and super-Poissonian distributions. This project was supported by the National Natural Science Foundation of China (No.11574173, 61474018) and the Fundamental Research Funds for the Central Universities (No. ZYGX2012J052).
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PII: DOI: Reference:
S0038-1098(16)30257-5 http://dx.doi.org/10.1016/j.ssc.2016.09.017 SSC13045
To appear in: Solid State Communications Received date: 14 June 2016 Revised date: 1 September 2016 Accepted date: 21 September 2016 Cite this article as: Sha-Sha Ke, Ling-E Miao, Zhen Guo, Yong Guo, Huai-Wu Zhang and Hai-Feng Lü, Quantum transport in a two-level quantum dot driven by coherent and stochastic fields, Solid State Communications, http://dx.doi.org/10.1016/j.ssc.2016.09.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Quantum transport in a two-level quantum dot driven by coherent and stochastic fields Sha-Sha Ke,1 Ling-E Miao,1 Zhen Guo,1 Yong Guo,2 Huai-Wu Zhang,1 and Hai-Feng L¨ u1 1
State Key Laboratory of Electronic Thin Films and Integrated Devices and School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China 2 Department of Physics, Tsinghua University, Beijing 100084, China We study theoretically the current and shot noise properties flowing through a two-level quantum dot driven by a strong coherent field and a weak stochastic field. The interaction x(t) between the quantum dot and the stochastic field is assumed to be a Gaussian-Markovian random process with zero mean value and correlation function x(t)x(t) = Dκe−κ|t−t | , where D and κ are the strength and bandwidth of the stochastic field, respectively. It is found that the stochastic field could enhance the resonant effect between the quantum dot and the coherent field, and generate new resonant points. At the resonant points, the state population difference between two levels is suppressed and the current is considerably enhanced. The zero-frequency shot noise of the current varies dramatically between sub- and super-Poissonian characteristics by tuning the stochastic field appropriately. PACS numbers: 72.25.Dc, 73.63.Nm, 72.70.+m
I.
INTRODUCTION
There have been extensive investigations focused on the effect of time-varying external fields on quantum transport in mesoscopic systems.1–5 All-optically controlled single electron transport in quantum dots are particularly attractive.6–13 The interaction between the electronic transport and the light-matter interaction leads to a rich phenomenology. For instance, in twolevel quantum dot10,11 and Λ-type double-dot12 systems driven by an optical field, it has been shown that the zero-frequency shot noise can be tuned between suband super-Poissonian distributions due to the dynamical channel blockade induced by the optical field. One of the most influential recent advances in the field of superconducting quantum circuits is the ability to coherently couple microwave photons in cavity to mesoscopic electronic conductors.14–28 The interesting resonance phenomena have been observed, as well as a modification of the cavity behavior due to finite bias transport in a quantum dot device.14–18 Very recently, the photon mediated interaction between distant quantum dot circuits has been demonstrated by using the cavity quantum electrodynamics architectures, where the two dots are separated by 80μm.19 Theoretically, it has been analyzed how the coupling to a common photon mode generates entanglement between distant charge qubits realized in double quantum dots and how this entanglement manifests in the transport properties of the system.22–24 The study of the photon-assisted tunneling through a quantum conductor is also extended to the case of nonclassical light, such as squeezed states and Fock states.28 It is found that the transport property of the conductor could serve as a nontrivial probe of the nonclassical microwave states. In the field of quantum optics, it has been a long standing issue to investigate the effect of laser noise on the resonance fluorescence and absorption spectra in the
atom-light interaction.29–37 Over the last two decades, it was revealed that laser noise can induce the relatively large intensity fluctuations of the resonance fluorescence of a macroscopic sample of atoms.29,30 The stochastic field was also proposed to achieve phase-dependent spectra in resonance fluorescence in a two-level atom to avoid preparing a squeezed vacuum.34 Another typical effect induced by a stochastic perturbation in atom systems is that collisional noise can unexpectedly create phase correlation between the neighbouring atomic dressed states.35 This phase correlation is responsible for quantum interference between dressed state transition channels, resulting in anomalous modifications of the resonance fluorescence spectra. When an external light field is applied to a quantum circuit, it is important to take account of the amplitude fluctuations of the interaction between the coherent field and quantum conductor. However, it remains unknown how the input of a stochastic field affects quantum transport in a mesoscopic conductor. In this paper we investigate the quantum transport through a two-level quantum dot driven by a strong coherent field and a weak stochastic field with a wide bandwidth. By adopting the Born-Markov and rotating-wave approximations, we eliminate the stochastic variable x(t) to obtain a master equation for the reduced density matrix of the quantum dot. It is shown that the stochastic field effectively forms a reservoir. We have a reservoir with tunable amplitude and frequency and it strongly affects such physical properties as the state population, the tunneling current, and the shot noise. The paper is organized as follows. In Sec. II we introduce the model Hamiltonian of the system. An effective master equation for the reduced density operator of the quantum dot is derived when the coherent driving field is much stronger than the stochastic part. In Sec. III we discuss the effect of the stochastic field on the transport property of the device. We find that the presence of the additionally
stochastic field may modulate electronic bunching and antibunching in the transport, leading to the shot noise varying between sub- and super-Poissonian distributions. Finally, a summary is presented in Sec. IV.
II.
MODEL AND FORMULA
The schematic setup studied here is shown in Fig. 1. The system consists of a two-level quantum dot connected to two electronic reservoirs by tunnel barriers. The Coulomb repulsion inside the dot is assumed to be so large that only single occupation is allowed at one time. The inelastic transitions from the upper to the lower state could be induced by the lattice vibrations at low temperatures. In analogy to resonance fluorescence in quantum optics, the two-level dot is bichromatically driven by a coherent field with a frequency ωL and a constant amplitude Ec as well as by a stochastic field with a central frequency ωs and a stochastically fluctuating amplitude Es (t). Thus, the electron in the dot is coherently delocalized between both levels performing photon-assisted Rabi oscillations. For simplicity, the spin-resolved physics is not taken into account here and we consider spinless electrons. Before introducing the model Hamiltonian, we briefly discuss the effect of other noises in nanodevices, such as the thermal noise, the background charge (1/f ) noise in the sample, and electrical noise in the gate voltages.38 Such noises could induce the fluctuations of the device parameters and give rise to decoherence of the system. Up to now, the fluctuations with 1/f spectral density of different variables and physical origin have been widely observed in various quantum nanodevices. The 1/f noise destroys quantum coherent dynamics and is considered as the main source of decoherence. Experimentally, it has been pointed out that the amplitude of low-frequency fluctuation in energy spacing is estimated to be the order of 1μeV,38,39 which is much smaller than the strength of the stochastic field considered in the following discussion. For the thermal noise (Johnson-Nyquist noise), it is generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. When limited to a finite bandwidth, thermal noise has a nearly Gaussian amplitude distribution. For an interacting nanodevice, the thermal noise is dominant at relatively low bias voltages.40
FIG. 1: A two-level quantum dot coupled to two electronic leads. Electrons in the dot oscillate between two levels driven by strong coherent field and a weak stochastic field.
We describe our system by the Hamiltonian H(t) =
εi d†i di +
i
Ω −iωL t † (e d2 d1 + eiωL t d†1 d2 ) 2
1 + x(t)[e−iωs t d†2 d1 + eiωs t d†1 d2 ] 2 ωQ a†Q aQ + λQ (d†2 d1 aQ + H.c.) + Q
+
kη
Q
εkη c†kη ckη
+
Vη (c†kη di + H.c.), (1)
kηi
where di (i = 1, 2) and ckη (η = L, R) are annihilation operators of electrons in the dot and in the leads, respectively. The fourth and fifth terms of Eq. (1) describe the phonon field induced by lattice vibrations and its interaction with the two-level dot respectively, where aQ is the annihilation operator for phonon.10 Here Ω = 2|d · eEc |/ is the Rabi frequency of the coherent field, and x(t) = 2|d · eEs (t)|/ represents the stochastic amplitude of the dot-stochastic-field interaction, which is assumed to be a Guassian-Markovian random process, with zero mean value and correlation functions29–33 x(t) = 0,
(2)
x(t)x(t ) = Dκe−κ|t−t | ,
(3)
where D is the strength of the stochastic process and κ can be associated with the bandwidth of the stochastic field. The last two terms of Eq. (1) describe the electrodes and the dot-electrode tunneling, where c†kη (ckη ) is the electron creation (annihilation) operator in the lead η with an energy εkη and Vη is the lead-dot coupling strength. In a frame rotating at the frequency ωL the master equation for the system is of the form ρ˙ = −i[Hd−c + Hd−s , ρ] + Ld (ρ) + Lp (ρ),
(4)
in which Δ Ω σz + (σ+ + σ− ), 2 2
(5)
1 x(t)[e−iδt d†2 d1 + eiδt d†1 d2 ], 2
(6)
γ (2σ− ρσ+ − σ+ σ− ρ − ρσ+ σ− ), 2
(7)
Hd−c =
Hd−s =
Lp (ρ) = and Ld (ρ) =
1 + Γ (2d† ρdi − di d†i ρ − ρdi d†i ) 2 iη ηi i +
1 − Γ (2di ρd†i − d†i di ρ − ρd†i di ). 2 iη ηi
(8)
Hd−c and Hd−s respectively describe the interaction of the quantum dot with the coherent and the stochastic fields. σz = d†2 d2 −d†1 d1 , σ+ = d†2 d1 and σ− = d†1 d2 are respectively Pauli matrice, raising and lowering operators. Here we assume ε2 = −ε1 = ε, then Δ = ε2 − ε1 − ωL = 2ε − ωL denotes the detuning of the dot’s resonance frequency and the frequency of coherent part of the driving field, δ = ωs − ωL is the frequency difference between the coherent and stochastic components of the driving field. The operator Lp (ρ) represents inelastic transitions from the upper to the lower state where γ = 2π|λε2 −ε1 |2 is the phonon spontaneous emission rate. The operator Ld (ρ) represents the one-way tunneling process of injection of electrons into quantum dot and the damping of the dot by the one-way electron tunneling out to the electrode. − Here, Γ+ ηi (Γηi )(η = L, R and i = 1, 2) describes the tunneling rate of electron of the state |i (i = 1, 2) into (out of) the dot from (into) the η lead. The temperature± dependent tunneling rates are defined as Γ± ηi = Γη fη (εi ), where Γη is the tunneling strength. It is assumed that the dot is symmetrically coupled to the external leads, i.e., ΓL = ΓR = Γ0 . Here, fη+ (ε) = {1 + e(ε−μη )/kB T }−1 is the Fermi distribution function of the lead η (η = L, R), and fη− (ε) = 1 − fη+ (ε). In the following discussion, the Fermi energy of the left lead is considered infinite, so no electrons can tunnel from the quantum dot to the left lead.
For simplicity, we assume the intensity of the coherent part is much greater than that of the stochastic field, and the bandwidth κ of the stochastic field is much greater than the tunneling rate Γ0 and the phonon spontaneous emission rate γ, i.e., Ω
√ Dκ,
κ Γ0 , γ.
(9)
One can then invoke standard perturbative techniques to eliminate the stochastic variable x(t). To do this we at first disregard the electronic tunneling and relaxation, respectively represented by Ld (ρ) and Lp (ρ), since these quantities undergo no change in the elimination procedure. Firstly, we perform a canonical transformation on the master Eq. (4) by ρ˜(t) = eiHd−c t ρe−iHd−c t . Then the master equation in this interaction picture takes the form ˜ d−s (t), ρ˜(t)], ρ˜˙ (t) = −i[H
(10)
˜ d−s (t) = 1 x(t)[e−iδt σ H ˜+ (t) + eiδt σ ˜− (t)]. 2
(11)
where
We now integrate Eq. (10) formally to give ρ˜(t) = ρ˜(0) −
i 2
+x(t )e
iδt
t
0
[x(t )e−iδt σ ˜+ (t )
σ ˜− (t ), ρ(t )]dt .
(12)
Substituting for ρ˜(t) inside the commutator in Eq. (10) leads to ρ˜˙ (t)
i = − [x(t)e−iδt σ ˜+ (t) + x(t)eiδt σ ˜− (t), ρ˜(0)] 2 t 1 [x(t)e−iδt σ ˜+ (t) + x(t)eiδt σ ˜− (t), (13) − 4 0
˜+ (t ) + x(t )eiδt σ ˜− (t ), ρ˜(t )]]dt . [x(t )e−iδt σ Because the the correlation time κ−1 of the stochastic field is very short compared to the radiative lifetime γ −1 of the quantum dot, one can eliminate the stochastic variable x(t).33 Substituting Eqs. (2) and (3) in, we have
1 i ˜+ (t) + x(t)eiδt σ ˜− (t), ρ˜(0)] − ρ˜˙ (t) = − [x(t)e−iδt σ 2 4
t
0
[x(t)e−iδt σ ˜+ (t) + x(t)eiδt σ ˜− (t),
˜+ (t ) + x(t )eiδt σ ˜− (t ), ρ˜(t )]]dt [x(t )e−iδt σ t 1 = − x(t)x(t ){[e−iδ(t−t ) σ ˜+ (t), [˜ σ− (t ), ρ˜(t )]] + [eiδ(t−t ) σ ˜− (t), [˜ σ+ (t ), ρ˜(t )]]}dt 4 0 Dκ t −κ(t−t ) −iδ(t−t ) = − e {[e σ ˜+ (t), [˜ σ− (t ), ρ˜(t )]] + [eiδ(t−t ) σ ˜− (t), [˜ σ+ (t ), ρ˜(t )]]}dt 4 0 Dκ t −(κ+iδ)τ = − {e [˜ σ+ (t), [˜ σ− (t − τ ), ρ(t ˜ − τ )]] + e−(κ−iδ)τ [˜ σ− (t), [˜ σ+ (t − τ ), ρ(t ˜ − τ )]]}dτ. 4 0 √ The assumption that Ω Dκ, κ γ, Γ0 validates the Markovian approximation, i.e., ρ˜(t − τ ) ≈ ρ˜(t).
(15)
We can also extend the upper limit of the integral to infinity by transformation ρ˜(t) back to the original picture via ρ = exp(−iHd−ct)˜ ρ(t) exp(iHd−c t). The resultant master equation for the reduced density operator ρ takes the form ρ˙ = −i[
where S+
with
= Dκ
0 †
∞
¯ + Δ)Ω ¯ − Δ)Ω (Ω (Ω Dκ 2ΔΩ − + [ 2 ¯ ¯ ¯ ], 4Ω κ − iδ κ − i(δ − Ω) κ − i(δ + Ω) Ω2 Ω2 Dκ 2Ω2 − − = ¯2 [ ¯ ¯ ], 4Ω κ − iδ κ − i(δ − Ω) κ − i(δ + Ω) ¯ + Δ)2 ¯ − Δ)2 (Ω (Ω Dκ 2Δ + + = ¯2 [ ¯ ¯ ], 4Ω κ − iδ κ − i(δ − Ω) κ − i(δ + Ω) (18)
B0 = B1 B2
† Ω (d d2 − d†1 d1 ) + (d†2 d1 + d†1 d2 ), ρ] 2 2 2
1 − ([d†1 d2 , [S+ , ρ]] + [d†2 d1 , [S− , ρ]]) 4 +Lp (ρ) + Ld (ρ),
¯= where Ω (16)
(14)
√ Ω2 + Δ2 .
The reduced density matrices in vector form could be chosen as ρ = (ρ00 , ρ11 , ρ22 , ρ12 , ρ21 ) and its time evaluation is governed by the master equations
dτ e−(k−iδ)τ e−iHd−c τ σ+ eiHd−c τ
ρ˙ = Wρ,
(19)
= (S− )
= B0 (d†2 d2 − d†1 d1 ) + B1 d†1 d2 + B2 d†2 d1 ,
(17)
where the matrix W are given by
⎞ + −(2ΓL + Γ+ Γ− Γ− 0 0 R1 + ΓR2 ) R1 R2 ⎜ ΓL + Γ+ −Γ− 2γ − 2Re(B2 ) 2B0∗ + iΩ/2 2B0 − iΩ/2 ⎟ R1 R1 + 2Re(B2 ) ⎟ ⎜ + − ∗ ⎟ W=⎜ Γ + Γ −2Re(B ) −Γ − 2γ + 2Re(B ) −2B − iΩ/2 −2B L 2 2 0 + iΩ/2 ⎟ 0 ⎜ R2 R2 ⎠ ⎝ 0 iΩ/2 −iΩ/2 −ΓN − γ + iΔ −2B1 0 −iΩ/2 iΩ/2 −2B1∗ −ΓN − γ − iΔ ⎛
+ − − and ΓN = (2ΓL + Γ+ R1 + ΓR2 + ΓR1 + ΓR2 )/2. Here we have considered that μL → ∞ and no electron tunnels from the dot to the left reservoir.
(20)
case is Iη = e
ˆ η ρ(0) ]k , [Γ
(21)
k
We can solve the master equation at the stationary case and obtain the transport property subsequently. In the sequential regime, the current Iη in the steady-state
ˆ η ρ(0) elements where the summation goes over all vector Γ (0) (k = 1, 2, ..., 5), ρ is the steady state solution of Eq. ˆ η are the matrixes of the current operator and (20), Γ
they are given by ⎛ ⎜ ˆL = ⎜ Γ ⎜ ⎝ ⎛ ⎜ ⎜ ˆR = ⎜ Γ ⎜ ⎝
0 ΓL ΓL 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎞ ⎟ ⎟ ⎟, ⎠
− 0 −Γ− R1 −ΓR2 + ΓR1 0 0 Γ+ 0 0 R2 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎞ ⎟ ⎟ ⎟. ⎟ ⎠
(22)
We are interested in the zero-frequency shot noise. It is well known that the noise power spectra can be expressed as the Fourier transform of the current-current correlation function9,10,12,13,40–44 ∞ SIi Ij (ω) = 2 dteiωt [Ii (t)Ij (0) − 2Ii Ij ] −∞
= 2Ii (t)Ij (0)ω − 2Ii ω Ij ω .
(23)
Here, Ii and Ij are the electrical currents across the i and j junctions and t is the time. Furthermore, the currentcurrent correlation function of the currents Ii and Ij in the ω-space can be expressed as
ˆ i Tˆ (ω)Γ ˆ j Tˆ (−ω)Γ ˆ j ρ(0) + Γ ˆ i ρ(0) , Ii (t)Ij (0)ω = Γ k
k
(24)
−1 where Tˆ(±ω) = ∓iωˆI − W and I is the unit matrix. For electron transport in the mesoscopic device, the current fluctuation is sensitively affected by different interactions, thus the shot noise can be enhanced or suppressed with respect to the Poissonian value. Due to current conservation, for the two terminal system, there is SLL = SRR = −SLR = −SRL . In the following, we use the component −SLR ≡ S to represent the noise of charge current and then the Fano factor F = S(ω = 0)/2eI is introduced to represent the deviation from Poissonian shot noise for which F = 1. III.
RESULTS AND DISCUSSION
In the following discussion, we focus on the quantum transport through the quantum dot driven by a strong coherent field, e.g., Ω kB T Γ0 . The restriction of the low-order perturbation theory requires that all the energy scales are much larger than the dot-lead coupling strength. In the calculation, the symmetric dot-lead coupling strengths ΓL,R = Γ0 are considered and the temperature kB T is taken as 10Γ0 . The energy unit is taken as 1meV. The Fermi energy of the left lead is considered infinite, so no electrons can tunnel from the dot to the left lead.
FIG. 2: The diagonal elements of the density matrix in the energy basis as a function of the detuning Δ for different frequency difference between the coherent field and stochastic field (a) δ = 0 and (b) δ = 2. Other parameters are taken as D = 0.1, κ = 0.05, Ω = 1, Γ0 = ΓL,R = 1×10−3 , γ = 1×10−4 , ε1 = −0.2, ε2 = 0.2, μR = 0, and kB T = 0.01. The energy unit is taken as 1meV.
In Fig. 2 we demonstrate the dependence of the populations in the quantum dot on the detuning Δ between the coherent field frequency and the dot energy difference, where the effect of frequency difference δ between the coherent field and the stochastic field is considered. In the resonant case that Δ ≈ 0, the electron in the dot performs rapid Rabi oscillation under strong coherent field. The steady-state population difference ρ11 −ρ22 approaches to zero. In the presence of the detuning Δ, the population difference appears and the empty state ρ00 is also suppressed. In the absence of the frequency difference δ = 0, the population difference is almost independent on the stochastic field strength D and it is given by ρ11 − ρ22 ≈
4(3γ + Γ0 )Δ2 . 4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2
(25)
The population difference ρ11 − ρ22 increases monotonically as a function of Δ and saturates in the large detuning case. The appearance of the detuning could suppress the Rabi resonance between two dot energy levels, and their competition determined the state population and coherence in the dot. It is shown in Fig. 2b that the frequency difference δ between the coherent field and the stochastic field could modulate the state populations in the dot considerably. Besides the resonant point, the population difference ρ11 − ρ22 reduces to a very small value at a certain detuning point in the presence of a frequency difference δ. This implies that a new resonant √ point is generated and its position is nearly about Δ = δ 2 − Ω2 . At this point, the population difference
can be simplified as ρ11 − ρ22
Δ2 16(3γ + Γ0 ) ≈ √ 2 . 7D Δ2 + Ω2 + Δ
factor in the strong driving field is given by
(26)
It is clearly shown that there is no population inversion in this case and ρ11 − ρ22 reduces to a small value for D γ, Γ0 . The transport property of the device sensitively relies on the state populations in the quantum dot. In the absence of a stochastic field, the quantum transport driven by a normal pumping field has been widely discussed in the previous literatures in detail.10 In the presence of a strong coherent field, we mainly concentrate on the transport in the large detuning case to investigate the effect of the stochastic field.
FIG. 3: (A) the current I and (b) noise Fano factor F as a function of the chemical potential μR in the right lead for different stochastic field strength D = 0, 0.005, 0.02, and 0.1. Other parameters are taken as those in Fig. 2 and Δ = 3, δ = 3.
In Fig. 3 we present the current and noise Fano factor as a function of the chemical potential μR in the right lead for different stochastic field strength D. By tuning the chemical potential μR in the right lead, we control the tunneling of electrons between the dot and the two leads. When μR lies between two dot energy levels, the transport is in the dynamical channel blockade regime, which is of special interest. For the nondriven case (Ω = 0), the occupation of the lower level blocks the transport through the upper level, leading to super-Poissonian noise. In the presence of a driving field, the dynamical channel blockade could be considerably suppressed by the process of pumping electrons from the lower to the upper level by means of the absorption of one photon. Correspondingly, the electronic noise turns out to be sub-Poissonian. For D = 0 and ε1 μR ε2 , the current and noise Fano
I ≈ F ≈
2Γ0 (2Γ0 + γ)Ω2 , 4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2 16(2Γ0 + 3γ)(Γ0 + 3γ)Δ4 + 41(2Γ0 + γ)2 Ω4
+
[4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2 ]2 4(30Γ20 + 101Γ0 γ + 38γ 2 )(Γ0 + 3γ)Δ2 Ω2 [4(Γ0 + 3γ)Δ2 + 7(2Γ0 + γ)Ω2 ]
2
.(27)
In the large detuning case Δ Ω and resonant case Δ = 0, we have FΔΩ ≈ 1 + FΔ=0 ≈
Γ0 , Γ0 + 3γ
41
0.837, 49
(28)
which gives the bound of the noise Fano factor. It is found that the shot noise varies between sub- and superPoissonian type as a function of the detuning Δ. It is illustrated in Fig. 3 that with the increase of μR , the transport windows are closed one by one as the chemical potential of the right lead scans the energy levels in the dot. The introduction of a stochastic field affects the character of the electron current and noise only in the regime that μR lies between two dot energy levels, i.e., ε1 μR ε2 . In this case, the stochastic field could enhance the resonant effect between the quantum dot and the coherent field, while it suppresses the dephasing effect induced by large detuning. Correspondingly, the current flowing through the dot is enhanced, and it is interesting that the shot noise is weakened as a noise field is introduced to the device. With the increase of the stochastic field strength D, the super-Poissonian shot noise becomes a sub-Poissonian type. The frequency difference δ between the driving field and the stochastic field is another tunable parameter in a real experiment. Fig. 4 exhibits the dependence of the current and noise Fano factor on the frequency difference δ in the regime of ε1 μR ε2 . It is shown in Fig. 4a that the current flowing through the dot is strongly enhanced for appropriate δ. Two resonant peaks appear in the current and they √ are located around the points ¯ where Ω ¯ = Ω2 + Δ2 . From Eq. (18), it is δ = ±Ω, obvious that the coefficients B0 , B1 and B2 are resonant when the central frequency of the stochastic field is tuned ¯ At these points, the frequency difference to δ = 0, ±Ω. δ compensates the detuning of the interaction between the driving field and the dot energy levels, where the resonant Rabi oscillation of the electron in the dot is re¯ it is indicated covered. At the resonance points δ = ±Ω, in Fig. 4 that a huge difference in the amplitudes of the resonance peaks appears. At these two points, the values of B1 are the same and they affects the state populations in a same way. However, the values of B0 and B2 at these two points are quite different, which results the difference in the amplitudes of the resonance peaks. At ¯ the state populations of two states are the point δ = Ω,
favorable to equally occupied. The suppression of population difference relieves the dynamical channel blockade effect, leading to a sub-Poissonian shot noise. Especially, ¯ could be simplified as the current at δ = ±Ω
−1 16(3γ + Γ0 )Δ2 Iδ=−Ω¯ ≈ 2Γ0 7 + , ¯ − Δ)2 4(γ + 2Γ0 )Ω2 + D(Ω −1 16(3γ + Γ0 )Δ2 Iδ=Ω¯ ≈ 2Γ0 7 + .(29) ¯ + Δ)2 4(γ + 2Γ0 )Ω2 + D(Ω It can be deduced that Iδ=Ω¯ > Iδ=−Ω¯ . The upper bound of Iδ=Ω¯ is 2Γ0 /7, while for Δ Ω, the lower bound of Iδ=−Ω¯ is 2Γ0 (γ + 2Γ0 )Ω2 /(3γ + Γ0 )Δ2 .
FIG. 4: (a) The current I and (b) noise Fano factor F as a function of the frequency difference δ between the driving field and the stochastic field for different stochastic field strength D = 0, 0.005, 0.02, and 0.1. Other parameters are taken as those in Fig. 3 while μR = 0.
It is demonstrated in Fig. 4a that a rather small stochastic field (D/Ω = 1/200) could induce the dramat-
1 2 3
4
5 6 7 8
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IV.
SUMMARY
In conclusion, we investigate the effect of a stochastic field on the quantum transport flowing through a twolevel quantum dot irradiated by a strong coherent field. It is found that the import of a stochastic field affects the character of the electron current and noise only in the dynamical channel blockade case, in which the Fermi energy of the drain reservoir lies between two dot energy levels. The stochastic field could enhance the resonant effect between the quantum dot and the coherent field, and suppress the population difference between two dot levels induced by large detuning. New resonant points could be produced when the central frequency of the stochastic field equals to the detuning approximately. At the resonant points, the population difference in the dot reduces to a small value. The current is considerably enhanced by the stochastic field. By tuning the strength and frequency of the stochastic field, the zero-frequency shot noise of the current varies dramatically between sub- and super-Poissonian distributions. This project was supported by the National Natural Science Foundation of China (No.11574173, 61474018) and the Fundamental Research Funds for the Central Universities (No. ZYGX2012J052).
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