Real-time counting of single electron tunneling through a T-shaped double quantum dot system
JunYan Luo, Shi-Kuan Wang, Xiao-Ling He, Xin-Qi Li, YiJing Yan
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Real–time counting of single electron tunneling through a T–shaped double quantumdot system
JunYan Luo,
1, 2, ∗ Shi-Kuan Wang, Xiao-Ling He, Xin-Qi Li,
2, 3, 4 and YiJing Yan † School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong SAR, China State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors,Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China Department of Physics, Beijing Normal University, Beijing 100875, China (Dated: September 28, 2018)Real–time detection of single electron tunneling through a T–shaped double quantum dot is sim-ulated, based on a Monte Carlo scheme. The double dot is embedded in a dissipative environment,and the presence of electrons on the double dot is detected with a nearby quantum point contact.We demonstrate directly the bunching behavior in electron transport, which leads eventually to asuper–Poissonian noise. Particularly, in the context of full counting statistics, we investigate theessential difference between the dephasing mechanisms induced by the quantum point contact detec-tion and the coupling to the external phonon bath. A number of intriguing noise features associatedwith various transport mechanisms are revealed.
PACS numbers: 72.70.+m, 73.23.Hk, 73.63.-b, 03.65.Ta
I. INTRODUCTION
To control and manipulate electronic dynamics innanoscale devices it requires knowledge of the involv-ing transport processes at single–electron level. Thespectrum of current fluctuations, which characterizes thedegree of correlation between charge transport events,serves as an essential tool superior to the average cur-rent in distinguishing various transport mechanisms.
Full counting statistics (FCS) of current has also beenmeasured, owing in particular to the development ofhighly sensitive on–chip detection of single–electron tun-neling technique. All statistical cumulants of the numberof transferred particles can now be extracted experimen-tally.Current fluctuations would obey a Poissonian processif the tunneling events were statistically independent.However, non-Poissonian fluctuation is in general a real-ity. In the case of transport through a localized state, thePauli exclusion principle suppresses the noise, leadingto a sub-Poissonian statistics. Systems of multiple non-local states such as coupled quantum dots are moreinteresting. The intrinsic quantum coherence and many–particle interactions there result in different sources ofcorrelations. The fascinating super-Poissionian noisethus occurs in various contexts, and thereby has beenattracting a wide interest recently.A representing system, which will be studied in thiswork, is a T–shaped double quantum dot (TDQD)system, as schematically shown in Fig. 1. The sys-tem is of particular interest, as it can be mapped to astructure of quantum well in presence of an impurity in-side, which has been investigated experimentally. Thesource and drain electrodes of the TDQD are in such aconfiguration that maximizes locality versus nonlocalitycontrast. In addition, the TDQD is also influenced by aninevitable dissipative environment. The nearby quantum I QPC
R L
QD1 QD2
FIG. 1: Schematics of electron transport through a TDQDsystem, monitored continuously by the QPC current that de-pends on the charge state for excess electrons on two dotsindividually. The TDQD is also coupled to an inevitable dis-sipative phonon environment (not shown explicitly). point contact (QPC) that serves as a charge detector isasymmetrically coupled to the dots. The current throughthe QPC depends on the charge state, ( n , n ), for excesselectrons on two dots individually. Electron tunnelingthrough the TDQD results in temporal changes in thecharge state, and thus fluctuations in the QPC current.In this work, we first simulate the real–time detectionof single electron tunneling through TDQD in experi-mental setup of Fig. 1. The real–time simulation will becarried out based on a Monte Carlo approach to quantummaster equation. In particular, we demonstrate trans-parently the electron bunching, i.e., the super–Poissoniannoise of charge transfer. The FCS is then studied, and theresult shows good agreement with those obtained fromMonte Carlo simulation. It therefore verifies the validityof our Monte Carlo method for real–time measurement.Particularly, it is found that the FCS is capable of dif-ferentiating between the dephasing effect induced by theQPC charge detection and that by the coupling to thedissipative bath environment. A number of intriguingnoise features will be revealed in association with vari-ous mechanisms, demonstrating thus the sensitivity andselectivity of cumulants to different transport properties.The paper is organized as follows. We describe theTDQD system and conditional quantum master equa-tion in Sec. II. The Monte Carlo simulation of real–timedetection of single electron tunneling is summarized inSec. III, together with the electron bunching behaviordemonstrated vividly for the system in study. Havingthe FCS theory reviewed in Sec. IV, we present the re-sults and discussions in Sec. V. Finally, we conclude inSec. VI.
II. QUANTUM MASTER EQUATION THEORYOF QUANTUM MEASUREMENTA. Model device
Let us start with the model setup for the QPC de-tection of single–electron tunneling through TDQD, assketched in Fig. 1. The total Hamiltonian consists of thecoupled dots system, the environment, and the couplingbetween them; i.e. H = H sys + H env + H sys − env . Theelectron Hamiltonian for the coupled dots system reads H sys = X σ (cid:0) ǫ ˆ Q zσ +Ω ˆ Q xσ (cid:1) + X l U l ˆ n l ↑ ˆ n l ↓ + U ′ ˆ n ˆ n , (1)where ˆ Q zσ ≡ d † σ d σ − d † σ d σ , ˆ Q xσ ≡ d † σ d σ + d † σ d σ ,ˆ n l = P σ ˆ n lσ , and ˆ n lσ = d † lσ d lσ , with d lσ ( d † lσ ) the elec-tron annihilation (creation) operator in the QD1 ( l = 1)or QD2 ( l = 2). Each quantum dot is assumed to haveonly one spin–degenerate level ( σ = ↑ or ↓ ) in the biaswindow. The level detuning between the two dots is ǫ = ǫ − ǫ . The interdot coupling strength is Ω. Theintradot and interdot Coulomb interactions, U l and U ′ ,are both assumed to be much larger than the Fermi lev-els. We shall be interested in the double-dot Coulombblockade regime, i.e. at most one electron resides inthe TDQD. It can be realized in experiments by propertuning the gate and bias voltages. The environment is of H env = h ph + P α h α + h QPC .It is composed of the phonon bath, the electron reser-voirs of the source and drain ( α = L and R) elec-trodes, as well as the QPC detector. Each of them ismodeled as a collection of noninteracting particles. Thephonon bath assumes h ph = P j ~ ω j ( p j + x j ). The elec-tron reservoirs are modeled with h α = P k,σ ǫ αk c † αkσ c αkσ and h QPC = P p,σ ǫ p c † pσ c pσ + P q,σ ǫ q c † qσ c qσ , respectively.These electronic parts are written in terms of electroncreation and annihilation operators in the α -electrodeand the QPC reservoirs states. The system–environment coupling can be written as H sys − env = X σ ˆ Q zσ F ph + X α,σ ( d † σ f ασ + f † ασ d σ )+ X s =0 , , ˆ n s F QPC s . (2)The first term describes the coupling with phonon bath,in which F ph ≡ P j λ j x j . This term is responsible for thedot level energy fluctuations. The effect of phonon bathon the double–dot system is characterized by the phononinteraction spectral density, J ph ( ω ) = P j | λ j | δ ( ω − ω j ).The second term describes the transfer coupling betweenQD1 and the leads, in which f ασ ≡ P k t αkσ c αkσ . Thethird term describes the interaction between the QPCdetector and the measured system, in which F QPC s ≡ P p,q,σ ( t spq c † pσ c qσ +H . c . ). The amplitude of electron tun-neling through the QPC depends on the TDQD chargestate ˆ n s = | s ih s | , with s = 0 for no excess electrons,and s = 1 or 2 for one excess electron in QD1 or QD2,respectively. In other words, the current through theQPC sensitively depends on the charge states of theTDQD, and thus can be used to measure single elec-tron tunneling events. The effects of these electron reser-voirs components on the double–dot system are charac-terized individually by their interaction spectral densi-ties: J α ( ω ) = P kσ | t αkσ | δ ( ω − ǫ αk ), and J QPC s ( ω ) = P p,q,σ | t spq | δ ( ω − ǫ pσ + ǫ qσ ), respectively.In the Coulomb blockade regime, together with largesource–drain voltage, the tunneling between QD1 andthe electrode α =L or R can be characterized by the rateΓ α ( ω ) = 2 π P kσ | t αkσ | δ ( ω − ǫ αk ). In what follows weadopt flat bands in the electrodes, which yields energy in-dependent couplings Γ α . Analogously, the coupling withthe QPC is described by the rate T s = 2 πg p g q | t spq | V QPC ,where V QPC is the QPC bias voltage, g p and g q are thedensity of states in the QPC reservoirs. We assume here-after the density of states to be constant, and t spq reser-voir states independent. Thus T s just depends on thesystem charge occupation state | s ih s | , with s = 0 beingfor zero excess electron, and s = 1 or 2 for one excesselectron in QD1 or QD2, respectively. For the QPC cur-rent, I s = e T s , we have I > I > I , as implied in thescheme of Fig. 1. B. Quantum master equation theory
To describe the quantum measurement, we exploitthe reduced density operator ρ ( n ) ( t ) of the TDQD sys-tem, for the specified number n electrons having passedthrough the QPC detector and being recorded up tothe given time. The related conditional quantum mas-ter equation can be derived, which is greatly simplifiedunder the Born–Markov approximation. Here,instead of using the “ n ”–resolved equation directly, itis convenient to introduce its χ –space counterpart via ρ ( χ, t ) ≡ P n e i nχ ρ ( n ) ( t ), where χ is the so–called count-ing field. Let the quantum master equation be formally ∂∂t ρ ( χ, t ) = − (i L + R χ ) ρ ( χ, t ) , (3)where L · ≡ [ H sys , · ] is the system Liouvillian, and R χ isthe dissipation superoperator to be specified soon. Here-after, we set unit of ~ = e = 1 for the Planck constantand electron charge, unless where is needed for clarity.To that end, let us recast the reduced density matrixin the vector notation, ρ = ( ρ , ρ , ρ , ρ , ρ , ρ ¯1¯1 , ρ ¯2¯2 , ρ ¯1¯2 , ρ ¯2¯1 ) T , (4)with ρ ss ′ ≡ h s ↑ | ρ | s ↑i and ρ ¯ s ¯ s ′ ≡ h s ↓ | ρ | s ↓i . There are 9 nonzero elements of the reduced density matrix inthe Coulomb blockade regime. The other 16 elements,describing coherence between different spin states, are allzeroes as the system–environment coupling consider heredoes not cause spin flip. The dissipation superoperator R χ in Eq. (3) is simply a 9 × R χ matrix is obtained explicitly R χ = L + T q ( χ ) − Γ R − Γ R − Γ L Γ R + T q ( χ ) 0 0 0 0 0 0 00 0 T q ( χ ) 0 0 0 0 0 00 Γ + Γ − Γ d + Γ − d − Γ L R + T q ( χ ) 0 0 00 0 0 0 0 0 T q ( χ ) 0 00 0 0 0 0 Γ + Γ − Γ d
00 0 0 0 0 Γ + Γ − d . (5)Here we have introducedΓ ± = π J ph (∆) (cid:20) ± ǫ ∆ coth (cid:16) ∆2 k B T (cid:17)(cid:21) , (6)Γ d = 12 (Γ R + γ ph + γ χ ) , (7)and γ ph = 2 π Ω ∆ J ph (∆) coth (cid:18) ∆2 k B T (cid:19) , (8) γ χ = ( p T − p T ) + 2 p T T q ( χ ) , (9)with q ( χ ) ≡ − e i χ , ∆ ≡ √ ǫ + 4Ω the Rabi frequencyof the double–dot system, k B the Boltzman constant, and T the temperature. The physical processes described bythe transfer elements in Eq. (5) are clear. Consider R χ , = − Γ L , R χ , = − Γ R , and their op-posite spin counterparts, as inferred from Eq. (5). Theseidentities agree with the sequential tunneling picture,with Γ L being the rate of electron tunneling from theleft electrode to system, and Γ R being that from systemto the right electrode; both via QD1.The parameter Γ ± [Eq. (6)] denotes the nonsecular el-ements for the population–to–coherence transfers: Γ + ≡ R χ , = R χ , , Γ − ≡ R χ , = R χ , , and their oppo-site spin counterparts, as denoted in Eq. (5). They arepurely due to the phonon bath environment.The parameter Γ d [Eq. (7)] is the total decoherencerate between two levels: Γ d ≡ R χ , = R χ , and theopposite spin counterparts, as denoted in Eq. (5). Thetotal decoherence rate is composed of not just Γ R dueto the electron depopulation to collector, but also γ ph [Eq. (8)] and γ χ [Eq. (9)], due to the phonon bath cou-pling and the QPC detection, respectively. Note that γ d ≡ γ χ =0 = ( √T − √T ) denotes the dephasing rateinduced by the existence of QPC ensemble.In the numerical demonstrations below, we set thephonon bath spectral density J ph ( ω ) = 2 ηωe − ω/ω c . Here,the dimensionless parameter η reflects the strength of dis-sipation and ω c is the Ohmic high energy cutoff. III. MONTE CARLO SIMULATION OF SINGLEELECTRON TUNNELING
Experimentally, the most intuitive method for measur-ing the FCS of electron transport is to count electronspassing one by one through the conductor. The real–time detection of single electron transport enables directevaluation of the probability distribution function of thenumber of electrons transferred through device within agiven time period. In addition to the current and the (0,1) I Q P C [ n A ] Time [ms] (b) T = 800mK (1,0) I Q P C [ n A ] (0,0)(0,1)(1,0) (0,0) (a) T =100mK
FIG. 2: Time traces of the QPC current fluctuations correspond to different charge states in the TDQD (shown on the right).The arrows indicate transitions where an electron is entering the QD1 from the left lead. (a) and (b) correspond to differenttemperatures T = 100 mK and T = 800 mK, respectively. Other parameters are: Γ R = 4Γ L = Ω = 20 kHz, ǫ = − η = 2 × − , and ω c = 0 . τ =0.002ms, such that the minimum characteristic time-scale of thesystem can be clearly resolved. shot noise, which are the first and second moments ofthis distribution, this method gives also access to higherorder cumulants.Here we outline the Monte Carlo simulation schemefor single electron tunneling through device. Let usstart with the quantum master equation (3) and denote L χ ≡ − i L − R χ there. The solution to the reduced statesimply reads ρ ( χ, t ) = e L χ δt ρ ( χ, t ), for an arbitrary ini-tial condition ρ ( χ, t ) = ρ ( t ) and finite time interval δt ≡ t − t . Its counterpart in the particle number “ n ”–space is obtained via the inverse Fourier transform ρ ( n ) ( t ) = Z π dχ π e L χ δt − i nχ ρ ( t ) ≡ U ( n, δt ) ρ ( t ) . (10)The involving propagator U ( n, δt ) is completely deter-mined by the dynamic structure of the master equation(3), regardless of the initial state. Therefore, we can nu-merically evaluate it by a “one–time task”, such as fastFourier transform, which results in an efficient real–timesimulation.Specifically, consider the evolution of state ρ ( t j ) at t j to ρ ( n j ) ( t j + τ ) at t j + τ : ρ ( n j ) ( t j + τ ) = U ( n j , τ ) ρ ( t j ), anddenote Pr( n j ) ≡ Tr[ ρ ( n j ) ( t j + τ )] that is the probabilityof having n j electrons passed through QPC during thetime interval [ t j , t j + τ ]. If the measurement is made butthe result is ignored, then the (mixture) state reads ρ ( t j + τ ) = X n j ρ ( n j ) ( t j + τ ) = X n j Pr( n j ) ρ c ( n j , t j + τ ) . (11)Here, ρ c ( n j , t j + τ ) = ρ ( n j ) ( t j + τ ) / Pr( n j ) is the nor-malized state, conditioned by the definite number of n j electrons having passed through QPC during [ t j , t j + τ ].The second equality of Eq. (11) implies that if we stochas-tically generate n j according to Pr( n j ) for each time in-terval [ t j , t j + τ ], and collapse the state definitely onto ρ c ( n j , t j + τ ), we have in fact simulated a particular re-alization for the selective state evolution conditioned onthe specific measurement results.For the output current in a particular real–time mea-surement, we have I QPC ( t ) = I ρ c00 + I ( ρ c11 + ρ c¯1¯1 )+ I ( ρ c22 + ρ c¯2¯2 )+ ξ ( t ) . (12)The first three terms determine the conditional evolutionof the charge state. The last term ξ ( t ) originates from theintrinsic noise of detector. Here, we consider in the diffu-sive regime, where ξ ( t ) is a Gaussian variable with zeromean value and the spectral density S ξ = 2 e h I QPC i , with h I QPC i the average stationary QPC current. Accordingly,we can stochastically generate n j , the number of elec-trons having passed through QPC during [ t j , t j + τ ], via n j = R t j + τt j dt ′ I QPC ( t ′ ) = [ I ρ c00 + I ( ρ c11 + ρ c¯1¯1 ) + I ( ρ c22 + ρ c¯2¯2 )] τ + dW ( t j ), where dW ( t j ) is the Wiener incrementduring [ t j , t j + τ ].A typical example of simulated real–time detector cur-rent I QPC is displayed in Fig. 2 for (a) T = 100 mK and(b) T = 800 mK, respectively. The temperatures aremuch smaller than the bias voltage across the TDQD, aswell as the interdot and the intradot charging energies.Thus, electrons are transported in one direction, i.e. fromthe left lead to the right one. The corresponding chargestates ( n , n ) of the TDQD, with n and n being theexcess electrons in QD1 and QD2, respectively, are shown C o un t s N T = C o un t s N (b) T = (a) FIG. 3: Statistical distribution of the number N of electronsentering the TDQD during a given measurement time spanof t c = 3 ms for (a) T = 100 mK, and (b) T = 800 mK,respectively. The other parameters are the same as in Fig. 2.The solid line shows the distribution calculated from Eq. (16). on the right. We choose rates of tunneling to the left andright leads as Γ L = 5 kHz and Γ R = 20 kHz, such that allthe tunneling rates are within the bandwidth ( ∼
30 kHz)of the QPC detector. The QPC conductance (withoutexcess electrons in TDQD) is G = 0 . e /h , and QPCbias voltage V QPC = 0 .
15 mV, which corresponds to aQPC current I ≈ .
12 nA. The conductance is assumedto decrease by 4% or 2%, when an electron occupies onthe QD1 or QD2, respectively. The corresponding QPCcurrents are I ≈ .
115 nA and I ≈ .
118 nA. The rateof dephasing due to QPC charge detection is then de-termined as γ d = e ( √ I − √ I ) ≈
75 kHz.In Fig. 2, each arrow indicates the tunneling of one elec-tron from the left lead into QD1. Accordingly, a chargestate transition (0,0) → (1,0) occurs. Remarkably, we ob-serve a clear signature of electron bunching, especiallywhen temperature is high, as shown in Fig. 2(b). Thisphenomenon can be understood in terms of dynamicalchannel blockade, as will be explained later in more de-tail. For a thorough analysis of the statistical propertiesfor electron transport through the TDQD device, we re-sort to the FCS. It is enabled by the probability distri-bution for the number N of electrons entering the QD1from the left lead (the down steps indicated by arrowsin Fig. 2) during a given time span t c , by counting thenumber of electrons from the time traces analogous tothe ones shown in Fig. 2. The obtained current distribu-tions for T = 100 mK and T = 800 mK are displayed byhistograms in Fig. 3(a) and (b), respectively, by repeat-ing this counting procedure on one thousand independenttraces with the measurement time span of t c = 3 ms. IV. FULL COUNTING STATISTICS THEORY
We consider hereafter the counting statistics of cur-rent through the TDQD device. The central quantityis then the probability that a given N electrons havepassed through the device during the counting measure-ment time span t c . This probability is related to theparticle–number–resolved reduced density operator as P ( N, t c ) ≡ Tr ρ ( N ) ( t c ), where the trace is over the systemdegrees of freedom. Note here “ N ” denotes the numberof electrons transferred through the device, rather thanthe QPC detector considered earlier. From the knowl-edge of these probabilities one can easily derive not onlythe current and noise, but all the cumulants of the cur-rent distribution. The associated cumulant generatingfunction (CGF) g ( ϕ ) reads e g ( ϕ ) = X N P ( N, t c ) e − i Nϕ , (13)where ϕ is the counting field on a specified TDQD lead.All cumulants of the current can be obtained from theCGF by performing derivatives with respect to the count-ing field h I k i = − ( − i ∂ ϕ ) k g ( ϕ ) | ϕ =0 . (14)The first three cumulants are related to the average cur-rent, the (zero-frequency) current noise, and the skew-ness, respectively.To evaluate the CGF, let us consider ̺ ( ϕ, t ) ≡ P N ρ ( N ) ( t ) e i Nϕ . Its equation of motion reads˙ ̺ ( ϕ, t ) ≡ L ϕ ̺ ( ϕ, t ) . (15)The involving generator L ϕ is completely determinedby the associated conditional master equation, whichis similar to Eq. (3) but with the number of electronspassing through the TDQD device being resolved. Theformal solution to Eq. (15) is ̺ ( ϕ, t ) = e L ϕ t ̺ ( ϕ, g ( ϕ ) = − ln { Tr ̺ ( ϕ, t c ) } . Actually, we are most interested in thezero-frequency limit, i.e. the counting time t c is muchlonger than the time of tunneling through the system.The CGF then simplifies to g ( ϕ ) = − λ min ( ϕ ) t c , where λ min ( ϕ ) is the minimal eigenvalue of L ϕ that satisfies λ min ( ϕ → → With the knowledge of CGF,the distribution function can be readily obtained via P ( N ) = Z π dϕ π e − g ( ϕ ) − i Nϕ . (16) V. RESULTS AND DISCUSSIONS
In what follows, we focus our analysis on electron tun-neling from the left lead to QD1. The relevant N isthen the number of electrons entering the dot from leftlead, and ϕ is the corresponding counting field. Note thesame calculations apply to the counting for the right lead.The numerical results for the probability distribution areplotted by the solid lines in Fig. 3. It shows a strikingagreement with the histograms obtained by Monte Carlosimulation; thus, it also verifies the validity of our MonteCarlo method for real–time measurement. The two dis-tributions in Fig. 3(a) and (b) are rather different. Thelatter shows a broader and more asymmetric distributionthan the former. We characterize the differences quanti-tatively based on the FCS analysis, as follows.Consider first the situation without electron-phononinteraction ( η = 0). The first current cumulant gives theaverage current h I i = 2Γ L Γ R / Γ eff , with Γ eff ≡ L + Γ R the total effective tunneling width. It is independentof level detuning ǫ , interdot coupling Ω, and QPC chargedetection induced dephasing γ d . In contrast to the cur-rent, valuable information can be extracted in the secondcumulant (zero-frequency shot noise). By expressing it interms of the Fano factor F ≡ h I i / h I i , we readily obtain F = 1 − L Γ R Γ + 2Γ Γ R ( γ d + Γ R ) + 4 ǫ ( γ d + Γ R )Γ Ω . (17)It thus allows us to get more knowledge than the currenton the processes involved in the electronic transport. InFig. 4(a), the Fano factor F is plotted against the leveldetuning ǫ for different QPC–induced dephasing rates γ d .Noticeably, a significantly enhanced Fano factor is ex-pected when the dot levels are far from resonance. In thiscase, electron transitions between the two dots are sup-pressed. For instance, if an electron is tunneled into QD2,it will dwell on it for a long time. In the strong Coulombblockade regime, the electron in QD2 will block the cur-rent until it is removed and tunneled out to the rightlead. Consequently, a mechanism of dynamical channelblockade is developed, and the system exhibitsstrong electron bunching behavior, which leads eventu-ally to a profound super–Poissonian noise.It is worth noting that a strong super–Poissonian noiseis more readily achieved for a small Ω [cf. Eq. (17)]. Inthis case, electron tunneling between the two dots issuppressed, which would enhance electron localization ofelectron in QD2. Electrons thus tend to be transferredin bunches. Remarkably, as Ω → Similar results were alsoreported in Ref. 39, where counting statistics for electrontransport through a double–dot Aharonov–Bohm inter-ferometer was investigated. However, in that case the di-vergence is closely related to a separation of the Hilbertspace of the double–dot into disconnected subspaces thatcontain the spin singlet and triplet states for double oc-cupancy.The QPC charge detection has a two–fold effect on themeasured system. On one hand, it causes dephasing be-tween the two dot–levels. Generally, the dephasing mech-anism leads to the suppression of noise, which explainsthe γ d –dependence of the Fano factor shown in Fig. 4,i.e. the noise is reduced with rising dephasing rate, par-ticularly in the regime far from resonance. On the otherhand, the measurement gives rise to the so–called quan-tum “Zeno” effect, which dominates in the regimeof large dephasing rate, and results in a strong dynamiccharge blockade behavior. The noise is finally enhancedwith increasing dephasing rate, as we have checked (notshown in Fig. 4). -6 -4 -2 0 2 4 60.00.51.01.52.02.53.03.5 d =0 d =25kHz d =50kHz d =75kHz d =100kHz F = I / I I / I (a) -6 -4 -2 0 2 4 6-0.6-0.4-0.20.00.20.40.6 S = (b) FIG. 4: (a) Fano factor and (b) normalized skewness ver-sus level detuning for different QPC–induced dephasing rates.The electron charge is set to be e = 1. The temperature is T =100mK, and the other parameters are the same as in Fig. 2,but in the absence of phonon bath; i.e., η = 0. The Fano factor studied so far proves to be much sen-sitive than the average current, it only reveals, however,limited information about the QPC charge detection in-duced dephasing mechanism. We thus expect more infor-mation to be extracted in the next order cumulant, i.e.the skewness. The numerical results for the normalizedskewness S ≡ h I i / h I i is displayed in Fig. 4(b) as a func-tion of level detuning ǫ . Without QPC charge detectioninduced dephasing, the skewness reaches the maximumwhen the dot levels are in resonance ( ǫ = 0). As the de-phasing rate grows, it turns into a local minimum. At theedges of the resonance a double maximum structure sym-metric around ǫ = 0 is observed. The local maxima shiftaway from resonance with increasing dephasing rates, asclearly demonstrated in Fig. 4(b).Now let us turn to the influence of phonon heat baththat induces dephasing between two dots. We will revealthe essential difference between the dephasing inducedvia QPC charge detection and that by phonon coupling.The former can be modified via the coupling between theTDQD and the QPC, while the latter is generated withemission and absorption of phonons and increases withrising temperature. Here, we limit our discussions to thetemperatures well below the Coulomb charging energiesand the bias voltage. Therefore the temperature actssolely due to the coupling to the phonon heat bath.The calculated Fano factor and normalized skewnessversus level detuning are plotted in Fig. 5 for differenttemperatures. Increasing temperature results in the en-hancement of phonon–bath induced dephasing, as theemission and absorption of phonons occur more fre-quently. Analogous to the QPC detection, it gives riseto a mechanism of localization in QD2, and consequentlyto a dynamical channel blockade. Eventually, electrontransport through QD1 occurs in bunches during lapsesof time when the QD2 is empty. This is confirmed bythe real–time trajectory [cf. Fig. 2(b)]. Thus the noise in- -6 -4 -2 0 2 4 60.51.01.52.0 T=100mK
T=300mK
T=500mK
T=800mK F = I / I I / I (a) -6 -4 -2 0 2 4 60.00.51.01.5 S = (b) FIG. 5: (a) Fano factor and (b) normalized skewness versuslevel detuning for different temperatures with γ d = 75 kHz.The electron charge is set to be e = 1. The other parametersare the same as in Fig. 2, including those of phonon heat bathof η = 2 × − and ω c = 0 . creases with rising temperature, as displayed in Fig. 5(a).If the TDQD is coupled only to the QPC, the cumu-lants are symmetric around ǫ = 0, as shown in Fig. 4.However, with non–zero coupling to the heat bath thenoise exhibits a clear asymmetry (see Fig. 5). This isessentially due to another consequences of phonon cou-pling: The phonon mediated transition can partially re-solve the dynamical charge blockade. For instance, atzero temperature and ǫ >
0, spontaneous phonon emis-sion can partially lift the localization caused by dephas-ing, and the spectrum turns out to be asymmetric. Simi-lar argument applies to the regime of finite temperature,where both phonon absorption and emission take place.The fact that emission is more likely than absorption ex-plains the observed asymmetry.The skewness of the current distribution, which wasnot explored in Ref. 24, is found to be more sensitive tothermal phonon bath–induced dephasing. Without cou-pling to the phonon bath, the skewness shows a symmet-ric double maximum structure. The spectral becomesasymmetric in the presence of electron–phonon interac-tion. With increasing temperature, the phonon absorp-tion takes place more frequently at ǫ < ǫ >
VI. SUMMARY
We have investigated electron transport through a T–shaped double quantum dot system by utilizing a quan-tum master equation approach. The major advantage ofthe present approach is its simplicity of treating prop-erly the dephasing mechanism of the QPC charge detec-tion and that due to an external heat bath. In addition,this approach has the merit of dealing with other sourcesof dephasing, such as that entailed by anti-resonances. Particularly, based on the Monte Carlo scheme, real-timedetection of single electron tunneling is simulated by ex-ploiting the sensitivity of a current, passing through anearby quantum point contact, to the fluctuating chargeon the quantum dots. Owing to the interplay betweenthe Coulomb interactions and the dephasing mechanisms,a strong bunching behavior in the charge transfer wasdetected, which leads eventually to a super–Poissoniannoise.Furthermore, full counting statistics of the transportcurrent is analyzed based on the probability distribu-tion, which is determined by ensemble average over alarge number of single trajectories. It is demonstratedthat the dephasing mechanism of the QPC charge detec-tion and that owing to the external heat bath give riseto distinct and intriguing features. It thereby enablesus to achieve a clear identification of different dephas-ing sources. Investigations of various processes involvedin the electronic transport through similar devices arehighly desirable in experiments.
Acknowledgments
Support from the National Natural Science Foundationof China under Grants No. 10904128, and the ResearchGrants Council of the Hong Kong Government (GrantNo. 604709) are gratefully acknowledged. ∗ Electronic address: [email protected] † Electronic address: [email protected] Y. M. Blanter and M. B¨uttiker, Phys. Rep. , 1 (2000). Y. V. Nazarov,
Quantum Noise in Mesoscopic Physics (Kluwer, Dordrecht, 2003). L. S. Levitov, H. W. Lee, and G. B. Lesovik, J. Math.Phys. , 4845 (1996). D. A. Bagrets and Y. V. Nazarov, Phys. Rev. B , 085316(2003). W. Lu, Z. Ji, L. Pfeiffer, K. W. West, and A. J. Rimberg,Nature , 422 (2003). J. Bylander, T. Duty, and P. Delsing, Nature , 361 (2005). R. Schleser, E. Ruh, T. Ihn, K. Ensslin, D. C. Driscoll, andA. C. Gossard, Appl. Phys. Lett. , 2005 (2004). L. M. K. Vandersypen, J. M. Elzerman, R. N. Schouten,L. H. W. van Beveren, R. Hanson, and L. P. Kouwenhoven,Appl. Phys. Lett. , 4394 (2004). S. Gustavsson, R. Leturcq, B. Simovic, R. Schleser, T. Ihn,P. Studerus, K. Ensslin, D. C. Driscoll, and A. C. Gossard,Phys. Rev. Lett. , 076605 (2006). T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama,Science , 1634 (2006). L. Y. Chen and C. S. Ting, Phys. Rev. B , 4534 (1991). W. G. van der Wiel, S. D. Franceschi, J. M. Elzerman,T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Rev.Mod. Phys. , 1 (2003). T.-S. Kimand and S. Hershfield, Phys. Rev. B , 245326(2001). P. S. Cornaglia and D. R. Grempel, Phys. Rev. B ,075305 (2005). I. Djuric, B. Dong, and H. L. Cui, Appl. Phys. Lett. ,032105 (2005). A. Nauen, I. Hapke-Wurst, F. Hohls, U. Zeitler, R. J. Haug,and K. Pierz, Phys. Rev. B , 161303 (2002). S.-K. Wang, J. S. Jin, and X.-Q. Li, Phys. Rev. B ,155304 (2007). J. Y. Luo, X.-Q. Li, and Y. J. Yan, Phys. Rev. B ,085325 (2007). J. Y. Luo, X.-Q. Li, and Y. J. Yan, J. Phys.: Cond. Matt. , 345215 (2008). K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Science , 1313 (2002). X. Q. Li, P. Cui, and Y. J. Yan, Phys. Rev. Lett. ,066803 (2005). X. Q. Li, J. Y. Luo, Y. G. Yang, P. Cui, and Y. J. Yan,Phys. Rev. B , 205304 (2005). T. Brandes, Phys. Rep. , 315 (2005). G. Kießlich, E. Sch¨oll, T. Brandes, F. Hohls, and R. J.Haug, Phys. Rev. Lett. , 206602 (2007). R. Aguado and T. Brandes, Phys. Rev. Lett. , 206601(2004). S. A. Gurvitz, Phys. Rev. B , 15215 (1997). C. W. Groth, B. Michaelis, and C. W. J. Beenakker, Phys.Rev. B , 125315 (2006). C. Flindt, T. Novotny, and A.-P. Jauho1, Europhys. Lett. , 475 (2005). G. Kießlich, P. Samuelsson, A. Wacker, and E. Sch¨oll,Phys. Rev. B , 033312 (2006). A. N. Korotkov and D. V. Averin, Phys. Rev. B , 165310(2001). S. A. Gurvitz, L. Fedichkin, D. Mozyrsky, and G. P.Berman, Phys. Rev. Lett. , 066801 (2003). J. Y. Luo, H. J. Jiao, F. Li, X.-Q. Li, and Y. J. Yan, J.Phys.: Cond. Matt. , 385801 (2009). A. Cottet, W. Belzig, and C. Bruder, Phys. Rev. Lett. ,206801 (2004). M. Gattobigio, G. Iannaccone, and M. Macucci, Phys. Rev.B , 115337 (2002). R. S´anchez, G. Platero, and T. Brandes, Phys. Rev. Lett. , 146805 (2007). G. Kießlich, H. Sprekeler, and E. Sch¨oll, Semicond. Sci.Technol. , S37 (2004). A. Cottet and W. Belzig, Europhys. Lett. , 405 (2004). F. Li, H. J. Jiao, J. Y. Luo, X.-Q. Li, and S. A. Gurvitz,Physica E , 1707 (2009). D. Urban and J. K¨onig, Phys. Rev. B , 165319 (2009). A. Braggio, C. Flindt, and T. Novotn´y, J. Stat. Mech.:Theory Exp. , P01048. L. E. F. F. Torres, H. M. Pastawski, and E. Medina, Eu-rophys. Lett.73