Counting statistics of coherent population trapping in quantum dots

Counting statistics of coherent population trapping in quantum dots

Groth, C.W.; Michaelis, B.D.; Beenakker, C.W.J.

Citation

Groth, C. W., Michaelis, B. D., & Beenakker, C. W. J. (2006). Counting statistics of coherent

population trapping in quantum dots. Physical Review B, 74(12), 125315.

doi:10.1103/PhysRevB.74.125315

Version:

Not Applicable (or Unknown)

License:

Leiden University Non-exclusive license

Downloaded from:

https://hdl.handle.net/1887/76585

Counting statistics of coherent population trapping in quantum dots

C. W. Groth, B. Michaelis, and C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

共Received 26 May 2006; published 22 September 2006兲

Destructive interference of single-electron tunneling between three quantum dots can trap an electron in a coherent superposition of charge on two of the dots. Coupling to external charges causes decoherence of this superposition, and in the presence of a large bias voltage each decoherence event transfers a certain number of electrons through the device. We calculate the counting statistics of the transferred charges, finding a crossover from sub-Poissonian to super-Poissonian statistics with increasing ratio of tunnel and decoherence rates. DOI:10.1103/PhysRevB.74.125315 PACS number共s兲: 73.50.Td, 73.23.Hk, 73.63.Kv

I. INTRODUCTION

The phenomenon of coherent population trapping, origi-nating from quantum optics, has recently been recognized as a useful and interesting concept in the electronic context as well.1,2_{An all-electronic implementation, proposed in Ref.3,}

is based on destructive interference of single-electron tunnel-ing between three quantum dots 共see Fig. 1兲. The trapped

state is a coherent superposition of the electronic charge in two of these quantum dots, so it is destabilized as a result of decoherence by coupling to external charges. In the limit of weak decoherence one electron is transferred on average through the device for each decoherence event.

In an experimental breakthrough,4,5 _{Gustavsson et al.}

have now reported real-time detection of single-electron tun-neling, obtaining the full statistics of the number of trans-ferred charges in a given time interval. The first two mo-ments of the counting statistics give the mean current and the noise power, and higher moments further specify the corre-lations between the tunneling electrons.6 _{This recent}

devel-opment provides a motivation to investigate the counting sta-tistics of coherent population trapping, going beyond the first moment studied in Ref.3.

Since the statistics of the decoherence events is Poisso-nian, one might surmise that the charge counting statistics would be Poissonian as well. In contrast, we find that charges are transferred in bunches instead of independently as in a Poisson process. The Fano factor共ratio of noise power and mean current兲 is three times the Poisson value in the limit of weak decoherence. We identify the physical origin of this super-Poissonian noise in the alternation of two decay pro-cesses共tunnel events and decoherence events兲 with very dif-ferent time scales—in accord with the general theory of Belzig.7 _{For comparable tunnel and decoherence rates the}

noise becomes sub-Poissonian, while the Poisson distribution is approached for strong decoherence.

The analysis of Ref.3 was based on the Lindblad master equation for electron transport,8,9_{which determines only the}

average number of transferred charges. The full counting sta-tistics can be obtained by an extension of the master equation.10–12 _{In spite of the added complexity, we have}

found analytical solutions for the second moment at any de-coherence rate and for the full distribution in the limit of weak or strong decoherence.

II. MODEL

The system under consideration, studied in Ref.3, is de-picted schematically in Fig. 1. It consists of three tunnel-coupled quantum dots connected to two electron reservoirs. In the limit of large bias voltage, which we consider here, electron tunneling from the source reservoir into the dots and from the dots into the drain reservoir is irreversible. We as-sume that a single level in each dot lies within range of the bias voltage. We also assume that due to Coulomb blockade there can be at most one electron in total in the three dots. The basis states, therefore, consist of the state兩0典 in which all dots are empty, and the states兩A典, 兩B典, and 兩C典 in which one electron occupies one of the dots.

The time evolution of the density matrix␳for the system is given by the Lindblad-type master equation,8,9

d␳ dt = − i关H,␳兴 +X=A,B,C,

兺

␾A,␾B,␾C

共

LX␳LX † −1₂LX † LX␳− 1 2␳LX † LX

兲

. 共1兲 The Hamiltonian H = T兩C典具A兩 + T兩C典具B兩 + H.c. 共2兲

is responsible for reversible tunneling between the dots, with tunnel rate T. For simplicity, we assume that the three energy levels in dots A, B, and C are degenerate and that the two tunnel rates from A to C and from B to C are the same. The quantum jump operators

LA=

冑

⌫兩A典具0兩, LB=

冑

⌫兩B典具0兩, LC=

冑

⌫兩0典具C兩, 共3兲 model irreversible tunneling out of and into the reservoirs, with a rate⌫ 共which we again take the same for each dot兲. Finally, the quantum jump operators

L_␾

X=

冑

⌫␾兩X典具X兩, X = A,B,C, 共4兲

model decoherence due to charge noise with a rate⌫_␾. As a basis for the density matrix we use the four states

兩e0典 = 2−1/2共兩A典 − 兩B典兲,

兩e1典 = 2−1/2共兩A典 + 兩B典兲,

兩e2典 = 兩C典, 兩e3典 = 兩0典. 共5兲

If the initial state is 兩0典具0兩 most of the coefficients of ␳ re-main zero. We collect the five nonzero real variables in a vector

v =共␳00,␳11,␳22,␳33,Im␳02兲T, 共6兲

whose time evolution can be expressed as

dv/dt = Xv, 共7兲 X =

冢

−⌫_␾/2 ⌫_␾/2 0 ⌫ − 23/2_T ⌫␾/2 −⌫␾/2 0 ⌫ 0 0 0 −⌫ 0 23/2T 0 0 ⌫ − 2⌫ 0 21/2T 0 − 21/2T 0 −⌫/2 − ⌫_␾

冣

. 共8兲

It is our goal to determine the full counting statistics, being the probability distribution P共n兲 of the number of transferred charges in time t. Irrelevant transients are removed by taking the limit t→⬁. The associated cumulant generating function F共␹兲 is related to P共n兲 by

exp关− F共␹兲兴 =

兺

n=0 ⬁

P共n兲exp共in␹兲. 共9兲

From the cumulants

Ck= −共− i⳵␹兲kF共␹兲兩␹=0 共10兲

we obtain the average current I = eC1/ t and the zero-frequency noise S = 2e2C2/ t, both in the limit t→⬁. The Fano factor is

defined as␣= C2/ C1.

As described in Refs. 11 and 12, in order to calculate F共␹兲 one multiplies coefficients of the rate matrix X which are associated with tunneling into one of the reservoirs共the right one in our case兲, by counting factors ei␹_{. This leads to the}

␹-dependent rate matrix

L共␹兲 =

冢

−⌫_␾/2 ⌫_␾/2 0 ⌫ − 23/2T ⌫␾/2 −⌫␾/2 0 ⌫ 0 0 0 −⌫ 0 23/2T 0 0 ⌫ei␹ − 2⌫ 0 21/2_{T 0 − 2}1/2_{T 0 −⌫/2 − ⌫} ␾

冣

. 共11兲

The cumulant generating function for t→⬁ can then be ob-tained from the eigenvalue⌳min共␹兲 of L共␹兲 with the smallest

absolute real part,11,12

F共␹兲 = − t⌳_min共␹兲. 共12兲 III. RESULTS

A. Fano factor

Low order cumulants can be calculated by perturbation theory in the counting parameter␹. The calculation is

out-lined in the Appendix. For the average current we find

I = 4e⌫T

2

⌫2_{+ 14T}2_{+ 2⌫⌫}

␾共1 + 2T2/⌫␾2兲

, 共13兲

in agreement with Ref.3. By calculating the noise power and dividing by the mean current we obtain the Fano factor

␣=关⌫4_{+ 148T}4_{+ 4⌫}2_共⌫ ␾ 2_{+ 4T}2_{+ 12T}4_/⌫ ␾ 2_{兲 + 共16T}2_{+ 2⌫}2_兲␤兴 ⫻关⌫2_{+ 14T}2_+␤兴−2_{, 共14兲}

GROTH, MICHAELIS, AND BEENAKKER PHYSICAL REVIEW B 74, 125315共2006兲

␤= 2⌫⌫␾共1 + 2T2/⌫␾2兲. 共15兲

In Fig.2the Fano factor has been plotted as a function of ⌫⌽/ T for three different values of⌫/T. The dependence of

the Fano factor on the decoherence rate is nonmonotonic, crossing over from super-Poissonian 共␣⬎1兲 to Poissonian 共␣= 1兲 via a region of sub-Poissonian noise 共␣⬍1兲. To ob-tain a better understanding of this behavior, we study sepa-rately the regions of weak and strong decoherence.

B. Weak decoherence

For decoherence rate⌫_␾⌫,T we have the limiting be-havior

I→ e⌫_␾, ␣→ 3 − ⌫_␾

冉

17 ⌫ +

⌫

T2

冊

. 共16兲

Hence one charge is transferred on average per decoherence event, but the Fano factor is three times the value for inde-pendent charge transfers.

There exists a simple physical explanation for this behav-ior. For zero decoherence the system becomes trapped in the state兩e0典. The system is untrapped by “decoherence events,”

which occur randomly at the rate⌫_␾ according to Poisson statistics. If⌫_␾is sufficiently small there is enough time for the system to decay into the trapped state between two sub-sequent events, so they can be viewed as independent. The super-Poissonian statistics appears because a single decoher-ence event can trigger the transfer of more than a single charge.

The probability of n electrons being transferred in total as a consequence of one decoherence event is

R1共n兲 = 1

2n+1, 共17兲

since a decoherence event projects the trapped state兩e0典 onto

itself or onto兩e1典 with equal probabilities 1/2 and each

elec-tron subsequently entering the dots has a 50% chance of getting trapped in the state兩e₀典.

The number of electrons which have been transferred due to exactly k decoherence events has distribution Rk_{共n兲, the} 共k−1兲th convolution of R1_{共n兲 with itself. We obtain}

Rk共n兲 = 1 2n+ki

兺

₀=0 n

兺

i₁=0 i₀ ¯

兺

i_k−2=0 i_k−3 1 = 1 2n+k

冉

n + k − 1 n

冊

. 共18兲 By definition, R0共n兲 =␦n,0=

再

1 for n = 0, 0 for n⬎ 0 共19兲 being the distribution of the transferred charges after no de-coherence events have occurred.

The decoherence events in a time t have a Poisson distri-bution,

P_Poisson共k兲 = e−t⌫␾_{共t⌫␾兲}k_{/k ! . 共20兲} Combining with Eq.共18兲 we find the probability that n

elec-trons have been transferred during a time t,

P共n兲 =

兺

k=0 ⬁ P_Poisson共k兲Rk_共n兲 =

兺

k=1 ⬁ e−t⌫␾共t⌫_␾兲k 2n+kk!

冉

n + k − 1 n

冊

+ e −t⌫␾␦_n,0. 共21兲

The corresponding cumulant generating function is

F共␹兲 = t⌫_␾− t⌫␾

2 − ei␹, 共22兲 which gives rise to the cumulants

C1= t⌫␾, C2= 3t⌫␾, C3= 13t⌫␾, 共23兲

in agreement with Eq.共16兲.

The probability distribution共22兲 has been found by Belzig

in a different model.7_{As shown in that paper, this}

superpo-sition of Poisson distributions with Fano factor 3 arises ge-nerically whenever there are two transport channels with very different transport rates共in our case slow transport via the trapped state 兩e0典, and fast transport via the untrapped

state兩e₁典兲.

C. Strong decoherence

We show that Poisson statistics of the transferred charges is obtained for strong decoherence. Consider the evolution equation 共7兲 of the system. For ⌫_␾⌫,T the coefficients X00, X01, X10, and X11will ensure thatv0is equal tov1after

a time which is short compared to the other characteristic times of the system. The trapped and the nontrapped states will be equally populated. Let us therefore define

v

⬘

=共␳00+␳11,␳22,␳33,Im␳02兲T 共24兲

and use␳00=␳11=v0

⬘

/ 2. The evolution ofv

⬘

is governed by dv

⬘

/ dt = X

⬘

v

⬘

, with X

⬘

=

冢

0 0 2⌫ − 23/2T 0 −⌫ 0 23/2T 0 ⌫ − 2⌫ 0 2−1/2T − 21/2T 0 −⌫/2 − ⌫_␾

冣

. 共25兲 The rate matrix L

⬘

共␹兲 is obtained by multiplying X₁₂

⬘

by the counting factor ei␹_{. An analytic expression can be found for} the smallest eigenvalue⌳_min

⬘

共␹兲 of L

⬘

共␹兲, leading to the cu-mulant generating function

F共␹兲 =2T 2 ⌫␾t共1 − e i␹_{兲 共26兲} of a Poisson distribution. IV. CONCLUSION

In conclusion, we have shown that coherent population trapping in a purely electronic system has a highly nontrivial statistics of transferred charges. Depending on the ratios of decoherence rate and tunnel rates, both super-Poissonian and sub-Poissonian statistics are possible. We have obtained ex-act analytical solutions for the crossover from sub- to super-Poissonian charge transfer, and have calculated the full dis-tribution in the limits of weak and strong decoherence. We hope that the rich behavior of this simple device will moti-vate experimental work along the lines of Refs. 4 and5. It might be also interesting to examine non-Markovian effects in this device along the lines of Ref.13.

ACKNOWLEDGMENTS

We thank L. Ament for useful discussions. This research was supported by the Dutch Science Foundation NWO/FOM. C.W.G. acknowledges support from the Cu-sanuswerk Foundation.

APPENDIX: DERIVATION OF THE FANO FACTOR To derive the result共14兲 for the Fano factor it is sufficient

to know the cumulant generating function to second order in

␹. The eigenvalues of the rate matrix L共␹兲 defined in Eq. 共11兲

have the expansion

␭ = ␭0+␭1␹+␭2␹2+ O共␹3兲. 共A1兲

We seek the eigenvalue with the smallest real part in absolute value. That eigenvalue has␭0= 0. We also express the

eigen-vector w corresponding to␭ and the matrix itself in a power series in␹,

w = w₀+ w₁␹+ w₂␹2+ O共␹3兲, 共A2兲

L = L0+ L1␹+ L2␹2+ O共␹3兲. 共A3兲

Inserting the above expansions into the eigenvalue equation

Lw =␭w yields the following relationships of, respectively,

zero, first, and second order:

L0w0= 0, 共A4兲

L1w0+ L0w1=␭1w0, 共A5兲 L2w0+ L1w1+ L0w2=␭2w0+␭1w1. 共A6兲

The coefficients Lkare known, while wkand␭kremain to be found by solving these equations sequentially. The first two cumulants then follow from

C1= − it␭1, C2= − 2t␭2. 共A7兲

In an analog way it is possible to calculate higher cumulants.

1_{T. Brandes, Phys. Rep. 408, 315共2005兲.}

2_{J. Siewert and T. Brandes, Adv. Solid State Phys. 44, 181共2004兲.} 3_{B. Michaelis, C. Emary, and C. W. J. Beenakker, Europhys. Lett.}

73, 677共2006兲.

4_{S. Gustavsson, R. Leturcq, B. Simovič, R. Schleser, T. Ihn, P.}

Studerus, K. Ensslin, D. C. Driscoll, and A. C. Gossard, Phys. Rev. Lett. 96, 076605共2006兲.

5_{S. Gustavsson, R. Leturcq, B. Simovič, R. Schleser, P. Studerus,}

T. Ihn, K. Ensslin, D. C. Driscoll, and A. C. Gossard, cond-mat/ 0605365共unpublished兲.

6_{Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1共2000兲.}

7_{W. Belzig, Phys. Rev. B 71, 161301共R兲 共2005兲.} 8_{Yu. V. Nazarov, Physica B 189, 57共1993兲.} 9_{S. A. Gurvitz, Phys. Rev. B 57, 6602共1998兲.}

10_{Mahn-Soo Choi, Francesco Plastina, and Rosario Fazio, Phys.}

Rev. Lett. 87, 116601共2001兲.

11_{D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. B 67, 085316}

共2003兲.

12_{G. Kießlich, P. Samuelsson, A. Wacker, and E. Schöll, Phys. Rev.}

B 73, 033312共2006兲.

13_{Alessandro Braggio, Jürgen König, and Rosario Fazio, Phys. Rev.}

Lett. 96, 026805共2006兲.

GROTH, MICHAELIS, AND BEENAKKER PHYSICAL REVIEW B 74, 125315共2006兲