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Effects of observation on quantum interference in a laterally coupled double quantum dot using a quantum dot charge sensor T. Kubo a,, Y. Tokura a,b, T. Hatano a, S. Amaha a, S. Teraoka a, S. Tarucha a,c a

Quantum Spin Information Project, ICORP-JST, Atsugi-shi, Kanagawa 243-0198, Japan NTT Basic Research Laboratories, NTT Coporation, Atsugi-shi, Kanagawa 243-0198, Japan c Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku,Tokyo 113-8656, Japan b

a r t i c l e in f o

a b s t r a c t

Article history: Received 14 September 2009 Received in revised form 13 November 2009 Accepted 13 November 2009 Available online 24 November 2009

We theoretically investigated the effects of the observation by a quantum dot charge sensor on an Aharonov–Bohm effect in a laterally coupled double quantum dot using the interpolative 2nd-order perturbation theory. In particular, we introduce the notion of the coherent indirect coupling, which characterizes the strength of the indirect coupling between two quantum dots via the reservoir. As the Coulomb interaction VS for the charge sensing increases, the linear conductance through a double quantum dot decreases because of the many-body correlation effect. The visibility of Aharonov–Bohm oscillation in the linear conductance behaves super-linearly for weak sensing interaction regime and sub-linearly for strong sensing interaction regime. & 2009 Elsevier B.V. All rights reserved.

Keywords: Laterally coupled double quantum dot Quantum dot charge sensor Aharonov–Bohm effect Visibility

1. Introduction Particle-wave duality is one of the most essential concepts in quantum theory and provides the impressive illustration of Bohr’s complementarity principle [1]. The wave characteristic arises if the particles are not detected in the different possible paths. By introducing the which-path detector [2], we can determine the actual path traversed by the particle necessarily involved with a coupling to an environment. As a result, the observation by the which-path detector gives rise to the dephasing. Mesoscopic systems can be often used to study the interplay between interference and dephasing of electrons. Nano-fabrication and low-temperature measuring techniques using semiconductors have enabled us to observe a variety of coherent effects of electrons such as Aharonov–Bohm (AB) [3–6], Fano [7], and Kondo effects [8–10]. The AB effect has been proven to be a convenient way to observe interference fringe in mesoscopic systems since it provides an experimentally straightforward way of controlling the phase. Recently, the AB effect is more intriguing in an AB interferometer containing a laterally coupled double quantum dot (DQD) [11–14], providing a refreshing subject for both theoretical and experimental studies. In particular, the quantum interference effects are sensitive to the inter-dot coherent coupling. In this work, we investigate the effects of observation on quantum interference in a laterally coupled DQD due to a Corresponding author.

E-mail address: [email protected] (T. Kubo). 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.11.059

coupling with a quantum dot charge sensor as shown in Fig. 1. In particular, we introduce the coherent indirect coupling, which characterizes the strength of the indirect coupling between two quantum dots (QDs) via the reservoir [16,11,17]. This notion is very essential for the quantum interference effect such as AB and Fano effects [11]. To analyze the Coulomb interaction effect, we employ the interpolative 2nd-order perturbation theory [15]. This paper is organized as follows. In Section 2, a standard tunneling Hamiltonian is employed to describe a laterally coupled triple QD and the expressions of the linear conductance are derived. In Section 3, we discuss the AB oscillations in the linear conductance, and we show the change in the visibility of the AB oscillation by the QD charge sensor. Finally, we summarize our results in Section 4.

2. Model and formulation We consider the AB interferometer containing two QDs with a capacitively coupled QD charge sensor as shown in Fig. 1. The source and drain reservoirs of the charge sensor (QD3) are separated from the reservoirs coupled to the two QDs (QD1 and QD2). We neglect the spin degree of freedom, and only a single energy level in each QD is assumed to be relevant. We model this system with the Hamiltonian H ¼ HR þ HDQD þ HS þ HI þ HT ;

ð1Þ

ARTICLE IN PRESS T. Kubo et al. / Physica E 42 (2010) 852–855

853

dependent linewidth functions are given by

CS ðfÞ ¼ CLS ðfÞ þ CRS ;

ð8Þ

CD ðfÞ ¼ CLD ðfÞ þ CRD ;

ð9Þ

where 0

GLS

C ðfÞ ¼ B @ aLS eif=2 GLS 0

aLS eif=2 GLS 0

LS

0

0

where HR describes the four reservoirs: X X HR ¼ enk cnyk cnk

ð2Þ

n A fLS;LD;RS;RDg k

deﬁning the Fermi seas of noninteracting electrons in the left source (LS), left drain (LD), right source (RS), and right drain (RD) reservoirs. Here enk is the electron energy with the wave number k in the reservoir n, and the operator cnk (cnyk ) annihilates (creates) an electron in the reservoir n. The Hamiltonian HDQD is HDQD ¼

2 X

ej dyj dj ;

ð3Þ

j¼1

where ej is the single-particle energy level, and dj (dyj ) annihilates (creates) an electron in the j-th QD (j ¼ 1; 2). HS represents the QD charge sensor (QD3) HS ¼ e

y 3 d3 d3 ;

ð4Þ d3 (dy3 )

where e3 is the energy level of QD3 and annihilation (creation) operator of the QD3. HI is the interaction between QD2 and QD3 HI ¼ VS n22 n33 :

ð5Þ

Here VS is the sensing interaction energy, namely inter-dot Coulomb interaction energy between QD2 and QD3, and we introduced the notation of njj ¼ dyj dj . HT is the tunneling Hamiltonian between the reservoirs and QDs X ð1Þ ð2Þ if=4 y y ½tLSk eif=4 cLSk d1 þ tLSk e cLSk d2 HT ¼ k ð1Þ if=4 y ð2Þ if=4 y þ tLDk e cLDk d1 þ tLDk e cLDk d2 ð3Þ y ð3Þ y þ tRSk cRSk d3 þtRDk cRDk d3 þ h:c: 2 X XX ½tnðjÞk ðfÞcnyk dj þ h:c:

n A fLS;LDg k j ¼ 1

þ

X

X ð3Þ y ½tnk cnk d3 þ h:c:

ð6Þ

n A fRS;RDg k

where tnðjÞk are the tunneling amplitudes and real parameters. The factors e 7 if=4 indicate the effect of the magnetic ﬂux (f ¼ 2pF=F0 is an AB phase in the AB interferometer, where F0 ¼ h=e is the magnetic ﬂux quantum). The linewidth functions are deﬁned by X n Gij ðe; fn Þ ¼ 2p tnðiÞk ðfn ÞtnðjÞk ðfn Þdðeenk Þ: ð7Þ k

Moreover, in the wide-band limit, we neglect the energy dependence of the linewidth functions. In our model, the ﬂux-

0

C

0 B ¼@0 0

0

0

0 C A;

0

0 0

RD

C

GRS

0 B ¼@0 0

1

C 0 A; 0

GLD

1

0

0

aLD eif=2 GLD 0

GLD

C ðfÞ ¼ B @ aLD eif=2 GLD 0 RS

C 0 A;

GLS

LD

Fig. 1. Schematic diagram of an Aharonov–Bohm interferometer containing two quantum dots (QD1 and QD2) with a quantum dot charge sensor (QD3). QD3 is capacitively coupled to QD2. We take into account the propagation of electrons in the reservoirs (LS and LD). sn12 is the propagation length, where n A LS; LD. VS is a sensing interaction. F is the magnetic ﬂux threading through an Aharonov–Bohm interferometer, and causes the Aharonov–Bohm effect.

1

1

0

0

0

0 C A;

0

ð10Þ

GRD

where the boldface notation indicates the 3 3 matrix. Here we introduce the coherent indirect coupling parameters an , which characterizes the strength of the indirect coupling between QD1 and QD2 via the reservoir n [11]. In general, the coherent indirect coupling parameters an are the function of the propagation length sn12 as shown in Fig. 1. In the DQD systems, jan j r 1. In the following, we choose the Fermi level as the origin of the energy. The linear conductance through a DQD is given by [18] Z e2 1 @fFD ðeÞ ~ LS Ga ðeÞC ~ LD g; de TrfGr ðeÞC ð11Þ GDQD ¼ h 1 @e n

~ is the renormalized linewidth function due to the where C electron–electron interaction, and fFD ðeÞ is the equilibrium Fermi– Dirac distribution function deﬁned by 1 fFD ðeÞ ¼ e=k T ; e B þ1

ð12Þ

and Gr ðeÞ and Ga ðeÞ are the retarded and advanced Green’s functions, respectively. Similarly, the linear conductance through a QD charge sensor is Z ~ RS G ~ RD e2 1 @fFD ðeÞ 2G de ImfGr33 ðeÞg; ð13Þ GS ¼ ~ RD ~ RS þ G h 1 @e G where Gr33 ðeÞ is the ð3; 3Þ matrix element of the retarded Green’s function. We consider the sensing interaction VS as the perturbation. To calculate the retarded Green’s function, we employ the 2nd-order perturbation theory. However, the formal 2nd-order perturbation theory exhibits the unphysical features except for the particle– hole symmetric condition because of the failure of the Friedel– Langreth sum rule. Then, we use the interpolative 2nd-order perturbation theory which yields the physically correct solution even in the region apart from the particle–hole symmetric condition [15]. Within the framework of this interpolative 2ndorder perturbation theory, we use the improved self-energy which yields the appropriate atomic limit. The retarded Green’s function is Gr ðeÞ ¼ ½fg r ðeÞg1 Rr ðeÞ1 ; r

ð14Þ r

where g ðeÞ is the retarded Green’s function for VS ¼ 0, and R ðeÞ is the improved retarded self-energy given by the interpolative 2ndorder perturbation approach [15]. The retarded self-energy is non-zero for (2,2) and (3,3) matrix elements since there is the sensing Coulomb interaction only between QD2 and QD3. Therefore, the linewidth function is renormalized only for Gr22 and Gr33 due to the sensing interaction. The real and imaginary parts of the retarded self-energy provide the energy level shift and the renormalization of the linewidth function, respectively.

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Fig. 3. Sensing interaction dependence of the visibility of the AB oscillations in the linear conductance for kB T=‘ G ¼ 0:1 (square mark) and kB T=‘ G ¼ 1 (triangle mark). The solid and dotted lines indicate the linear and parabolic ﬁtting lines.

Fig. 2. AB oscillations in the linear conductance through a DQD for several sensing interaction energies. The solid, broken, and dotted lines indicate the situation of VS =‘ G ¼ 0, VS =‘ G ¼ 1, and VS =‘ G ¼ 2, respectively.

3. Theoretical results In the following discussions, we consider the situation when

e1 ¼ e2 ¼ e3 ¼ 0, aS ¼ aD ¼ 0:5, and GLS ¼ GLD ¼ GRS ¼ GRD ¼ G=2. In Fig. 2, we plot the AB oscillations of the linear conductance through a DQD for several sensing interaction energies for kB T=‘ G ¼ 0:1. The solid, broken, and dotted lines indicate the situation of VS ¼ 0, VS =‘ G ¼ 1, and VS =‘ G ¼ 2, respectively. The period of all AB oscillations is 2p. As the sensing Coulomb interaction energy VS increases, the linear conductance through a DQD decreases since it disturbs the electrons to enter the QD2 because of the many-body correlation effect. Next we examine how the visibility of the AB oscillations in the linear conductance through a DQD changes as the sensing Coulomb interaction energy increases. We deﬁne the visibility v of the AB oscillations in the linear conductance through a DQD as v¼

min Gmax DQD GDQD min Gmax DQD þ GDQD

;

ð15Þ

min where the Gmax DQD and GDQD are the maximal and minimal values of the AB oscillations in the linear conductance through a DQD, respectively. In Fig. 3, we plot the sensing interaction dependence of the visibility of the AB oscillations in the linear conductance for kB T=‘ G ¼ 0:1 and kB T=‘ G ¼ 1. As shown in Fig. 3, for weak sensing interaction regime, the visibility exhibits the parabolic (ﬁtted by the dotted line) behavior. However, as the sensing interaction increases, the visibility behaves linearly (ﬁtted by the solid line), and ﬁnally shows the sub-linear dependence. In higher temperature, the parabolic behavior is observed in wider interaction region.

4. Discussions and conclusions In this paper, we theoretically studied how the AB oscillations in the linear conductance through a laterally coupled DQD are

affected by a coupling with a QD charge sensor. We calculated the AB oscillations in the linear transport through a DQD using the interpolative 2nd-order perturbation theory based on Green’s function method. As the sensing Coulomb interaction between one of the DQD and the QD charge sensor increases, we found that the visibility of the AB oscillations in the linear conductance through the DQD decreases monotonically. In particular, the visibility behaves parabolically for weak interaction regime, and sub-linearly for strong interaction regime. Our result for the temperature dependence is similar to the previous theoretical works for the single QD [19,20]. The linear behavior of the visibility was shown in the which-path experiment by Buks et al. [2]. However, the Coulomb interaction introduces not only the dephasing but also the energy level shifts more for QD2, which also modiﬁes the interference condition, resulting in the change of visibility. In particular, the linear and sub-linear behaviors are dominated by the Hartree level-shift in Fig. 3. To extract only dephasing effect, we have to compensate the level renormalization [21].

Acknowledgments We would like to thank O. Entin-Wohlman, A. Aharony, and S. Sasaki for useful discussions and valuable comments. This work was ﬁnancially supported by Grants-in-Aid for Scientiﬁc Research S (No. 19104007), B (No. 18340081), and by Special Coordination Funds for Promoting Science and Technology. S.T. acknowledges support from QuEST program (BAA-0824). References [1] N. Bohr, Nature 121 (1928) 580. [2] E. Buks, R. Schuster, M. Heiblum, D. Mahalu, V. Umansky, Nature 391 (1998) 871. [3] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. [4] A. Yacoby, M. Heiblum, D. Mahalu, H. Shtrikman, Phys. Rev. Lett. 74 (1995) 4047. [5] A.W. Holleitner, C.R. Decker, H. Qin, K. Eberl, R.H. Blick, Phys. Rev. Lett. 87 (2001) 256802. [6] T. Hatano, M. Stopa, W. Izumida, T. Yamaguchi, T. Ota, S. Tarucha, Physica E 22 (2004) 534. [7] K. Kobayashi, H. Aikawa, S. Katsumoto, Y. Iye, Phys. Rev. Lett. 88 (2002) 256806. [8] S. Sasaki, et al., Nature (London) 405 (2000) 764. [9] W.G. van der Wiel, et al., Science 289 (2000) 2105. [10] T. Kubo, Y. Tokura, S. Tarucha, Phys. Rev. B 77 (2008) 041305(R).

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