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Geometric phase for a coupled two quantum dot system
Optics Communications 284 (2011) 2919–2922
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Geometric phase for a coupled two quantum dot system Amitabh Joshi a,⁎, Shoukry S. Hassan b a b
Department of Physics, Eastern Illinois University, Charleston, IL 61920, United States Department of Mathematics, College of Science, University of Bahrain, P O Box 32038, Bahrain
a r t i c l e
i n f o
Article history: Received 4 June 2010 Received in revised form 20 December 2010 Accepted 21 January 2011 Available online 21 February 2011
a b s t r a c t The adiabatic geometric phase is calculated in a coupled two quantum dot system, which is entangled through Förster interaction. This phase is then utilized for implementing basic quantum logic gate operation useful in quantum information processing. Such gates based on geometric phase provide fault-tolerant quantum computing. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The exponential faster speed of quantum computers over their classical counterparts has motivated several research areas in quantum computation and quantum information processing in recent years [1–3]. Semiconductor quantum dot (QD) system is one of the practical physical systems for quantum information processing in which, the exciton constitutes an alternative for the usual two-level system [4–7]. When a system having more than one quantum dot is considered then the coupling and the interaction between quantum dots become important. A prominent interaction called Förster interaction responsible for the transfer of exciton from one QD to another QD has been utilized in producing quantum teleportation, optical switching, entangled Bell states, and GHZ states [8,9]. Another interaction due to static exciton–exciton dipole coupling was used to generate entangled few exciton state via ultra fast laser sequences [4]. During a cyclic evolution of any quantum mechanical system described by a Hamiltonian, the associated wave function of the system acquires a geometrical phase (Berry phase) in addition to the dynamical phase. Such an adiabatic geometric phase is accumulated in an instantaneous eigenstate of an adiabatically evolving Hamiltonian which is periodic in the parametric space [10]. The quantum computing and quantum information processing tasks require implementation of quantum logic gate operations. Such operations are based on the dynamic evolution of the quantum system or the pure geometric based operations. The geometric based operation is quite a promising approach for the implementation of built-in faulttolerant quantum logic gates. The geometric phase based quantum logic gates have intrinsic advantages over the dynamic phase based counterparts as they are insensitive to starting state distributions, the ⁎ Corresponding author. Tel.: +1 217 581 5950; fax: +1 217 581 8548. E-mail addresses: [email protected] (A. Joshi), [email protected] (S.S. Hassan). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.01.066
path shape and the passage rate to traverse the closed path and thus robust against dephasing and significantly higher in fidelity [10-14]. The normal procedure of geometric logic gate operation is to drive the qubit to have an appropriate adiabatic cyclic operation. Such schemes are proposed for the trapped ions [12], NMR systems [13] and semiconductor nano circuits and SQUIDS in microcavites [14]. Investigations of Berry phase for coupled quantum dots have been a center of attention by several researchers and the effect of environment temperature on it is also studied [15]. In another study the Berry phase in a bipartite system was investigated including coupling within subsystems. It was shown that as the coupling constants tend to infinity all the geometric phases go to zero [16]. The effect of Dzyaloshinnski–Moriya interaction was also included in such study and the sudden change in the Berry phase for weak fields was reported [17]. The SWAP operation in a two-qubit anisotropic XXZ model in the presence of an inhomogeneous magnetic field was studied to establish the range of anisotropic parameter within which the SWAP operation was feasible [18]. Implementation of nonadiabatic geometrical quantum gates within semiconductor quantum dots exploiting excitonic degrees of freedom has been discussed. Also, the effect of geometric phases induced by either classical or quantum electric fields acting on single electron spins in quantum dots in the presence of spin–orbit coupling has been studied [19]. In a feasible quantum dot model, the geometric phase of the quantum dot system in nonunitary evolution was calculated and the effect of environment parameters on the phase value was investigated [20]. In this work we rigorously investigate adiabatic geometric phases of two coupled quantum dots (QDs) considered as two spin-1/2 system including Förster interaction between them. For this system the adiabatic Berry's phase can be used to implement conditional phase shifts and thus realization of quantum logic gate operations. The importance of Förster interaction will be investigated in this work. We will show that Förster interaction can be exploited to generate the adiabatic geometric phase which then can be used to
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implement quantum logic gates. The Förster–Dexter resonant energy transfer has been studied in sensitized luminescence of solids in which an excited sensitizer atom can transfer its excitation to a neighboring acceptor atom through a virtual photon. Some interesting work on energy transfer using this mechanism for quantum dot system has been reported [21,22]. The paper is organized as follows. In section 2 we present the physical model with theoretical description, where two coupled QDs interact through the dipole-dipole interactions. Section 3 is devoted to describe the adiabatic evolution of this system and calculation of Berry's phase operation of quantum logic gates. In Section 4 we give a summary. 2. The model We consider two coupled QDs situated at some distance from each other such that they can interact through dipole–dipole interaction. Each QD has a ground state |0〉 and a first excited state |1〉. The two dots could be of different sizes and hence their dipole-coupling strengths may be slightly different. One can write down the interaction Hamiltonian in the computational basis {|00〉, |01〉, |10〉, |11〉}, where first and second digits are referring to QDI and QDII, respectively. The situation can be described by the following phenomenological Hamiltonian [22] 0
ω0 B 0 B H ðt Þ = @ 0 0
0 ω0 + ω2 W ⁎ ðt Þ 0
0 W ðt Þ ω0 + ω1 0
1 0 C 0 C; A 0 ω0 + ω1 + ω2 + Fz
−iγn ðt Þ
jξn ðt Þ〉 = e
jξn ð0Þ〉;
ð1Þ
3. Geometric phase and quantum logic gates In this case we need only four computational basis states of Hamiltonian (Eq. 1) defined as |00〉, |10〉, |01〉, and |11〉. We call them as qubit states of couple QD system. Any instantaneous eigenstate of the Hamiltonian can be represented as ð2Þ
(j = 1,2) where n = 1,2,3,4 and basis states are eigenstates of S(j) z operators. The instantaneous normalized eigenstates are listed as follows [22]: jξ1 〉 = j00〉; jξ2 〉 = a2 j01〉 + a1 eiαðtÞ j10〉; ð3Þ jξ3 〉 = −a1 j01〉 + a2 eiαðt Þ j10〉; jξ4 〉 = j11〉; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi 2 0 1+4 W . in which a1 = 12 1 + Q1 , a2 = 12 1− Q1 , and Q = Δ0 The quantity Δ0 =ω1 −ω2 is the difference between exciton creation energy for dot I and that for dot II and the corresponding eigen energies are λ1 =0, λ2 = ω1 − Δ20 ð1 + Q Þ, λ3 = ω1 − Δ20 ð1−Q Þ, and λ4 =ω1 + ω2 +Fz.
ð4Þ
where the total phase γn(t) has two components D
G
γn ðt Þ = γn ðt Þ + γn :
ð5Þ
The geometric phase γGn , given by γGn = ∫ 2π 0 dΦ〈ξn(Φ)| ∂ Φξn(Φ)〉 is independent of the detailed variation of Φ with time. For Berry phase, evolution of Φ (which is α(t) here) is to be done adiabatically. The unitary transformation in a cycle is diag(eiγ1, eiγ2, eiγ3, eiγ4) written in the basis {|ξ1〉, |ξ2〉, |ξ3〉,|ξ4〉} and γn are the total phases as given in Eq. (5). The dynamical phase is given by γD n = − λnt and for the states |ξ1〉 and |ξ4〉, geometrical phases are γG1 = γG4 = 0. The evolutions of other states |ξ2〉 and |ξ3〉 contain both dynamical and geometric phase and the geometric phase is given by the solid angle subtended by the qubit trajectory on the Bloch sphere. We give some outline to estimate this geometric phase. Here the qubit states |00〉 and |11〉 factor out from other two states |01〉 and |10〉. Hence we have a sub-space expanded by the vectors |10〉 and |01〉 for which state vectors given by |ξ2〉 and |ξ3〉 can be written in a slightly different form as jξ2 〉 = sinðθ = 2Þj01〉 + cosðθ = 2Þe
where ω0 denotes the ground state energy of QDs, the exciton frequency in the first (second) QD is ω1 (ω2), Fz is the strength of static exciton–exciton dipole interaction energy, which is the diagonal interaction and is the direct Coulomb binding energy between two excitons, one located on each dot [22]. W(t) = W0e− iα(t) is the strength of Förster interaction between two QDs whose time variation/cyclicity is determined by the variable α(t) (which takes value from 0 to 2π during adiabatic cyclic evolution). The Förster interaction is off-diagonal and therefore induces the transfer of an exciton from one QD to another. For simplicity we measure all energies from ground state and thus set ω0 = 0 in the subsequent discussion.
jξn ðt Þ〉 = xn j00〉 + yn j01〉 + zn j10〉 + wn j11〉;
In the adiabatic limit each eigenstate evolves as
iαðt Þ
j10〉;
iαðt Þ
j ξ3 〉 = −cosðθ = 2Þj01〉 + sinðθ = 2Þe j10〉; ð6Þ 1 = 2 Δ0 p1ffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi and sinðθ = 2Þ = such that cosðθ=12= 2Þ = 2 1 + Δ20 + 4W02 Δ0 1 pffiffi 1− pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . If the qubit starts in an eigen state |10〉 or |01〉 2 2 2 Δ0 + 4W0
of Hamiltonian H(0), then it remains throughout in one of the instantaneous eigenstates (Eq. 6). Under adiabatic variation of the Hamiltonian, the initial superposition of state such as p1ffiffi2 ½ j10 N + j01 N will undergo the evolution as [13] i 1 1 h −iγ −iγ pffiffiffi j10〉 + j01〉→ pffiffiffi e 2 jξ2 〉 + e 3 jξ3 〉 ; 2 2
ð7Þ
where the total phases γ2 and γ3 (containing both dynamical and geometrical phases) are given by Eq. (4). The dynamical phase can be made to cancel out using a spin-echo or refocusing type of technique. The basic idea of this technique is to apply the cyclic evolution twice. The second cyclic evolution (preceded by fast π pulses that swaps the basis states |10〉 and |01〉) is performed by retracing the first but in an opposite direction so the total dynamical phase factors in two cycles cancel out but the geometric phase factors get added up as the solid angle is preserved in either direction (see detail in Ref.[13]) . We call this scheme as a ‘two-cycle’ scheme, which is using two cycles with an opposite direction of evolution of parameter W(t). We do not go to the details for such a calculation but stress that by doing things properly only the Berry 1 phases appear h inG the final evolution i and the state pffiffi2 ½ j10〉 + j01〉 G evolves to p1ffiffi2 e2iγ j10〉 + e−2iγ j01〉 under the two cycle scheme [13]. The geometric phase associated with the isolated states |00〉 and |11〉 is zero as discussed above. Hence the unitary transformation in the basis {|00〉, |10〉, |01〉, |11〉} is then given by 2iγG −2iγG UB = diag 1; e ; e ;1 ;
ð8Þ
where γG is Berry phase given by the solid angle enclosed by the qubit's trajectory on the Bloch's sphere [10,13] 2
3
Δ0 G 6 7 ffi5: γ = π½1−cosðθÞ = π41− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ20 + 4W02
ð9Þ
A. Joshi, S.S. Hassan / Optics Communications 284 (2011) 2919–2922
Fig. 1. Schematics of the closed loop for implementing qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π/8 gate on a Bloch sphere. The analogous generalized Rabi frequency is
Δ20 + 4W02 . The solid angle for the cone is
Ω(C) = 2π(1 − cosθ). The other details of implementation are given in the text.
Clearly, the expression of geometric phase turns out to be different from the expression of dynamical phase mentioned above. It is r straightforward to show that UB can act as π/8 gate ffiffiffiffiffiffiffiffiffi 31 Δ0 . The closed loop for implementing such gate is when W0 = 900 illustrated on a Bloch sphere (Fig. 1). The ‘two-cycle’ scheme [13] of 22 2 gate operation can be described by UB = π T C Tπ T C T in which T is tipping of Bloch vector through angle θ by introduction of Förster interaction, C is 2π operation of phase α, π is pi-pulse operation, and bar denotes reverse operation. The dynamical phase in the entire operation for all the eigenstates turns out to be a global phase factor of exp[i(ω1 + ω2)t] (assuming Fz to be very small) and not physical and hence the gate operation is given by Eq. (8). It is easy to construct other quantum logic gates based on pure geometric transformation. We propose the two-qubit gate in the equivalence class [23] of the controlled-Z gate, i.e., it only differs from the controlled-Z gate by local operations to two qubits. The relative phase in unitary transformation UB or the gate operation is given by Φz = 4πcos(θ). The local invariants of this gate are G1 = (1/2) (1+cosΦz) and G2 =(1+(1/2)cosΦz) and hence those of controlled-Z gate are given by G1 =0 and G2 =0 demanding cosθ =1/4 and can be easily achieved with experimental parameters. The advantage of such quantum logic gates based on pure geometric transformation is due to the robustness of the geometric phase. In quantum evolutions, holonomies and geometrical phases are naturally occurring, which are robust to local errors [10-13]. The form of dependence of geometric phase on detuning Δ0 and the strength of Förster interaction bring a natural type of fault tolerance not present in the non-geometric conditional phase gate, which is quite similar to the one discussed for the NMR system [13]. Note that any fluctuation of the parameters such as detuning or Förster interaction will result into errors that reside only in the dynamical phase since the solid angle of the loop is preserved on average if the fluctuations are sufficiently random. Removal of dynamical phase can give insensitivity to parameter fluctuations. So the robustness (fault-tolerance) of geometric-phase based quantum gates is due to the cancelation of dynamical phase over two consecutive cycles [10-13]. Geometric phase based quantum computing is normally less susceptible to the environmental noise. It is decoherence caused by the environment, that constraints implementation of quantum gates. This is true for the coupled quantum dot system also. In a recent work, Berry phase of two coupled quantum dots was calculated in the frame
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work of cavity quantum electrodynamics [15]. The interaction of the phonon field with quantum dots was also taken into account in this work. It was shown that the Berry phase changes near a critical temperature determined by the resonance condition of the cavity and thus the Berry phase before and after critical temperature maintains some fixed values. This work suggests that in our case also despite the presence of decoherence mechanism due to the environment we can maintain a fixed Berry phase up to a certain temperature range and can do the quantum information processing. It is possible to get further physical insight into the effect of decoherence on geometric phases with the help of Ref. [20], which may be applicable to our study. According to [20] the physical picture emerging out is that for the large decoherence the geometric phase becomes vanishingly small during the evolution of the Bloch vector over the Bloch-sphere in a closed path. However, when there is moderate decoherence, the implementation of geometric-phase quantum gate is feasible [20] and we can achieve such a moderate decoherence in our system. In our scheme, the value of Förster interaction W depends on the dot sizes and the confinement potential. Also, W can be modulated by applying an electric field. The measured dipole values for CdSe dots range in 0.9 to 5.2 A and the corresponding W goes as 0.02 to 0.6 meV leading to energy transfer rate of several tens of picoseconds [22]. This time is short enough to be useful for any quantum information processing using geometric-phase based quantum gates because the decoherence times as large as a few nanoseconds have been observed in QDs [22]. Arrays of strongly interacting individual molecules may also behave as an excellent system for QIP tasks using Förster interaction. In such systems the typical Förster interaction strength is 8.3 mev and the transfer times are of the order of 500 ps. Finally, we would like to give a basic outline for realization of the above work experimentally. The theoretical understanding of geometric phase for a pure quantum state is quite clear and has been experimentally demonstrated in NMR, single-photon and two-photon interferometry [24]. When decoherence is present in the system then it leads to the preparation of mixed states. For the mixed state, geometric phase has been interpreted in terms of quantum interferometry. The Förster interaction is controlled by the external electric field. When field amplitude is large, the Förster interaction remains suppressed. By decreasing field amplitude, this interaction can be initiated. The Förster interaction is a sensitive function of applied electric field as the electric field caused separation of electrons and holes to increase and thus overlap integral of electron–hole decreases, consequently reduces the Förster interaction. Initially, one starts with state |1, 0〉 by selectively exciting QD-I. When the Förster interaction is turned on, the system evolves into the eigenstate |ξ2〉 or |ξ3〉. This initiates tipping operation of the Bloch vector through an angle θ (defined after Eq. (6)) with α at zero. One can stop the evolution by applying an external electric field when the system is in one of these eigenstates (which are entangled states). Adiabatic variation of α(t) from 0 to 2π at this stage results into acquisition of the adiabatic geometric phase. This can be achieved by varying the phase of the external field that controls the Förster interaction parameter [22]. For this purpose one needs to apply an adiabatic sweep which will provide circular motion in the parametric space of W. In order to cancel the dynamical phase one has to move back in the reverse path also (i.e., the two-cycle scheme to cancel dynamical phase as described above). The essential ingredients of a quantum interferometer, which can eventually be implemented in the experiment are described as follows. The quantum system of two QDs undergoes a series of unitary evolutions after which the probability of finding system in one of its eigenstates becomes an oscillatory function [25,26] of the control parameter, i.e., the Förster interaction. Such oscillation in probability is similar to an optical interference pattern [25]. The shift of interference pattern is a function of the geometric phases acquired by the double QDs system during the unitary evolutions, as well as purity (which depends on the decoherence) of the internal states of the double QDs system. Thus geometric phase can be directly
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measured from the shift of the interference pattern. At the end of interferometric operation the tomography will be required to construct the components of density matrix. Such geometric phase has been experimentally measured in NMR and single photon interferometry [26]. On the lines of these experiments one can implement our proposal of experiment with QDs. 4. Summary We have considered a Hamiltonian system of two quantum dots coupled through their dipole–dipole Förster interaction and also via the static exciton–exciton dipole interaction energy. It is assumed the Hamiltonian has a time-variation of the Förster interaction parameter. In an adiabatic limit of this variation, the geometric Berry phase has been calculated. Based on the idea of two consecutive cyclic evolution, the net dynamic phase can be canceled while the net geometric phase is doubled in the final unitary transformation of the system. We can extend this idea for using several interacting quantum dots to construct three-qubit and multi-qubit entangling gates and hence to generate nqubit 2-dimensional graph (cluster) state entanglement which will be useful to prepare a practical quantum information processing system. Acknowledgements Many useful discussions with Min Xiao is gratefully acknowledged. We acknowledge the funding support from Research Corporation. References [1] G. Benenti, G. Casati, G. Strini, Prinicples of Quantum Computation and Information, Vol. I, World Scientific, Singapore, 2004. [2] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, England, 2000. [3] D. Bouwmeester, A.K. Ekert, A. Zeilinger, The Physics of Quantum Information, Springer-Verlag, Berlin, 2000. [4] E. Biolatti, R.C. Iotti, P. Zanardi, F. Rossi, Phys. Rev. Lett. 85 (2000) 5647. [5] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83 (1999) 4. [6] Daniel Loss, David P. DiVincenzo, Phys. Rev. A 57 (1998) 120.
[7] T. Pellizzari, S.A. Gardiner, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 75 (1995) 3788. [8] J.H. Reina, L. Quiroga, N.F. Johnson, Phys. Rev. A 62 (2000) 012305; L. Quiroga, N.F. Johnson, Phys. Rev. Lett. 83 (1999) 2270. [9] J.H. Reina, N.F. Johnson, Phys. Rev. A 63 (2000) 012303; J. Gea-Banacloche, M. Mumba, M. Xiao, Phys. Rev. B 74 (2006) 165330; A. Joshi, B. Anderson, M. Xiao, Phys. Rev. B 75 (2007) 125304. [10] M.V. Berry, Proc. R. Soc. Lond. Ser. 292 (1984) 45; A. Shappere, F. Wilczek, Geometric Phases in Physics, World Scientific, Singapore, 1989; E. Sjqvist, Physics 1 (2008) 35. [11] P. Zanardi, M. Rasetti, Phys. Lett. A 264 (1999) 94; S.L. Zhu, Z.D. Wang, Phys. Rev. Lett. 91 (2003) 187902; A. Joshi, M. Xiao, Phys. Lett. A 359 (2006) 390. [12] L.M. Duan, J.I. Cirac, P. Zoller, Science 292 (2001) 1695. [13] J.A. Jones, V. Vedral, A. Ekert, G. Castagnoli, Nature 403 (2000) 869; A. Ekert, M. Ericsson, P. Hayden, H. Inamori, J.A. Jones, D.K.L. Oi, V. Vedral, J. Mod. Opt. 47 (2000) 2501; A. Nazir, T.P. Spiller, W.J. Munro, Phys. Rev. A 65 (2002) 042303; S. Siddiqui, J. Gea-Banacloche, Phys. Rev. A 74 (2006) 052337. [14] P. Solinas, P. Zanardi, N. Zanghi, F. Rossi, Phys. Rev. B 67 (2003) 121307 (R); P. Zhang, Z.D. Wang, J.D. Sun, C.P. Sun, Phys. Rev. A 71 (2005) 042301. [15] K. Yang, et al., Chin. Phys. Lett. 20 (2003) 991; D.M. Tong, et al., Mod. Phys. Lett. B 18 (2004) 1433; X.Z. Yuan, K.D. Zhu, Phys. Rev. B 74 (2006) 073309. [16] H.Y. Sun, L.C. Wang, X.X. Yi, Phys. Lett. A 370 (2007) 119. [17] Y. Zhou, G.F. Zhang, Opt. Commun. 281 (2008) 5278. [18] Y. Zhou, G.F. Zhang, F.H. Yang, S. Feng, Phys. Rev. A 75 (2007) 062304. [19] P. Solinas, P. Zanardi, N. Zanghy, F. Rossi, Phys. Rev. A 67 (2003) 052309; P. San-Jose, B. Scharfenberger, G. Schon, A. Shnirman, G. Zarand, Phys. Rev. B 77 (2008) 045305. [20] S. Yin, D.M. Tong, Phys. Rev. A 79 (2009) 044303. [21] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, V.I. Klimov, Phys. Rev. Lett. 89 (2002) 186802. [22] Our model is based on the Hamiltonian discussed in:. B.W. Lovett, J.H. Reina, A. Nazir, B. Kothari, G.A.D. Briggs, Phys. Lett. A 315 (2003) 136; B.W. Lovett, J.H. Reina, A. Nazir, G.A.D. Briggs, Phys. Rev. B 68 (2003) 205319; see also:. S. Oh, Z. Huang, U. Peskin, S. Kais, Phys. Rev. A 78 (2008) 0621068 where a slightly different Hamiltonian is considered. [23] F. Brito, D.P. DiVincenzo, R.H. Koch, and M. Steffen, arXive:0709.1478v2[quantph] 19 Mar 2008. [24] D. Suter, K.T. Mueller, A. Pines, Phys. Rev. Lett. 60 (1988) 1218; P.G. Kwiat, R.Y. Chiao, Phys. Rev. Lett. 66 (1991) 588; J. Brendel, W. Dultz, W. Martienssen, Phys. Rev. A 52 (1995) 2551. [25] E. Sjöqvist, et al., Phys. Rev. Lett. 85 (2000) 2845. [26] J. Du, et al., Phys. Rev. Lett. 91 (2003) 100403; M. Ericsson, et al., Phys. Rev. Lett. 94 (2005) 050401; A. Ghosh, A. Kumar, Phys. Lett. A 349 (2006) 27.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Geometric phase for a coupled two quantum dot system Amitabh Joshi a,⁎, Shoukry S. Hassan b a b
Department of Physics, Eastern Illinois University, Charleston, IL 61920, United States Department of Mathematics, College of Science, University of Bahrain, P O Box 32038, Bahrain
a r t i c l e
i n f o
Article history: Received 4 June 2010 Received in revised form 20 December 2010 Accepted 21 January 2011 Available online 21 February 2011
a b s t r a c t The adiabatic geometric phase is calculated in a coupled two quantum dot system, which is entangled through Förster interaction. This phase is then utilized for implementing basic quantum logic gate operation useful in quantum information processing. Such gates based on geometric phase provide fault-tolerant quantum computing. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The exponential faster speed of quantum computers over their classical counterparts has motivated several research areas in quantum computation and quantum information processing in recent years [1–3]. Semiconductor quantum dot (QD) system is one of the practical physical systems for quantum information processing in which, the exciton constitutes an alternative for the usual two-level system [4–7]. When a system having more than one quantum dot is considered then the coupling and the interaction between quantum dots become important. A prominent interaction called Förster interaction responsible for the transfer of exciton from one QD to another QD has been utilized in producing quantum teleportation, optical switching, entangled Bell states, and GHZ states [8,9]. Another interaction due to static exciton–exciton dipole coupling was used to generate entangled few exciton state via ultra fast laser sequences [4]. During a cyclic evolution of any quantum mechanical system described by a Hamiltonian, the associated wave function of the system acquires a geometrical phase (Berry phase) in addition to the dynamical phase. Such an adiabatic geometric phase is accumulated in an instantaneous eigenstate of an adiabatically evolving Hamiltonian which is periodic in the parametric space [10]. The quantum computing and quantum information processing tasks require implementation of quantum logic gate operations. Such operations are based on the dynamic evolution of the quantum system or the pure geometric based operations. The geometric based operation is quite a promising approach for the implementation of built-in faulttolerant quantum logic gates. The geometric phase based quantum logic gates have intrinsic advantages over the dynamic phase based counterparts as they are insensitive to starting state distributions, the ⁎ Corresponding author. Tel.: +1 217 581 5950; fax: +1 217 581 8548. E-mail addresses: [email protected] (A. Joshi), [email protected] (S.S. Hassan). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.01.066
path shape and the passage rate to traverse the closed path and thus robust against dephasing and significantly higher in fidelity [10-14]. The normal procedure of geometric logic gate operation is to drive the qubit to have an appropriate adiabatic cyclic operation. Such schemes are proposed for the trapped ions [12], NMR systems [13] and semiconductor nano circuits and SQUIDS in microcavites [14]. Investigations of Berry phase for coupled quantum dots have been a center of attention by several researchers and the effect of environment temperature on it is also studied [15]. In another study the Berry phase in a bipartite system was investigated including coupling within subsystems. It was shown that as the coupling constants tend to infinity all the geometric phases go to zero [16]. The effect of Dzyaloshinnski–Moriya interaction was also included in such study and the sudden change in the Berry phase for weak fields was reported [17]. The SWAP operation in a two-qubit anisotropic XXZ model in the presence of an inhomogeneous magnetic field was studied to establish the range of anisotropic parameter within which the SWAP operation was feasible [18]. Implementation of nonadiabatic geometrical quantum gates within semiconductor quantum dots exploiting excitonic degrees of freedom has been discussed. Also, the effect of geometric phases induced by either classical or quantum electric fields acting on single electron spins in quantum dots in the presence of spin–orbit coupling has been studied [19]. In a feasible quantum dot model, the geometric phase of the quantum dot system in nonunitary evolution was calculated and the effect of environment parameters on the phase value was investigated [20]. In this work we rigorously investigate adiabatic geometric phases of two coupled quantum dots (QDs) considered as two spin-1/2 system including Förster interaction between them. For this system the adiabatic Berry's phase can be used to implement conditional phase shifts and thus realization of quantum logic gate operations. The importance of Förster interaction will be investigated in this work. We will show that Förster interaction can be exploited to generate the adiabatic geometric phase which then can be used to
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implement quantum logic gates. The Förster–Dexter resonant energy transfer has been studied in sensitized luminescence of solids in which an excited sensitizer atom can transfer its excitation to a neighboring acceptor atom through a virtual photon. Some interesting work on energy transfer using this mechanism for quantum dot system has been reported [21,22]. The paper is organized as follows. In section 2 we present the physical model with theoretical description, where two coupled QDs interact through the dipole-dipole interactions. Section 3 is devoted to describe the adiabatic evolution of this system and calculation of Berry's phase operation of quantum logic gates. In Section 4 we give a summary. 2. The model We consider two coupled QDs situated at some distance from each other such that they can interact through dipole–dipole interaction. Each QD has a ground state |0〉 and a first excited state |1〉. The two dots could be of different sizes and hence their dipole-coupling strengths may be slightly different. One can write down the interaction Hamiltonian in the computational basis {|00〉, |01〉, |10〉, |11〉}, where first and second digits are referring to QDI and QDII, respectively. The situation can be described by the following phenomenological Hamiltonian [22] 0
ω0 B 0 B H ðt Þ = @ 0 0
0 ω0 + ω2 W ⁎ ðt Þ 0
0 W ðt Þ ω0 + ω1 0
1 0 C 0 C; A 0 ω0 + ω1 + ω2 + Fz
−iγn ðt Þ
jξn ðt Þ〉 = e
jξn ð0Þ〉;
ð1Þ
3. Geometric phase and quantum logic gates In this case we need only four computational basis states of Hamiltonian (Eq. 1) defined as |00〉, |10〉, |01〉, and |11〉. We call them as qubit states of couple QD system. Any instantaneous eigenstate of the Hamiltonian can be represented as ð2Þ
(j = 1,2) where n = 1,2,3,4 and basis states are eigenstates of S(j) z operators. The instantaneous normalized eigenstates are listed as follows [22]: jξ1 〉 = j00〉; jξ2 〉 = a2 j01〉 + a1 eiαðtÞ j10〉; ð3Þ jξ3 〉 = −a1 j01〉 + a2 eiαðt Þ j10〉; jξ4 〉 = j11〉; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi 2 0 1+4 W . in which a1 = 12 1 + Q1 , a2 = 12 1− Q1 , and Q = Δ0 The quantity Δ0 =ω1 −ω2 is the difference between exciton creation energy for dot I and that for dot II and the corresponding eigen energies are λ1 =0, λ2 = ω1 − Δ20 ð1 + Q Þ, λ3 = ω1 − Δ20 ð1−Q Þ, and λ4 =ω1 + ω2 +Fz.
ð4Þ
where the total phase γn(t) has two components D
G
γn ðt Þ = γn ðt Þ + γn :
ð5Þ
The geometric phase γGn , given by γGn = ∫ 2π 0 dΦ〈ξn(Φ)| ∂ Φξn(Φ)〉 is independent of the detailed variation of Φ with time. For Berry phase, evolution of Φ (which is α(t) here) is to be done adiabatically. The unitary transformation in a cycle is diag(eiγ1, eiγ2, eiγ3, eiγ4) written in the basis {|ξ1〉, |ξ2〉, |ξ3〉,|ξ4〉} and γn are the total phases as given in Eq. (5). The dynamical phase is given by γD n = − λnt and for the states |ξ1〉 and |ξ4〉, geometrical phases are γG1 = γG4 = 0. The evolutions of other states |ξ2〉 and |ξ3〉 contain both dynamical and geometric phase and the geometric phase is given by the solid angle subtended by the qubit trajectory on the Bloch sphere. We give some outline to estimate this geometric phase. Here the qubit states |00〉 and |11〉 factor out from other two states |01〉 and |10〉. Hence we have a sub-space expanded by the vectors |10〉 and |01〉 for which state vectors given by |ξ2〉 and |ξ3〉 can be written in a slightly different form as jξ2 〉 = sinðθ = 2Þj01〉 + cosðθ = 2Þe
where ω0 denotes the ground state energy of QDs, the exciton frequency in the first (second) QD is ω1 (ω2), Fz is the strength of static exciton–exciton dipole interaction energy, which is the diagonal interaction and is the direct Coulomb binding energy between two excitons, one located on each dot [22]. W(t) = W0e− iα(t) is the strength of Förster interaction between two QDs whose time variation/cyclicity is determined by the variable α(t) (which takes value from 0 to 2π during adiabatic cyclic evolution). The Förster interaction is off-diagonal and therefore induces the transfer of an exciton from one QD to another. For simplicity we measure all energies from ground state and thus set ω0 = 0 in the subsequent discussion.
jξn ðt Þ〉 = xn j00〉 + yn j01〉 + zn j10〉 + wn j11〉;
In the adiabatic limit each eigenstate evolves as
iαðt Þ
j10〉;
iαðt Þ
j ξ3 〉 = −cosðθ = 2Þj01〉 + sinðθ = 2Þe j10〉; ð6Þ 1 = 2 Δ0 p1ffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi and sinðθ = 2Þ = such that cosðθ=12= 2Þ = 2 1 + Δ20 + 4W02 Δ0 1 pffiffi 1− pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . If the qubit starts in an eigen state |10〉 or |01〉 2 2 2 Δ0 + 4W0
of Hamiltonian H(0), then it remains throughout in one of the instantaneous eigenstates (Eq. 6). Under adiabatic variation of the Hamiltonian, the initial superposition of state such as p1ffiffi2 ½ j10 N + j01 N will undergo the evolution as [13] i 1 1 h −iγ −iγ pffiffiffi j10〉 + j01〉→ pffiffiffi e 2 jξ2 〉 + e 3 jξ3 〉 ; 2 2
ð7Þ
where the total phases γ2 and γ3 (containing both dynamical and geometrical phases) are given by Eq. (4). The dynamical phase can be made to cancel out using a spin-echo or refocusing type of technique. The basic idea of this technique is to apply the cyclic evolution twice. The second cyclic evolution (preceded by fast π pulses that swaps the basis states |10〉 and |01〉) is performed by retracing the first but in an opposite direction so the total dynamical phase factors in two cycles cancel out but the geometric phase factors get added up as the solid angle is preserved in either direction (see detail in Ref.[13]) . We call this scheme as a ‘two-cycle’ scheme, which is using two cycles with an opposite direction of evolution of parameter W(t). We do not go to the details for such a calculation but stress that by doing things properly only the Berry 1 phases appear h inG the final evolution i and the state pffiffi2 ½ j10〉 + j01〉 G evolves to p1ffiffi2 e2iγ j10〉 + e−2iγ j01〉 under the two cycle scheme [13]. The geometric phase associated with the isolated states |00〉 and |11〉 is zero as discussed above. Hence the unitary transformation in the basis {|00〉, |10〉, |01〉, |11〉} is then given by 2iγG −2iγG UB = diag 1; e ; e ;1 ;
ð8Þ
where γG is Berry phase given by the solid angle enclosed by the qubit's trajectory on the Bloch's sphere [10,13] 2
3
Δ0 G 6 7 ffi5: γ = π½1−cosðθÞ = π41− qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ20 + 4W02
ð9Þ
A. Joshi, S.S. Hassan / Optics Communications 284 (2011) 2919–2922
Fig. 1. Schematics of the closed loop for implementing qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π/8 gate on a Bloch sphere. The analogous generalized Rabi frequency is
Δ20 + 4W02 . The solid angle for the cone is
Ω(C) = 2π(1 − cosθ). The other details of implementation are given in the text.
Clearly, the expression of geometric phase turns out to be different from the expression of dynamical phase mentioned above. It is r straightforward to show that UB can act as π/8 gate ffiffiffiffiffiffiffiffiffi 31 Δ0 . The closed loop for implementing such gate is when W0 = 900 illustrated on a Bloch sphere (Fig. 1). The ‘two-cycle’ scheme [13] of 22 2 gate operation can be described by UB = π T C Tπ T C T in which T is tipping of Bloch vector through angle θ by introduction of Förster interaction, C is 2π operation of phase α, π is pi-pulse operation, and bar denotes reverse operation. The dynamical phase in the entire operation for all the eigenstates turns out to be a global phase factor of exp[i(ω1 + ω2)t] (assuming Fz to be very small) and not physical and hence the gate operation is given by Eq. (8). It is easy to construct other quantum logic gates based on pure geometric transformation. We propose the two-qubit gate in the equivalence class [23] of the controlled-Z gate, i.e., it only differs from the controlled-Z gate by local operations to two qubits. The relative phase in unitary transformation UB or the gate operation is given by Φz = 4πcos(θ). The local invariants of this gate are G1 = (1/2) (1+cosΦz) and G2 =(1+(1/2)cosΦz) and hence those of controlled-Z gate are given by G1 =0 and G2 =0 demanding cosθ =1/4 and can be easily achieved with experimental parameters. The advantage of such quantum logic gates based on pure geometric transformation is due to the robustness of the geometric phase. In quantum evolutions, holonomies and geometrical phases are naturally occurring, which are robust to local errors [10-13]. The form of dependence of geometric phase on detuning Δ0 and the strength of Förster interaction bring a natural type of fault tolerance not present in the non-geometric conditional phase gate, which is quite similar to the one discussed for the NMR system [13]. Note that any fluctuation of the parameters such as detuning or Förster interaction will result into errors that reside only in the dynamical phase since the solid angle of the loop is preserved on average if the fluctuations are sufficiently random. Removal of dynamical phase can give insensitivity to parameter fluctuations. So the robustness (fault-tolerance) of geometric-phase based quantum gates is due to the cancelation of dynamical phase over two consecutive cycles [10-13]. Geometric phase based quantum computing is normally less susceptible to the environmental noise. It is decoherence caused by the environment, that constraints implementation of quantum gates. This is true for the coupled quantum dot system also. In a recent work, Berry phase of two coupled quantum dots was calculated in the frame
2921
work of cavity quantum electrodynamics [15]. The interaction of the phonon field with quantum dots was also taken into account in this work. It was shown that the Berry phase changes near a critical temperature determined by the resonance condition of the cavity and thus the Berry phase before and after critical temperature maintains some fixed values. This work suggests that in our case also despite the presence of decoherence mechanism due to the environment we can maintain a fixed Berry phase up to a certain temperature range and can do the quantum information processing. It is possible to get further physical insight into the effect of decoherence on geometric phases with the help of Ref. [20], which may be applicable to our study. According to [20] the physical picture emerging out is that for the large decoherence the geometric phase becomes vanishingly small during the evolution of the Bloch vector over the Bloch-sphere in a closed path. However, when there is moderate decoherence, the implementation of geometric-phase quantum gate is feasible [20] and we can achieve such a moderate decoherence in our system. In our scheme, the value of Förster interaction W depends on the dot sizes and the confinement potential. Also, W can be modulated by applying an electric field. The measured dipole values for CdSe dots range in 0.9 to 5.2 A and the corresponding W goes as 0.02 to 0.6 meV leading to energy transfer rate of several tens of picoseconds [22]. This time is short enough to be useful for any quantum information processing using geometric-phase based quantum gates because the decoherence times as large as a few nanoseconds have been observed in QDs [22]. Arrays of strongly interacting individual molecules may also behave as an excellent system for QIP tasks using Förster interaction. In such systems the typical Förster interaction strength is 8.3 mev and the transfer times are of the order of 500 ps. Finally, we would like to give a basic outline for realization of the above work experimentally. The theoretical understanding of geometric phase for a pure quantum state is quite clear and has been experimentally demonstrated in NMR, single-photon and two-photon interferometry [24]. When decoherence is present in the system then it leads to the preparation of mixed states. For the mixed state, geometric phase has been interpreted in terms of quantum interferometry. The Förster interaction is controlled by the external electric field. When field amplitude is large, the Förster interaction remains suppressed. By decreasing field amplitude, this interaction can be initiated. The Förster interaction is a sensitive function of applied electric field as the electric field caused separation of electrons and holes to increase and thus overlap integral of electron–hole decreases, consequently reduces the Förster interaction. Initially, one starts with state |1, 0〉 by selectively exciting QD-I. When the Förster interaction is turned on, the system evolves into the eigenstate |ξ2〉 or |ξ3〉. This initiates tipping operation of the Bloch vector through an angle θ (defined after Eq. (6)) with α at zero. One can stop the evolution by applying an external electric field when the system is in one of these eigenstates (which are entangled states). Adiabatic variation of α(t) from 0 to 2π at this stage results into acquisition of the adiabatic geometric phase. This can be achieved by varying the phase of the external field that controls the Förster interaction parameter [22]. For this purpose one needs to apply an adiabatic sweep which will provide circular motion in the parametric space of W. In order to cancel the dynamical phase one has to move back in the reverse path also (i.e., the two-cycle scheme to cancel dynamical phase as described above). The essential ingredients of a quantum interferometer, which can eventually be implemented in the experiment are described as follows. The quantum system of two QDs undergoes a series of unitary evolutions after which the probability of finding system in one of its eigenstates becomes an oscillatory function [25,26] of the control parameter, i.e., the Förster interaction. Such oscillation in probability is similar to an optical interference pattern [25]. The shift of interference pattern is a function of the geometric phases acquired by the double QDs system during the unitary evolutions, as well as purity (which depends on the decoherence) of the internal states of the double QDs system. Thus geometric phase can be directly
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A. Joshi, S.S. Hassan / Optics Communications 284 (2011) 2919–2922
measured from the shift of the interference pattern. At the end of interferometric operation the tomography will be required to construct the components of density matrix. Such geometric phase has been experimentally measured in NMR and single photon interferometry [26]. On the lines of these experiments one can implement our proposal of experiment with QDs. 4. Summary We have considered a Hamiltonian system of two quantum dots coupled through their dipole–dipole Förster interaction and also via the static exciton–exciton dipole interaction energy. It is assumed the Hamiltonian has a time-variation of the Förster interaction parameter. In an adiabatic limit of this variation, the geometric Berry phase has been calculated. Based on the idea of two consecutive cyclic evolution, the net dynamic phase can be canceled while the net geometric phase is doubled in the final unitary transformation of the system. We can extend this idea for using several interacting quantum dots to construct three-qubit and multi-qubit entangling gates and hence to generate nqubit 2-dimensional graph (cluster) state entanglement which will be useful to prepare a practical quantum information processing system. Acknowledgements Many useful discussions with Min Xiao is gratefully acknowledged. We acknowledge the funding support from Research Corporation. References [1] G. Benenti, G. Casati, G. Strini, Prinicples of Quantum Computation and Information, Vol. I, World Scientific, Singapore, 2004. [2] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, England, 2000. [3] D. Bouwmeester, A.K. Ekert, A. Zeilinger, The Physics of Quantum Information, Springer-Verlag, Berlin, 2000. [4] E. Biolatti, R.C. Iotti, P. Zanardi, F. Rossi, Phys. Rev. Lett. 85 (2000) 5647. [5] A. Imamoglu, D.D. Awschalom, G. Burkard, D.P. DiVincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83 (1999) 4. [6] Daniel Loss, David P. DiVincenzo, Phys. Rev. A 57 (1998) 120.
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