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Effect of quantum lattice fluctuations on quantum coherent oscillations in a coherently driven quantum dot-cavity system
Physics Letters A 314 (2003) 380–385 www.elsevier.com/locate/pla
Effect of quantum lattice fluctuations on quantum coherent oscillations in a coherently driven quantum dot-cavity system Ka-Di Zhu a,b,∗ , Wai-Sang Li b a Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China b Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hong Kong, China
Received 11 February 2003; received in revised form 12 June 2003; accepted 13 June 2003 Communicated by P.R. Holland
Abstract The quantum coherent oscillations in a coherently driven quantum dot-cavity system with the presence of strong exciton– phonon interactions are investigated theoretically in a fully quantum treatment. It is shown that even at zero temperature, the strong exciton–phonon interactions still affect the quantum coherent oscillations significantly. 2003 Elsevier B.V. All rights reserved. PACS: 71.35.Cc; 73.21.La; 03.67.Lx
Quantum computations have the potential to perform tasks exponentially faster than the classical computations. The most important role to this speedup is the entanglement due to the superposition of states in quantum bits (qubits) [1–3]. But the requisite quantum coherence is very fragile, and can be destroyed by the interactions with the environment or other noise sources. Therefore, qubits must be sufficiently isolated from the environment so that they can maintain their coherence throughout the computation. However, for the quantum computations based on solid-state qubits, the qubits will unavoidably be influenced by the lattice vibrations, even at zero temperature. It is therefore of great importance to prevent or minimize the influence of environmental noise in the practical realization of quantum information processing. On the other side, in recent years a semiconductor quantum dot (QD) embedded in a semiconductor microcavity becomes a novel basic system for quantum information processing [4–8]. In such systems, excitons in the quantum dots constitute an alternative two-level system instead of the usual two-level atomic systems. In general, these small quantum dots are characterized by strong excitonphonon interactions [9–12]. Thus the effects of exciton–phonon interactions and exciton-exciton interactions will play an important role in this quantum dot-cavity system and also make natural difference from that in the usual two-level atom-cavity system. How the quantum decoherences due to the exciton–phonon interactions and exciton– exciton interactions to affect the quantum information processing based on the QD cavity-QED is one of the most hottest research subjects in current quantum information science. More recently, Wilson-Rae and Imamoglu [13] * Corresponding author.
E-mail address: [email protected] (K.-D. Zhu). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00973-3
K.-D. Zhu, W.-S. Li / Physics Letters A 314 (2003) 380–385
381
have shown that for super-Ohmic spectral functions the main role of exciton–phonon interactions is the reduction of the quantum dot-cavity coupling strength, but the influence of the external driving field has not been considered. On the other hand, excitonic Rabi oscillations in single quantum dots are observed in recent experiments [14–17]. Zrenner et al. [18] have demonstrated that such excitonic coherent oscillations in the quantum dot two-level system can be converted into deterministic photocurrents and found that this can function as an optically triggered single-electron turnstile device. The observation of such nonclassical oscillations would be a first step towards the quantum information processing in semiconductor nanostructures. In this Letter, we will further study the influence of the quantum lattice fluctuations on excitonic Rabi oscillations in such quantum dot-cavity systems with the external driving field. The exact results of the nonclassical Rabi oscillations with the presence of the quantum lattice fluctuations are obtained analytically. We find that the quantum Rabi oscillations dressed by quantum lattice fluctuations will still persist with a new single-photon Rabi frequency ge−λ/2 where g is the single–photon Rabi frequency without exciton–phonon interactions and λ is the Huang–Rhys factor which corresponds to the exciton– phonon couplings. Further, the influence of the external driving field on the coherent oscillations is also presented. In what follows, we assume a simple two-level model for a semiconductor quantum dot which consists of the electronic ground state |1 and the lowest-energy electron-hole (exciton) state |2. Such a quantum dot is placed inside a high-Q single mode cavity. We consider that this quantum dot via the exciton interacts with a single mode of the radiation field of frequency ωs and is coherently driven by an external quantized field of frequency ωc . Here the external driving field is treated quantum-mechanically so that the intensity of the field is unnecessarily assumed to be strong enough to render the photon fluctuation effects negligible. As usual this two-level system can be characterized by the pseudospin-1/2 operators S ± , S z and the cavity field (the external field) is characterized by the annihilation and creation operators as (ac ) and as+ (ac+ ) with Bose commutation relation [as , as+ ] = 1 ([ac , ac+ ] = 1). In this model, we no longer consider the exciton–phonon interaction as a perturbation, but take into account the new eigenstates resulting from the strong coupling of exciton with the bulk acoustic phonons. The total Hamiltonian for this coupled photon–exciton–phonon system in the dipole and rotating-wave approximations is written as (h¯ = 1) [13] H = ωex S z + 1/2 + ωs as+ as + ωc ac+ ac + gs S + as + S − as+ ωq bq+ bq + (S z + 1/2) Mq bq+ + bq , + gc S + ac + S − ac+ + (1) q
q
where ωex is the exciton frequency, gs is the coupling constant of the exciton-cavity photon and gc is the coupling constant of the exciton and the external field photon. bq+ (bq ) is the creation (annihilation) operator of the phonon (with momentum q and frequency ωq ). The last term is the exciton–phonon interaction characterized by the matrix elements Mq . For the sake of simplicity, we also assume that the off-diagonal exciton–phonon interactions are negligible if the energy separation between the states in the quantum dots is greater than 20 meV when the temperature is low enough (T < 50 K) [12,13]. For an InGaAs quantum dot, the energy separation is about 65 meV from the ground-state transition [19]. We first apply a canonical transformation to the Hamiltonian (1) [20] H = exp(S)H exp(−S),
(2)
where the generator S is Mq + S = S z + 1/2 bq − bq . ωq q
(3)
The transformed Hamiltonian is given by H = H0 + HI ,
(4)
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where
ωq bq+ bq , H0 = (ωex − ∆) S z + 1/2 + ωs as+ as + ωc ac+ ac +
(5)
q
HI = gs S + X+ as + S − Xas+ + gc S + X+ ac + S − Xac+ , where ∆ = q Mq2 /ωq and Mq X = exp − bq+ − bq , X+ = X−1 . ω q q
(6)
(7)
In what follows, it is convenient to work in the interaction picture with H0 . The Hamiltonian, in the interaction picture, is given by
H = eiH0 t HI e−iH0 t .
(8)
Using eαA Be−αA = B + α[A, B] +
α2 A, [A, B] + · · · , 2!
(9)
it can be readily seen that +a
eiωs as e
st
iωc ac+ ac t
+a
as e−iωs as
st
−iωc ac+ ac t
= as e−iωs t ,
ac e = ac e i(ωex −∆)S z t + −i(ωex −∆)S z t
−iωc t
(10)
,
(11)
+ i(ωex −∆)t
(12)
S e =S e ,
Mq + iωq t + + −iωq t b e ωq bq bq t X exp −i ωq bq bq t = exp − − bq e exp i . ωq q q q q
e
Combining Eqs. (4)–(6), (8) and (10)–(13), we have H = gs S + X+ (t)as eiδs t + S − X(t)as+ e−iδs t + gc S + X+ (t)ac eiδc t + S − X(t)ac+ e−iδc t , where δs = ωex − ∆ − ωs , δc = ωex − ∆ − ωc and
Mq + iωq t −iωq t X(t) = exp − b e − bq e , ωq q q
X+ (t) = X−1 (t).
(13)
(14)
(15)
The transformed Hamiltonian (Eq. (14)) from Eq. (1) is exact without making any further approximation. In what follows, for the sake of analytical simplicity, we only consider the case ωex − ∆ − ωs = ωex − ∆ − ωc = δ, and gs = gc = g [21,22]. Therefore, the Hamiltonian (14) becomes H = g S + X+ (t)eiδt (as + ac ) + S − X(t)e−iδt as+ + ac+ . (16) It is useful to define the normal-mode operators by [22] 1 A = √ (ac + as ), 2 1 B = √ (ac − as ). 2 These are annihilation operators just like as and ac and obey the Bose commutation relations, B, B + = 1. A, A+ = 1,
(17) (18)
(19)
K.-D. Zhu, W.-S. Li / Physics Letters A 314 (2003) 380–385
Moreover, the normal-mode operators commute with each other, [A, B] = 0, A, B + = 0.
383
(20)
In terms of these operators, the Hamiltonian (16) becomes independent of the antisymmetric normal-mode combination and is given by √ Heff = 2 g S + X+ (t)eiδt A + S − X(t)e−iδt A+ . (21) Next we proceed to solve the equation of motion for |ψ(t), i.e., d i ψ(t) = Heff ψ(t) . (22) dt In general, the state vector |ψ(t) is a linear combination of the states |1, m|ph and |2, m|ph. Here |2, m is the state in which the quantum dot is in the excited state |2 (i.e., with the presence of the exciton) and the symmetric normal mode of the field has m excitation number (A+ A|m = m|m). A similar description exists for the state |1, m. |ph is the phonon state, but here for the sake of analytical simplicity, we only consider the case of the quantum dot at zero temperature so that only quantum lattice fluctuations give rise to the decoherence. In this case, the phonon state is in the vacuum state |ph = |00 . . .0 . . . (for the phonon number operator bq+ bq , bq+ bq |n1 n2 . . . nq . . . = nq |n1 n2 . . . nq . . . and |n1 n2 . . . nq . . . = (bq+ )nq · · · (b2+ )n2 (b1+ )n1 |00 . . .0 . . ., where |00 . . .0 . . . is the vacuum state [23]). For the finite temperatures, the solution becomes somewhat more complicated so that we have to employ the perturbation methods. Consequently, an extension of this work to the finite temperatures will be presented elsewhere. As we are using the interaction picture, we use the slowly varying probability amplitudes c1,m,ph(t) and c2,m,ph (t). The state vector is therefore [24] c1,m,ph (t)|1, m|ph + c2,m,ph (t)|2, m|ph . |ψ(t) = (23) m
The interaction Hamiltonian (21) can only cause transitions between the states |1, m + 1|ph and |2, m|ph. We therefore consider the evolution of the amplitudes c1,m+1,ph(t) and c2,m,ph(t). The equations of motion for the probability amplitudes c2,m,ph (t) and c1,m+1,ph(t) are obtained by first substituting for |ψ(t) and Heff from Eqs. (23) and (21) in Eq. (22) and then projecting the resulting equations onto ph|m, 2| and ph|m + 1, 1|, respectively. We then obtain √ √ √ 1 d c2,m,ph (t) = −i 2 g m + 1 e− 2 λ eiδt c1,m+1,ph(t) = −ig0 2m + 2 eiδt c1,m+1,ph (t), (24) dt √ √ √ 1 d c1,m+1,ph(t) = −i 2 g m + 1 e− 2 λ e−iδt c2,m,ph(t) = −ig0 2m + 2 e−iδt c2,m,ph(t), (25) dt M 2 where g0 = ge−λ/2 and λ = q ωqq is the Huang–Rhys factor which corresponds to the exciton–phonon √ √ interactions [25]. It is obvious that the excitonic vacuum Rabi splitting is 2 2 g0 = 2 2 ge−λ/2 . The above coupled set of equations can be solved exactly subject to certain initial conditions. If initially the quantum dot is in the excited state |2 (i.e., with the presence of the exciton) then c2,m,ph (0) = cm (0) and c1,m+1,ph(0) = 0. Here cm (0) is the probability amplitude for the symmetric normal mode of the field at time t = 0. We then obtain
Ωm t Ωm t iδ c2,m,ph(t) = cm (0) cos (26) sin − eiδt /2, 2 Ωm 2 √ √
2 2 ig0 m + 1 Ωm t −iδt /2 sin , c1,m+1,ph(t) = −icm (0) (27) e Ωm 2 where 2 = δ 2 + 8g02 (m + 1). Ωm
(28)
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In the following, we consider the case in which initially both external field and cavity field are in coherent states, |βc and |αs , respectively. According to Ref. [22], we have 1 1 |αs |βc = √ (β + α) √ (β − α) . (29) 2 2 A
B
Therefore, the probability of finding the quantum dot in the excited state (i.e., with the presence of the exciton) at t = 0 is given by Pe (t) =
∞
|cm (0)|2 cos2 (Ωm t/2) + (δ/Ωm )2 sin2 (Ωm t/2) ,
(30)
m=0
where
|α + β|2 m 1 . |cm (0)| = e (31) 2 m! If we consider the case of exact resonance, i.e., δ = 0 which corresponds to the external field and the cavity field are all on resonance with the zero-phonon line, then we get the following expression for Pe (t):
∞ √ 1 |α + β|2 m 1 −|α+β|2 /2 cos 2g0 2m + 2 t . Pe (t) = (32) 1+e 2 2 m! 2
−|α+β|2 /2
m=0
It should be noted here that g0 = ge−λ/2 . This analytical result indicates that the effect of quantum lattice fluctuations on the excitonic Rabi oscillations is just added a factor of exp(−λ/2) on the exciton–photon coupling constant g, but the coherent oscillations can still persist like those in the two-level atom-cavity systems. It should be noted that the couplings of exciton and phonons will not become too large so that the QD-cavity system is not in the strong-coupling regime. For CdSe quantum dots the Huang–Rhys factor λ ≈ 1 [11], for self-organized InAs/GaAs quantum dots λ ≈ 0.015 [10] and for other semiconductor quantum dots such as GaAs [14] and InGaAs [15] λ is even more small. As a result, the QD-cavity systems based on these quantum dots can all remain in the strong-coupling regime so that the excitonic Rabi oscillations can be observed in these systems [14,15]. From the analytical expression we can also see that the coherent oscillations can be modified by manipulating the amplitude (β) of the external driving field. In conclusion, by using a fully quantum treatment, we have investigated the influence of strong exciton–phonon interactions on the quantum Rabi oscillations in a coherently driven quantum dot in a high-Q single-mode cavity. The exact results for the nonclassical Rabi oscillations at zero temperature are obtained analytically. It is found that the coherent oscillations dressed by quantum lattice fluctuations can persist with the coupling constant ge−λ/2 . Our analytical results thus indicate that even at zero temperature, the strong exciton–phonon interactions still affect the quantum coherent oscillations significantly. This gives a natural limitation to the quantum information processing based on quantum dot cavity-QED. The analytical results also show that the coherent oscillations can be modified by manipulating the amplitude of the external driving field. It should be emphasized that for an InGaAs quantum dot ensemble, an exciton lifetime is 1ns and a dephasing time of the zero-phonon line at 10 K is 500 ps [17,19], so finite temperature effects (especially for the temperature T > 50 K [12,19]) would rapidly overtake the kinds of the “vacuum” excitations discussed in this Letter. Finally, we hope that this work will stimulate more theoretical and experimental works which will be helpful for a better understanding of quantum dot-cavity systems in quantum information processing.
Acknowledgements The part of this work was supported by the 2000’s Distinguished Youth Foundation of the Ministry of Education, National Natural Science Foundation of China (No. 10274051), Research Grant Committee of The Hong Kong
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Polytechnic University (No. G.42.37.T380) and the Ph.D. Training Foundation of the Ministry of Education. The authors also thank the referee to bring their attention to thinking about the finite temperature effects and the other important effects in the quantum dot-cavity systems.
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Effect of quantum lattice fluctuations on quantum coherent oscillations in a coherently driven quantum dot-cavity system Ka-Di Zhu a,b,∗ , Wai-Sang Li b a Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China b Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hong Kong, China
Received 11 February 2003; received in revised form 12 June 2003; accepted 13 June 2003 Communicated by P.R. Holland
Abstract The quantum coherent oscillations in a coherently driven quantum dot-cavity system with the presence of strong exciton– phonon interactions are investigated theoretically in a fully quantum treatment. It is shown that even at zero temperature, the strong exciton–phonon interactions still affect the quantum coherent oscillations significantly. 2003 Elsevier B.V. All rights reserved. PACS: 71.35.Cc; 73.21.La; 03.67.Lx
Quantum computations have the potential to perform tasks exponentially faster than the classical computations. The most important role to this speedup is the entanglement due to the superposition of states in quantum bits (qubits) [1–3]. But the requisite quantum coherence is very fragile, and can be destroyed by the interactions with the environment or other noise sources. Therefore, qubits must be sufficiently isolated from the environment so that they can maintain their coherence throughout the computation. However, for the quantum computations based on solid-state qubits, the qubits will unavoidably be influenced by the lattice vibrations, even at zero temperature. It is therefore of great importance to prevent or minimize the influence of environmental noise in the practical realization of quantum information processing. On the other side, in recent years a semiconductor quantum dot (QD) embedded in a semiconductor microcavity becomes a novel basic system for quantum information processing [4–8]. In such systems, excitons in the quantum dots constitute an alternative two-level system instead of the usual two-level atomic systems. In general, these small quantum dots are characterized by strong excitonphonon interactions [9–12]. Thus the effects of exciton–phonon interactions and exciton-exciton interactions will play an important role in this quantum dot-cavity system and also make natural difference from that in the usual two-level atom-cavity system. How the quantum decoherences due to the exciton–phonon interactions and exciton– exciton interactions to affect the quantum information processing based on the QD cavity-QED is one of the most hottest research subjects in current quantum information science. More recently, Wilson-Rae and Imamoglu [13] * Corresponding author.
E-mail address: [email protected] (K.-D. Zhu). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00973-3
K.-D. Zhu, W.-S. Li / Physics Letters A 314 (2003) 380–385
381
have shown that for super-Ohmic spectral functions the main role of exciton–phonon interactions is the reduction of the quantum dot-cavity coupling strength, but the influence of the external driving field has not been considered. On the other hand, excitonic Rabi oscillations in single quantum dots are observed in recent experiments [14–17]. Zrenner et al. [18] have demonstrated that such excitonic coherent oscillations in the quantum dot two-level system can be converted into deterministic photocurrents and found that this can function as an optically triggered single-electron turnstile device. The observation of such nonclassical oscillations would be a first step towards the quantum information processing in semiconductor nanostructures. In this Letter, we will further study the influence of the quantum lattice fluctuations on excitonic Rabi oscillations in such quantum dot-cavity systems with the external driving field. The exact results of the nonclassical Rabi oscillations with the presence of the quantum lattice fluctuations are obtained analytically. We find that the quantum Rabi oscillations dressed by quantum lattice fluctuations will still persist with a new single-photon Rabi frequency ge−λ/2 where g is the single–photon Rabi frequency without exciton–phonon interactions and λ is the Huang–Rhys factor which corresponds to the exciton– phonon couplings. Further, the influence of the external driving field on the coherent oscillations is also presented. In what follows, we assume a simple two-level model for a semiconductor quantum dot which consists of the electronic ground state |1 and the lowest-energy electron-hole (exciton) state |2. Such a quantum dot is placed inside a high-Q single mode cavity. We consider that this quantum dot via the exciton interacts with a single mode of the radiation field of frequency ωs and is coherently driven by an external quantized field of frequency ωc . Here the external driving field is treated quantum-mechanically so that the intensity of the field is unnecessarily assumed to be strong enough to render the photon fluctuation effects negligible. As usual this two-level system can be characterized by the pseudospin-1/2 operators S ± , S z and the cavity field (the external field) is characterized by the annihilation and creation operators as (ac ) and as+ (ac+ ) with Bose commutation relation [as , as+ ] = 1 ([ac , ac+ ] = 1). In this model, we no longer consider the exciton–phonon interaction as a perturbation, but take into account the new eigenstates resulting from the strong coupling of exciton with the bulk acoustic phonons. The total Hamiltonian for this coupled photon–exciton–phonon system in the dipole and rotating-wave approximations is written as (h¯ = 1) [13] H = ωex S z + 1/2 + ωs as+ as + ωc ac+ ac + gs S + as + S − as+ ωq bq+ bq + (S z + 1/2) Mq bq+ + bq , + gc S + ac + S − ac+ + (1) q
q
where ωex is the exciton frequency, gs is the coupling constant of the exciton-cavity photon and gc is the coupling constant of the exciton and the external field photon. bq+ (bq ) is the creation (annihilation) operator of the phonon (with momentum q and frequency ωq ). The last term is the exciton–phonon interaction characterized by the matrix elements Mq . For the sake of simplicity, we also assume that the off-diagonal exciton–phonon interactions are negligible if the energy separation between the states in the quantum dots is greater than 20 meV when the temperature is low enough (T < 50 K) [12,13]. For an InGaAs quantum dot, the energy separation is about 65 meV from the ground-state transition [19]. We first apply a canonical transformation to the Hamiltonian (1) [20] H = exp(S)H exp(−S),
(2)
where the generator S is Mq + S = S z + 1/2 bq − bq . ωq q
(3)
The transformed Hamiltonian is given by H = H0 + HI ,
(4)
382
K.-D. Zhu, W.-S. Li / Physics Letters A 314 (2003) 380–385
where
ωq bq+ bq , H0 = (ωex − ∆) S z + 1/2 + ωs as+ as + ωc ac+ ac +
(5)
q
HI = gs S + X+ as + S − Xas+ + gc S + X+ ac + S − Xac+ , where ∆ = q Mq2 /ωq and Mq X = exp − bq+ − bq , X+ = X−1 . ω q q
(6)
(7)
In what follows, it is convenient to work in the interaction picture with H0 . The Hamiltonian, in the interaction picture, is given by
H = eiH0 t HI e−iH0 t .
(8)
Using eαA Be−αA = B + α[A, B] +
α2 A, [A, B] + · · · , 2!
(9)
it can be readily seen that +a
eiωs as e
st
iωc ac+ ac t
+a
as e−iωs as
st
−iωc ac+ ac t
= as e−iωs t ,
ac e = ac e i(ωex −∆)S z t + −i(ωex −∆)S z t
−iωc t
(10)
,
(11)
+ i(ωex −∆)t
(12)
S e =S e ,
Mq + iωq t + + −iωq t b e ωq bq bq t X exp −i ωq bq bq t = exp − − bq e exp i . ωq q q q q
e
Combining Eqs. (4)–(6), (8) and (10)–(13), we have H = gs S + X+ (t)as eiδs t + S − X(t)as+ e−iδs t + gc S + X+ (t)ac eiδc t + S − X(t)ac+ e−iδc t , where δs = ωex − ∆ − ωs , δc = ωex − ∆ − ωc and
Mq + iωq t −iωq t X(t) = exp − b e − bq e , ωq q q
X+ (t) = X−1 (t).
(13)
(14)
(15)
The transformed Hamiltonian (Eq. (14)) from Eq. (1) is exact without making any further approximation. In what follows, for the sake of analytical simplicity, we only consider the case ωex − ∆ − ωs = ωex − ∆ − ωc = δ, and gs = gc = g [21,22]. Therefore, the Hamiltonian (14) becomes H = g S + X+ (t)eiδt (as + ac ) + S − X(t)e−iδt as+ + ac+ . (16) It is useful to define the normal-mode operators by [22] 1 A = √ (ac + as ), 2 1 B = √ (ac − as ). 2 These are annihilation operators just like as and ac and obey the Bose commutation relations, B, B + = 1. A, A+ = 1,
(17) (18)
(19)
K.-D. Zhu, W.-S. Li / Physics Letters A 314 (2003) 380–385
Moreover, the normal-mode operators commute with each other, [A, B] = 0, A, B + = 0.
383
(20)
In terms of these operators, the Hamiltonian (16) becomes independent of the antisymmetric normal-mode combination and is given by √ Heff = 2 g S + X+ (t)eiδt A + S − X(t)e−iδt A+ . (21) Next we proceed to solve the equation of motion for |ψ(t), i.e., d i ψ(t) = Heff ψ(t) . (22) dt In general, the state vector |ψ(t) is a linear combination of the states |1, m|ph and |2, m|ph. Here |2, m is the state in which the quantum dot is in the excited state |2 (i.e., with the presence of the exciton) and the symmetric normal mode of the field has m excitation number (A+ A|m = m|m). A similar description exists for the state |1, m. |ph is the phonon state, but here for the sake of analytical simplicity, we only consider the case of the quantum dot at zero temperature so that only quantum lattice fluctuations give rise to the decoherence. In this case, the phonon state is in the vacuum state |ph = |00 . . .0 . . . (for the phonon number operator bq+ bq , bq+ bq |n1 n2 . . . nq . . . = nq |n1 n2 . . . nq . . . and |n1 n2 . . . nq . . . = (bq+ )nq · · · (b2+ )n2 (b1+ )n1 |00 . . .0 . . ., where |00 . . .0 . . . is the vacuum state [23]). For the finite temperatures, the solution becomes somewhat more complicated so that we have to employ the perturbation methods. Consequently, an extension of this work to the finite temperatures will be presented elsewhere. As we are using the interaction picture, we use the slowly varying probability amplitudes c1,m,ph(t) and c2,m,ph (t). The state vector is therefore [24] c1,m,ph (t)|1, m|ph + c2,m,ph (t)|2, m|ph . |ψ(t) = (23) m
The interaction Hamiltonian (21) can only cause transitions between the states |1, m + 1|ph and |2, m|ph. We therefore consider the evolution of the amplitudes c1,m+1,ph(t) and c2,m,ph(t). The equations of motion for the probability amplitudes c2,m,ph (t) and c1,m+1,ph(t) are obtained by first substituting for |ψ(t) and Heff from Eqs. (23) and (21) in Eq. (22) and then projecting the resulting equations onto ph|m, 2| and ph|m + 1, 1|, respectively. We then obtain √ √ √ 1 d c2,m,ph (t) = −i 2 g m + 1 e− 2 λ eiδt c1,m+1,ph(t) = −ig0 2m + 2 eiδt c1,m+1,ph (t), (24) dt √ √ √ 1 d c1,m+1,ph(t) = −i 2 g m + 1 e− 2 λ e−iδt c2,m,ph(t) = −ig0 2m + 2 e−iδt c2,m,ph(t), (25) dt M 2 where g0 = ge−λ/2 and λ = q ωqq is the Huang–Rhys factor which corresponds to the exciton–phonon √ √ interactions [25]. It is obvious that the excitonic vacuum Rabi splitting is 2 2 g0 = 2 2 ge−λ/2 . The above coupled set of equations can be solved exactly subject to certain initial conditions. If initially the quantum dot is in the excited state |2 (i.e., with the presence of the exciton) then c2,m,ph (0) = cm (0) and c1,m+1,ph(0) = 0. Here cm (0) is the probability amplitude for the symmetric normal mode of the field at time t = 0. We then obtain
Ωm t Ωm t iδ c2,m,ph(t) = cm (0) cos (26) sin − eiδt /2, 2 Ωm 2 √ √
2 2 ig0 m + 1 Ωm t −iδt /2 sin , c1,m+1,ph(t) = −icm (0) (27) e Ωm 2 where 2 = δ 2 + 8g02 (m + 1). Ωm
(28)
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In the following, we consider the case in which initially both external field and cavity field are in coherent states, |βc and |αs , respectively. According to Ref. [22], we have 1 1 |αs |βc = √ (β + α) √ (β − α) . (29) 2 2 A
B
Therefore, the probability of finding the quantum dot in the excited state (i.e., with the presence of the exciton) at t = 0 is given by Pe (t) =
∞
|cm (0)|2 cos2 (Ωm t/2) + (δ/Ωm )2 sin2 (Ωm t/2) ,
(30)
m=0
where
|α + β|2 m 1 . |cm (0)| = e (31) 2 m! If we consider the case of exact resonance, i.e., δ = 0 which corresponds to the external field and the cavity field are all on resonance with the zero-phonon line, then we get the following expression for Pe (t):
∞ √ 1 |α + β|2 m 1 −|α+β|2 /2 cos 2g0 2m + 2 t . Pe (t) = (32) 1+e 2 2 m! 2
−|α+β|2 /2
m=0
It should be noted here that g0 = ge−λ/2 . This analytical result indicates that the effect of quantum lattice fluctuations on the excitonic Rabi oscillations is just added a factor of exp(−λ/2) on the exciton–photon coupling constant g, but the coherent oscillations can still persist like those in the two-level atom-cavity systems. It should be noted that the couplings of exciton and phonons will not become too large so that the QD-cavity system is not in the strong-coupling regime. For CdSe quantum dots the Huang–Rhys factor λ ≈ 1 [11], for self-organized InAs/GaAs quantum dots λ ≈ 0.015 [10] and for other semiconductor quantum dots such as GaAs [14] and InGaAs [15] λ is even more small. As a result, the QD-cavity systems based on these quantum dots can all remain in the strong-coupling regime so that the excitonic Rabi oscillations can be observed in these systems [14,15]. From the analytical expression we can also see that the coherent oscillations can be modified by manipulating the amplitude (β) of the external driving field. In conclusion, by using a fully quantum treatment, we have investigated the influence of strong exciton–phonon interactions on the quantum Rabi oscillations in a coherently driven quantum dot in a high-Q single-mode cavity. The exact results for the nonclassical Rabi oscillations at zero temperature are obtained analytically. It is found that the coherent oscillations dressed by quantum lattice fluctuations can persist with the coupling constant ge−λ/2 . Our analytical results thus indicate that even at zero temperature, the strong exciton–phonon interactions still affect the quantum coherent oscillations significantly. This gives a natural limitation to the quantum information processing based on quantum dot cavity-QED. The analytical results also show that the coherent oscillations can be modified by manipulating the amplitude of the external driving field. It should be emphasized that for an InGaAs quantum dot ensemble, an exciton lifetime is 1ns and a dephasing time of the zero-phonon line at 10 K is 500 ps [17,19], so finite temperature effects (especially for the temperature T > 50 K [12,19]) would rapidly overtake the kinds of the “vacuum” excitations discussed in this Letter. Finally, we hope that this work will stimulate more theoretical and experimental works which will be helpful for a better understanding of quantum dot-cavity systems in quantum information processing.
Acknowledgements The part of this work was supported by the 2000’s Distinguished Youth Foundation of the Ministry of Education, National Natural Science Foundation of China (No. 10274051), Research Grant Committee of The Hong Kong
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Polytechnic University (No. G.42.37.T380) and the Ph.D. Training Foundation of the Ministry of Education. The authors also thank the referee to bring their attention to thinking about the finite temperature effects and the other important effects in the quantum dot-cavity systems.
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