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Quantum Information and Computation, Vol. 14, No. 15&16 (2014) 1372–1382 c Rinton Press
GENERATION OF QUANTUM CORRELATION IN QUANTUM DOTS BY LYAPUNOV CONTROL METHODS
DA-WEI LUO Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University Hangzhou 310027, People’s Republic of China JING-BO XUa Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University Hangzhou 310027, People’s Republic of China
Received February 17, 2014 Revised April 10, 2014 We present a scheme to generate steady-state quantum correlations in quantum dots. The shape of the control field is obtained by using the Lyapunov control method, and the controlled state is found to approach the target state monotonically. We also explore the possibility of replacing the continuous control field with a train of discreet rectangular pulses, which is much easier to implement experimentally. The discretized control field is found to be still able to drive the initial state to the target state with very small errors, which suggests that the Lyapunov based control is still effective under practical limitations. Keywords: Quantum entanglement, Control theory, Quantum dots Communicated by: I Cirac & G Milburn
1
Introduction
It is well known that the field of quantum information and quantum computation has received a lot of research interest due to its potential applications. Quantum entanglement, as a special kind of quantum correlation, has been considered as a valuable resource for many quantum information tasks, such as quantum teleportation and quantum cryptography [1]. The generation and manipulation of entanglement is therefore a problem of fundamental interest in quantum information processing. However, to efficiently accomplish quantum information tasks, free unitary evolution under a time-independent Hamiltonian is generally not good enough. Therefore, the efficient control of quantum mechanical process has become a very important research focus over the years. Several control methods has been proven to be very successful in achieving the desired time evolution of quantum mechanical systems, such as the dynamic pulse control to combat decoherence [2]. The Lyapunov function plays a very important role in control design, and this method has been shown to be very effective for spin systems as well as cavity QED systems connected with optical fibers [3, 4]. Experimentally, several physical systems for the realization of quantum information processing has been put forward, including cavity QED, quantum dots, nuclear magnetic [email protected]
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onance (NMR) and spin systems [1]. Among those proposals, the semiconductor quantum dots are a particularly promising candidate since they are solid-based, have great scaling potentials and can be finely-controlled [5, 6, 7, 8, 9]. With present experimental equipment, it has become a standard technology to confine single-electron charges in quantum dots as required, and can be measured relatively easily [10]. Since entanglement is essential for a wide range of quantum information and computation tasks, the generation of entanglement is of particular experimental and theoretical interest [11, 12, 13, 14, 15, 16, 17]. In Ref. [12, 13], the entanglement generation on two and three identical, equally spaced quantum dots with classical laser is studied. The quantum dots system under consideration has a resonant energy transfer process between dots known as the F¨orster process. However, the achieved fidelity of the generated entanglement is found to be highly oscillating, so very precise timing is needed to extract high quality entanglement, and this big fluctuating behavior is not desirable for the practical implementation of quantum information tasks. In this paper, we use the Lyapunov control method to drive the system of quantum dots into a stationary state that is maximally entangled. We give a brief outline of the Lyapunov based control method and then apply it to two coupled quantum dots and generate a maximally entangled Bell state. System consisting of three coupled quantum dots are also considered, and it is found that the three-body W state can also be generated using the Lyapunov control. Due to experimental limitations, the continuous control field may be difficult to generate, and we finally proposed a simplified discreet pulse sequence based on the Lyapunov control method, and find that the control is still effective. 2
The model and the Lyapunov-based control methods
Quantum dots are generally considered as a type of artificial atoms for quantum optics [18]. In this paper, we study a system of N identical, equally spaced quantum dots containing no net charge and radiated by a classical field. The formation of single excitons within the individual quantum dots with inter-dot hopping can be described by the Hamiltonian [12, 13]
H
=
N N W X † ǫ X † ep ep − hp h†p + ep hp′ ep′ h†p 2 p=1 2 ′ p,p =1
+hp e†p′ h†p′ ep
i
+ E(t)
N X p=1
e†p h†p
∗
+ E (t)
N X
h p ep ,
p=1
(1) where e†p and h†p are the electron and hole creation operators, ǫ is the quantum dot band gap, W is the inter-dot Coulomb interaction, and E(t) is the laser field strength. In reality, the band gap ǫ and inter-dot Coulomb interaction W may be different across different quantum dots. However, we can approximately treat both as uniform, which allows us to analytically study the system and its control method. This idealized treatment [12] can be a valid description of the quantum dot system. The operators are fermion and obey the anti-commutation rules {ep , e†p′ } = {hp , h†p′ } = δp,p′ . Using the standard approach, we introduce the following
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Generation of quantum correlation in quantum dots by Lyapunov control methods
operators J+ =
PN † † J− = p=1 p=1 ep hp , Jz = 12 e†p ep − hp h†p ,
PN
h p ep (2)
which are easily verified to be the usual angular momentum operators. The Hamiltonian Eq. (1) can now be expressed using the angular momentum operators and reads H = ǫJz + W J 2 − Jz2 +E(t)J+ + E ∗ (t)J− ,
(3)
In Ref. [12], a rectangular radiation pulse shape E(t) was proposed as an experimentally achievable way to generate entangled states in quantum dots. In this paper, we substitute this with the pulse shape determined by the Lyapunov control theory to study how to apply the control method presented in this paper to generate the entangled target state in a quantum dot system and compare our approach with that of Ref. [12]. The Hamiltonian above is generally time-dependent. However, if we consider a laser field of the shape E(t) = E0 e−iωt , we can eliminate the time dependence in the rotating frame using the rule |ψir
=
R† |ψis
Hr
=
R† HR + i(∂t R† )R
R
=
e−iωJz t
(4)
where the subscripts r and s stand for the rotating frame and the Schr¨odinger frame respec∂ |ψ(t)ir = Hr |ψ(t)ir with tively. After the rotation, the time evolution is given by i ∂t Hr = ∆Jz + W (J 2 − Jz2 ) + E0 J+ + E0∗ J− ,
(5)
where ∆ = ǫ − ω is the detuning. The Hamiltonian now is time-independent in the rotating frame, and can be easily used to solve the wave function’s time evolution using the standard approach [13, 12] in each fixed-J subspace since [J 2 , Hr ] = 0. The wave function obtained can then be directly transformed back to the Schr¨odinger picture using the relationship |ψis = R|ψir since R is unitary. However, as illustrated in [13, 12], applying the laser field of the form E0 e−iωt results in a highly oscillating fidelity between the system state and target state, which is undesirable for quantum information tasks because highly accurate timing is required to obtain the target state due to the big time fluctuation, which is a big experimental obstacle. In [14], a Gaussian pulse is suggested, which can generate Bell and GHZ steady states at resonance by avoiding the level-crossing. Here, we propose a scheme to monotonically generate maximally entangled states using the Lyapunov control method, which we find to be effective both at resonance and with non-zero detuning ∆. The Lyapunov control method [3, 4] is a type of feedback control in that the external control parameter is determined by the system’s current state. For a general bilinear Hamiltonian system, the time evolution is given by the Liouville equation ρ(t) ˙ = −i[H0 + f (t)Hc , ρ(t)],
(6)
where H0 is the system Hamiltonian, Hc the control Hamiltonian, both assumed to be timeindependent, and f (t) is the time-dependent, real-valued control parameter. The control
D-W Luo and J-B Xu
Fidelity 1.0 0.8 0.6 0.4 0.2 0.0 0
5
10 15 20 Concurrence
25
30
5
10 15 20 25 Lyapunov field
30
t HsL
1.0 0.8 0.6 0.4 0.2 0.0 0
5
10
15
20
25
30
t HsL
t HsL
-0.2 -0.4 -0.6 Fig. 1. (Color online) The achieved fidelity(panel (a)), concurrence(panel (b)) and the Lyapunov field(panel (c)) as a function of time is plotted with A = 0.5, W = 1 and κ = 1.2 for the generation of Bell State in two coupled quantum dots at resonance(Solid red line). The dashed blue line corresponds to the generated state under a laser field E(t) = e−iωt .
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Generation of quantum correlation in quantum dots by Lyapunov control methods
Fidelity 1.0 0.8 0.6 0.4 0.2 0.0 0
5
10 15 20 Concurrence
25
30
5
10 15 20 25 Lyapunov field
30
t HsL
1.0 0.8 0.6 0.4 0.2 0.0 0
t HsL
0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6
5
10
15
20
25
30
t HsL
Fig. 2. (Color online) The achieved fidelity(panel (a)), concurrence(panel (b)) and the Lyapunov field(panel (c)) as a function of time for the generation of Bell State in two coupled quantum dots with non-zero detuning(Solid red line). The dashed blue line corresponds to the generated state under a laser field E(t) = e−iωt .
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problem is then to find such f (t) that any initial state ρ0 converges to and stays at the target state ρd as t → ∞. Since the time evolution of any Hamiltonian system is unitary, we require the spectrum of the initial and target states coincide. Practically, the system Hamiltonian H0 can not be turned off, so as a generalization, the target state is allowed to evolve according to the system Hamiltonian H0 alone, ρd˙(t) = −i[H0 , ρd (t)]. If the target state commutes with H0 we have a stationary target state. To obtain the time-dependent control parameter f (t), we first define a function d(ρ(t), ρd (t)) to be the Hilbert-Schmidt distance between the target state ρd (t) and ρ(t), the state the system is currently in: 1 d(ρ(t), ρd (t)) = Tr[(ρ(t) − ρd (t))2 ], (7) 2 and it can be shown that d(ρ(t), ρd (t)) ≥ 0 with equality if and only if ρ(t) = ρd (t). Taking its derivative against time and substituting in the Liouville equation, we have ˙ d(ρ(t), ρd (t)) = −f (t)Tr [ρd (t)[−iHc , ρ(t)]] .
(8)
If we choose the control parameter to be f (t) = κTr [ρd (t)[−iHc , ρ(t)]] ,
(9)
where κ > 0 is the control field coupling parameter, then d˙ ≤ 0. Therefore, d(ρ(t), ρd (t)) is a Lyapunov function by definition for the system described by (6). The control task ρ(t) → ρd (t) can be now transformed into the requirement that d(ρ(t), ρd (t)) → 0. As a direct consequence of the LaSalle invariance principle, for the target state to be asymptotically stable the Hamiltonians have to satisfy certain conditions [3, 4]. The fist is that H0 should be strongly regular, which means that it should have distinct gaps between any pair of energy levels Ei , Ej ∀i 6= j ∈ {1 . . . dim(H0 )}. The second condition is that Hc should be fully connected, which translates to that in the basis that H0 is diagonal, all off-diagonal elements of Hc should be non-vanishing so that all transitions between all energy levels are possible. However, for real physical systems that do not meet those requirements, it’s still possible to find subspaces where the Lyapunov control is still effective. We now illustrate its use via two examples where we generate maximum entanglement among multiple quantum dots. As we shall find out, Lyapunov control can drive the quantum dots to a maximally entangled steady-state as opposed to the highly oscillating state achieved by applying a single-frequency laser field. 3
Generation of quantum correlations on quantum dots
For two interacting quantum dots, the Hamiltonian is usually expressed in the space spanned by the spin singlet and triplet states. Furthermore, because the J = 0 and J = 1 subspaces are totally separated, we can focus on a smaller space under the basis |ai = |0, 0i, |bi = √ (|10i + |01i)/ 2 and |ci = |1, 1i, which corresponds to the vacuum, single-exciton and biexciton states respectively. Then, in the usual Schr¨odinger picture, the Hamiltonians read 0 1 0 W +ǫ 0 0 √ 2W 0 , Hc = 2A 1 0 1 (10) H0 = 0 0 1 0 0 0 W −ǫ
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Generation of quantum correlation in quantum dots by Lyapunov control methods
where A is the overall external field amplitude. It’s clear that while Hc is not fully connected, the Bell state |bi is an eigenstate of H0 . Therefore, if we set the target state to be the maximally entangled Bell state ρd (0) = |bihb|, then the target state is stationary and steady-state entanglement is now possible if we can find some subspace of the system where the Lyapunov control is effective. Moreover, because unitary evolutions do not alter the eigenvalues of the density matrix, we require the initial state of the system to be a pure state. Substituting the Hamiltonians into Eq. (9), we find that for the control to be effective, the initial state should have non-zero complex projection along |bi. As we find out, the control is still effective for the initial state |ψ(0)i = [1, 10−10 i, 0]T up to an overall normalizing factor. To calculate the amount of entanglement we use Wootters’s concurrence [19]. The concurrence C varies from C = 0 for completely separable states to C = 1 for maximally entangled state. For any two qubits, the concurrence can be explicitly calculated from their density matrix ρAB : C(ρAB ) = max(0,
p
λ1 −
p
λ2 −
p
λ3 −
p
λ4 ),
where λ1 . . . λ4 are the eigenvalues of ρ(σy ⊗ σy )ρ∗ (σy ⊗ σy ) in decreasing order. The achieved fidelity and entanglement along with the external control field are plotted in Figs 1 and 2 for system at resonance and with detuning, respectively. We can see that the achieved fidelity and entanglement is much better than the state generated by the conventional laser field E(t) = e−iωt , and it is done on a much shorter time scale. In general, the Lyapunov control theory ensures that for all systems satisfying the controllability conditions discussed in Section II, the target state can be effectively reached. Because the system still have a unitary time evolution when a Lyapunov control function is applied, the initial state and the target state cannot have different spectrum. For mixed states, that means the eigenvalues of the density operator should be the same, and, since all pure states have the same spectrum, all initial pure states can be effectively driven to a target pure state for controllable systems satisfying the conditions given in Section II. We have chosen the initial state to be |ψ(0)i = [1, x0 , x0 ]T /N , where N is an overall normalize factor, and plot the achieved fidelity as a function of time and initial state parameter x0 in Fig. 3. It can be observed for other initial states, the Lyapunov function is still able to drive the initial state to the target maximally entangled state.
Fig. 3. The achieved fidelity as a function of time and initial state parameter x0 . It can be observed for other initial states, the Lyapunov function is still able to drive the initial state to the target maximally entangled state.
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Next we consider the generation of multi-body entanglement on three equally spaced quantum dots. We will focus on the J = 3/2 subspace√as it is the only optically active one √ [12]. In the basis |ai = |000i, |bi = [|100i + |010i + |001i]/ 3, |ci = [|110i + |101i + |011i]/ 3 and |di = |111i, the Hamiltonians read 3 0 0 0 −3 0 0 0 W 0 7 0 0 + ǫ 0 −1 0 0 ; H0 = 0 0 1 0 2 0 0 7 0 2 0 0 0 3 0 0 0 3 √ 3 0 0 √0 3 0 2 √0 . (11) Hc = A 0 3 2 √0 0 0 3 0 Again, we see that while Hc is not fully connected, one of the eigenstates of H0 is the W state [20], which is a more robust form of three-body entanglement than the GHZ state since the tracing out of any qubit still leaves a maximally two-body entanglement. Substituting the new Hamiltonians into Eq. (9), we can obtain the time-dependent control field for the generation of the W state. Taking the initial state to be |ψ(0)i = [1, 10−3 i, 10−3 , 0]T , we plot the achieved fidelity and the distance to the target state along with the time-dependent field in Fig 4. It’s quite clear from Fig 4 that we are able to generate a high-fidelity W state on a short time scale, and can approach the target state monotonically, as opposed to the highly oscillating state generated under a simple harmonic laser field. As illustrated in the previous examples, we can still find the Lyapunov control field that is effective for non-ideal systems. However, the control field needs to be continuously varying. Experimentally, the generation of such laser fields may be quite challenging with current technology [21]. We will now seek to simplify the control field to a train of pulses while still maintaining the effectiveness of the Lapunov control. The simplified field is proposed to be a series of step functions. On each time interval [(N − 1)∆t, N ∆t], the simplified control field is taken to be constant, and we only take the feedback at each period point N ∆t. We plot the achieved fidelity for the Bell state and the simplified control field in Fig 5. It is obvious that we are still able to generate the steady-state entanglement only a little slower, while reducing the control field to a much more simple form. In Ref. [22, 23, 24], delocalized qubits coupled to bosonic reservoirs and noisy environments are considered to study the influence of decoherence effects. Under decoherence, entangled state will generally lose its quantum coherence and display complex decoherence and disentanglement behaviors. Such behaviors can be related to the spatial separation of the qubits [23], global and local noises [24] and may be eliminated by encoding of quantum information in decoherence free subspaces [22]. It is interesting to explore how the control method presented in this paper performs with environmental decoherence. 4
Summary
In this paper we use the Lyapunov control methods to generate quantum correlation on quantum dots systems, and find out that while the systems are not ideal for the control requirements, we are still able to find subspaces where the control is still effective. Steadystate entanglement can be generated under this control paradigm on a short time scale which
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Generation of quantum correlation in quantum dots by Lyapunov control methods
Fidelity 1.0 0.8 0.6 0.4 0.2 0.0 0
5
10
15 20 Distance
25
30
5
10 15 20 25 Lyapunov field
30
t HsL
1.0 0.8 0.6 0.4 0.2 0.0
t HsL
0.5 5
10
15
20
25
30
t HsL
-0.5 -1.0 Fig. 4. (Color online) The achieved fidelity(panel (a)), distance to the target state(panel (b)) and the Lyapunov field(panel (c)) as a function of time for the generation of W State in three coupled quantum dots(Solid red line). We take W = 1, ǫ = 1.2, A = 0.5 and k = 1.2. The dashed blue line corresponds to the generated state under a laser field E(t) = e−iωt and the dash-dotted orange line corresponds to free evolution under zero field.
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Fidelity 1.0 0.8 0.6 0.4 0.2 5
10
5
15 20 25 Lyapunov field 10
15
20
30
25
t HsL
30
-0.2 -0.4 -0.6 Fig. 5. (Color online) (a) The achieved fidelity of the Bell state. (b) The Lyapunov field as a function of time for the generation of Bell state in two coupled quantum dots. The red lines correspond to a continuously varying control field and the blue lines correspond to the simplified control pulses.
has advantage over the highly oscillating state obtained by applying simple laser fields. Taking the experimental generation of laser fields into consideration, we find out that the control field can be simplified into a train of laser pulses, and the effectiveness of the control is still maintained. The Lyapunov control method is a robust control method whose convergence and stability properties have been extensively studied and has proven to be highly effective. It is interesting to study how other control methods compare with the Lyapunov control method outlined in our paper. Our results in the present paper may be realized experimentally in the near future. Acknowledgements This project was supported by the National Natural Science Foundation of China (Grant No. 11274274). References
1. Michael A. Nielsen and Isaac L. Chuang (2000), Quantum Computation and Quantum Information, Cambridge University Press. 2. Lorenza Viola and Seth Lloyd (1998), Dynamical suppression of decoherence in two-state quantum systems, Phys. Rev. A, 58(4), 2733–2744. 3. Xiaoting Wang and S. G. Schirmer (2009), Entanglement generation between distant atoms by lyapunov control, Phys. Rev. A, 80, 042305.
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4. D. D’lessandro (2007), Introduction to quantum control and dynamics, CRC Press. 5. Alex Greilich, Samuel G. Carter, Danny Kim, Allan S. Bracker, and Daniel Gammon (2011), Optical control of one and two hole spins in interacting quantum dots, Nat. Photon, 5(11), 702– 708. 6. Danny Kim, Samuel G. Carter, Alex Greilich, Allan S. Bracker, and Daniel Gammon (2011), Ultrafast optical control of entanglement between two quantum-dot spins, Nat. Phys, 7(3), 223– 229. 7. Wolfgang Langbein (2006),Quantum dots: All spins under control, Nat. Mater, 5(7), 519–520. 8. M. N. Makhonin, K. V. Kavokin, P. Senellart, A. Lematre, A. J. Ramsay, M. S. Skolnick, and A. I. Tartakovskii (2011), Fast control of nuclear spin polarization in an optically pumped single quantum dot, Nat. Mater, 10(11), 844–848. 9. David Press, Thaddeus D. Ladd, Bingyang Zhang, and Yoshihisa Yamamoto (2008), Complete quantum control of a single quantum dot spin using ultrafast optical pulses, Nature, 456(7219), 218–221. 10. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen (2007), Spins in few-electron quantum dots, Rev. Mod. Phys., 79, 1217–1265. 11. M. Abdel-Aty, M.R.B. Wahiddin, and A.-S.F. Obada (2007), Generation of three-qubit entangled states using coupled multi-quantum dots, Laser Physics Letters, 4(5): 399–403. 12. Luis Quiroga and Neil F. Johnson (1999), Entangled bell and greenberger-horne-zeilinger states of excitons in coupled quantum dots, Phys. Rev. Lett., 83, 2270–2273. 13. John H. Reina, Luis Quiroga, and Neil F. Johnson (2000), Quantum entanglement and information processing via excitons in optically driven quantum dots, Phys. Rev. A, 62, 012305. 14. Ping Zhang, C. K. Chan, Qi-Kun Xue, and Xian-Geng Zhao (2003), Quantum entanglement of excitons in coupled quantum dots, Phys. Rev. A, 67,012312. 15. A. S.-F. Obada and M. Abdel-Aty (2007), Quantitative aspects of entanglement in optically driven quantum dots, Phys. Rev. B, 75, 195310. 16. Magnus Radmark, Marcin Wiesniak, Marek Zukowski, and Mohamed Bourennane (2013),Experimental multilocation remote state preparation, Phys. Rev. A, 88, 032304. 17. Shang-Yu Huang, Hsi-Sheng Goan, Xin-Qi Li, and G. J. Milburn (2013), Generation and stabilization of a three-qubit entangled w state in circuit qed via quantum feedback control, Phys. Rev. A, 88, 062311. 18. Rudolf Bratschitsch and Alfred Leitenstorfer (2006), Quantum dots: Artificial atoms for quantum optics, Nat. Mater, 5(11):855–856. 19. William K. Wootters (1998), Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett., 80(10), 2245–2248. 20. W. Dur, G. Vidal, and J. I. Cirac (2000), Three qubits can be entangled in two inequivalent ways, Phys. Rev. A, 62, 062314. 21. X X Yi, S L Wu, Chunfeng Wu, X L Feng, and C H Oh (2011), Time-delay effects and simplified control fields in quantum lyapunov control, Journal of Physics B, 44(19), 195503. 22. Stefan Borghoff and Rochus Klesse (2008), Decoherence of encoded quantum registers, Phys. Rev. A, 77, 012306. 23. Roland Doll, Martijn Wubs, Peter Hanggi, and Sigmund Kohler (2007), Incomplete pure dephasing of n-qubit entangled w states, Phys. Rev. B, 76, 045317. 24. T. Yu and J. H. Eberly (2003), Qubit disentanglement and decoherence via dephasing, Phys. Rev. B, 68, 165322.
GENERATION OF QUANTUM CORRELATION IN QUANTUM DOTS BY LYAPUNOV CONTROL METHODS
DA-WEI LUO Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University Hangzhou 310027, People’s Republic of China JING-BO XUa Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University Hangzhou 310027, People’s Republic of China
Received February 17, 2014 Revised April 10, 2014 We present a scheme to generate steady-state quantum correlations in quantum dots. The shape of the control field is obtained by using the Lyapunov control method, and the controlled state is found to approach the target state monotonically. We also explore the possibility of replacing the continuous control field with a train of discreet rectangular pulses, which is much easier to implement experimentally. The discretized control field is found to be still able to drive the initial state to the target state with very small errors, which suggests that the Lyapunov based control is still effective under practical limitations. Keywords: Quantum entanglement, Control theory, Quantum dots Communicated by: I Cirac & G Milburn
1
Introduction
It is well known that the field of quantum information and quantum computation has received a lot of research interest due to its potential applications. Quantum entanglement, as a special kind of quantum correlation, has been considered as a valuable resource for many quantum information tasks, such as quantum teleportation and quantum cryptography [1]. The generation and manipulation of entanglement is therefore a problem of fundamental interest in quantum information processing. However, to efficiently accomplish quantum information tasks, free unitary evolution under a time-independent Hamiltonian is generally not good enough. Therefore, the efficient control of quantum mechanical process has become a very important research focus over the years. Several control methods has been proven to be very successful in achieving the desired time evolution of quantum mechanical systems, such as the dynamic pulse control to combat decoherence [2]. The Lyapunov function plays a very important role in control design, and this method has been shown to be very effective for spin systems as well as cavity QED systems connected with optical fibers [3, 4]. Experimentally, several physical systems for the realization of quantum information processing has been put forward, including cavity QED, quantum dots, nuclear magnetic [email protected]
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D-W Luo and J-B Xu
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onance (NMR) and spin systems [1]. Among those proposals, the semiconductor quantum dots are a particularly promising candidate since they are solid-based, have great scaling potentials and can be finely-controlled [5, 6, 7, 8, 9]. With present experimental equipment, it has become a standard technology to confine single-electron charges in quantum dots as required, and can be measured relatively easily [10]. Since entanglement is essential for a wide range of quantum information and computation tasks, the generation of entanglement is of particular experimental and theoretical interest [11, 12, 13, 14, 15, 16, 17]. In Ref. [12, 13], the entanglement generation on two and three identical, equally spaced quantum dots with classical laser is studied. The quantum dots system under consideration has a resonant energy transfer process between dots known as the F¨orster process. However, the achieved fidelity of the generated entanglement is found to be highly oscillating, so very precise timing is needed to extract high quality entanglement, and this big fluctuating behavior is not desirable for the practical implementation of quantum information tasks. In this paper, we use the Lyapunov control method to drive the system of quantum dots into a stationary state that is maximally entangled. We give a brief outline of the Lyapunov based control method and then apply it to two coupled quantum dots and generate a maximally entangled Bell state. System consisting of three coupled quantum dots are also considered, and it is found that the three-body W state can also be generated using the Lyapunov control. Due to experimental limitations, the continuous control field may be difficult to generate, and we finally proposed a simplified discreet pulse sequence based on the Lyapunov control method, and find that the control is still effective. 2
The model and the Lyapunov-based control methods
Quantum dots are generally considered as a type of artificial atoms for quantum optics [18]. In this paper, we study a system of N identical, equally spaced quantum dots containing no net charge and radiated by a classical field. The formation of single excitons within the individual quantum dots with inter-dot hopping can be described by the Hamiltonian [12, 13]
H
=
N N W X † ǫ X † ep ep − hp h†p + ep hp′ ep′ h†p 2 p=1 2 ′ p,p =1
+hp e†p′ h†p′ ep
i
+ E(t)
N X p=1
e†p h†p
∗
+ E (t)
N X
h p ep ,
p=1
(1) where e†p and h†p are the electron and hole creation operators, ǫ is the quantum dot band gap, W is the inter-dot Coulomb interaction, and E(t) is the laser field strength. In reality, the band gap ǫ and inter-dot Coulomb interaction W may be different across different quantum dots. However, we can approximately treat both as uniform, which allows us to analytically study the system and its control method. This idealized treatment [12] can be a valid description of the quantum dot system. The operators are fermion and obey the anti-commutation rules {ep , e†p′ } = {hp , h†p′ } = δp,p′ . Using the standard approach, we introduce the following
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Generation of quantum correlation in quantum dots by Lyapunov control methods
operators J+ =
PN † † J− = p=1 p=1 ep hp , Jz = 12 e†p ep − hp h†p ,
PN
h p ep (2)
which are easily verified to be the usual angular momentum operators. The Hamiltonian Eq. (1) can now be expressed using the angular momentum operators and reads H = ǫJz + W J 2 − Jz2 +E(t)J+ + E ∗ (t)J− ,
(3)
In Ref. [12], a rectangular radiation pulse shape E(t) was proposed as an experimentally achievable way to generate entangled states in quantum dots. In this paper, we substitute this with the pulse shape determined by the Lyapunov control theory to study how to apply the control method presented in this paper to generate the entangled target state in a quantum dot system and compare our approach with that of Ref. [12]. The Hamiltonian above is generally time-dependent. However, if we consider a laser field of the shape E(t) = E0 e−iωt , we can eliminate the time dependence in the rotating frame using the rule |ψir
=
R† |ψis
Hr
=
R† HR + i(∂t R† )R
R
=
e−iωJz t
(4)
where the subscripts r and s stand for the rotating frame and the Schr¨odinger frame respec∂ |ψ(t)ir = Hr |ψ(t)ir with tively. After the rotation, the time evolution is given by i ∂t Hr = ∆Jz + W (J 2 − Jz2 ) + E0 J+ + E0∗ J− ,
(5)
where ∆ = ǫ − ω is the detuning. The Hamiltonian now is time-independent in the rotating frame, and can be easily used to solve the wave function’s time evolution using the standard approach [13, 12] in each fixed-J subspace since [J 2 , Hr ] = 0. The wave function obtained can then be directly transformed back to the Schr¨odinger picture using the relationship |ψis = R|ψir since R is unitary. However, as illustrated in [13, 12], applying the laser field of the form E0 e−iωt results in a highly oscillating fidelity between the system state and target state, which is undesirable for quantum information tasks because highly accurate timing is required to obtain the target state due to the big time fluctuation, which is a big experimental obstacle. In [14], a Gaussian pulse is suggested, which can generate Bell and GHZ steady states at resonance by avoiding the level-crossing. Here, we propose a scheme to monotonically generate maximally entangled states using the Lyapunov control method, which we find to be effective both at resonance and with non-zero detuning ∆. The Lyapunov control method [3, 4] is a type of feedback control in that the external control parameter is determined by the system’s current state. For a general bilinear Hamiltonian system, the time evolution is given by the Liouville equation ρ(t) ˙ = −i[H0 + f (t)Hc , ρ(t)],
(6)
where H0 is the system Hamiltonian, Hc the control Hamiltonian, both assumed to be timeindependent, and f (t) is the time-dependent, real-valued control parameter. The control
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Fig. 2. (Color online) The achieved fidelity(panel (a)), concurrence(panel (b)) and the Lyapunov field(panel (c)) as a function of time for the generation of Bell State in two coupled quantum dots with non-zero detuning(Solid red line). The dashed blue line corresponds to the generated state under a laser field E(t) = e−iωt .
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problem is then to find such f (t) that any initial state ρ0 converges to and stays at the target state ρd as t → ∞. Since the time evolution of any Hamiltonian system is unitary, we require the spectrum of the initial and target states coincide. Practically, the system Hamiltonian H0 can not be turned off, so as a generalization, the target state is allowed to evolve according to the system Hamiltonian H0 alone, ρd˙(t) = −i[H0 , ρd (t)]. If the target state commutes with H0 we have a stationary target state. To obtain the time-dependent control parameter f (t), we first define a function d(ρ(t), ρd (t)) to be the Hilbert-Schmidt distance between the target state ρd (t) and ρ(t), the state the system is currently in: 1 d(ρ(t), ρd (t)) = Tr[(ρ(t) − ρd (t))2 ], (7) 2 and it can be shown that d(ρ(t), ρd (t)) ≥ 0 with equality if and only if ρ(t) = ρd (t). Taking its derivative against time and substituting in the Liouville equation, we have ˙ d(ρ(t), ρd (t)) = −f (t)Tr [ρd (t)[−iHc , ρ(t)]] .
(8)
If we choose the control parameter to be f (t) = κTr [ρd (t)[−iHc , ρ(t)]] ,
(9)
where κ > 0 is the control field coupling parameter, then d˙ ≤ 0. Therefore, d(ρ(t), ρd (t)) is a Lyapunov function by definition for the system described by (6). The control task ρ(t) → ρd (t) can be now transformed into the requirement that d(ρ(t), ρd (t)) → 0. As a direct consequence of the LaSalle invariance principle, for the target state to be asymptotically stable the Hamiltonians have to satisfy certain conditions [3, 4]. The fist is that H0 should be strongly regular, which means that it should have distinct gaps between any pair of energy levels Ei , Ej ∀i 6= j ∈ {1 . . . dim(H0 )}. The second condition is that Hc should be fully connected, which translates to that in the basis that H0 is diagonal, all off-diagonal elements of Hc should be non-vanishing so that all transitions between all energy levels are possible. However, for real physical systems that do not meet those requirements, it’s still possible to find subspaces where the Lyapunov control is still effective. We now illustrate its use via two examples where we generate maximum entanglement among multiple quantum dots. As we shall find out, Lyapunov control can drive the quantum dots to a maximally entangled steady-state as opposed to the highly oscillating state achieved by applying a single-frequency laser field. 3
Generation of quantum correlations on quantum dots
For two interacting quantum dots, the Hamiltonian is usually expressed in the space spanned by the spin singlet and triplet states. Furthermore, because the J = 0 and J = 1 subspaces are totally separated, we can focus on a smaller space under the basis |ai = |0, 0i, |bi = √ (|10i + |01i)/ 2 and |ci = |1, 1i, which corresponds to the vacuum, single-exciton and biexciton states respectively. Then, in the usual Schr¨odinger picture, the Hamiltonians read 0 1 0 W +ǫ 0 0 √ 2W 0 , Hc = 2A 1 0 1 (10) H0 = 0 0 1 0 0 0 W −ǫ
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where A is the overall external field amplitude. It’s clear that while Hc is not fully connected, the Bell state |bi is an eigenstate of H0 . Therefore, if we set the target state to be the maximally entangled Bell state ρd (0) = |bihb|, then the target state is stationary and steady-state entanglement is now possible if we can find some subspace of the system where the Lyapunov control is effective. Moreover, because unitary evolutions do not alter the eigenvalues of the density matrix, we require the initial state of the system to be a pure state. Substituting the Hamiltonians into Eq. (9), we find that for the control to be effective, the initial state should have non-zero complex projection along |bi. As we find out, the control is still effective for the initial state |ψ(0)i = [1, 10−10 i, 0]T up to an overall normalizing factor. To calculate the amount of entanglement we use Wootters’s concurrence [19]. The concurrence C varies from C = 0 for completely separable states to C = 1 for maximally entangled state. For any two qubits, the concurrence can be explicitly calculated from their density matrix ρAB : C(ρAB ) = max(0,
p
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where λ1 . . . λ4 are the eigenvalues of ρ(σy ⊗ σy )ρ∗ (σy ⊗ σy ) in decreasing order. The achieved fidelity and entanglement along with the external control field are plotted in Figs 1 and 2 for system at resonance and with detuning, respectively. We can see that the achieved fidelity and entanglement is much better than the state generated by the conventional laser field E(t) = e−iωt , and it is done on a much shorter time scale. In general, the Lyapunov control theory ensures that for all systems satisfying the controllability conditions discussed in Section II, the target state can be effectively reached. Because the system still have a unitary time evolution when a Lyapunov control function is applied, the initial state and the target state cannot have different spectrum. For mixed states, that means the eigenvalues of the density operator should be the same, and, since all pure states have the same spectrum, all initial pure states can be effectively driven to a target pure state for controllable systems satisfying the conditions given in Section II. We have chosen the initial state to be |ψ(0)i = [1, x0 , x0 ]T /N , where N is an overall normalize factor, and plot the achieved fidelity as a function of time and initial state parameter x0 in Fig. 3. It can be observed for other initial states, the Lyapunov function is still able to drive the initial state to the target maximally entangled state.
Fig. 3. The achieved fidelity as a function of time and initial state parameter x0 . It can be observed for other initial states, the Lyapunov function is still able to drive the initial state to the target maximally entangled state.
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Next we consider the generation of multi-body entanglement on three equally spaced quantum dots. We will focus on the J = 3/2 subspace√as it is the only optically active one √ [12]. In the basis |ai = |000i, |bi = [|100i + |010i + |001i]/ 3, |ci = [|110i + |101i + |011i]/ 3 and |di = |111i, the Hamiltonians read 3 0 0 0 −3 0 0 0 W 0 7 0 0 + ǫ 0 −1 0 0 ; H0 = 0 0 1 0 2 0 0 7 0 2 0 0 0 3 0 0 0 3 √ 3 0 0 √0 3 0 2 √0 . (11) Hc = A 0 3 2 √0 0 0 3 0 Again, we see that while Hc is not fully connected, one of the eigenstates of H0 is the W state [20], which is a more robust form of three-body entanglement than the GHZ state since the tracing out of any qubit still leaves a maximally two-body entanglement. Substituting the new Hamiltonians into Eq. (9), we can obtain the time-dependent control field for the generation of the W state. Taking the initial state to be |ψ(0)i = [1, 10−3 i, 10−3 , 0]T , we plot the achieved fidelity and the distance to the target state along with the time-dependent field in Fig 4. It’s quite clear from Fig 4 that we are able to generate a high-fidelity W state on a short time scale, and can approach the target state monotonically, as opposed to the highly oscillating state generated under a simple harmonic laser field. As illustrated in the previous examples, we can still find the Lyapunov control field that is effective for non-ideal systems. However, the control field needs to be continuously varying. Experimentally, the generation of such laser fields may be quite challenging with current technology [21]. We will now seek to simplify the control field to a train of pulses while still maintaining the effectiveness of the Lapunov control. The simplified field is proposed to be a series of step functions. On each time interval [(N − 1)∆t, N ∆t], the simplified control field is taken to be constant, and we only take the feedback at each period point N ∆t. We plot the achieved fidelity for the Bell state and the simplified control field in Fig 5. It is obvious that we are still able to generate the steady-state entanglement only a little slower, while reducing the control field to a much more simple form. In Ref. [22, 23, 24], delocalized qubits coupled to bosonic reservoirs and noisy environments are considered to study the influence of decoherence effects. Under decoherence, entangled state will generally lose its quantum coherence and display complex decoherence and disentanglement behaviors. Such behaviors can be related to the spatial separation of the qubits [23], global and local noises [24] and may be eliminated by encoding of quantum information in decoherence free subspaces [22]. It is interesting to explore how the control method presented in this paper performs with environmental decoherence. 4
Summary
In this paper we use the Lyapunov control methods to generate quantum correlation on quantum dots systems, and find out that while the systems are not ideal for the control requirements, we are still able to find subspaces where the control is still effective. Steadystate entanglement can be generated under this control paradigm on a short time scale which
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-0.5 -1.0 Fig. 4. (Color online) The achieved fidelity(panel (a)), distance to the target state(panel (b)) and the Lyapunov field(panel (c)) as a function of time for the generation of W State in three coupled quantum dots(Solid red line). We take W = 1, ǫ = 1.2, A = 0.5 and k = 1.2. The dashed blue line corresponds to the generated state under a laser field E(t) = e−iωt and the dash-dotted orange line corresponds to free evolution under zero field.
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-0.2 -0.4 -0.6 Fig. 5. (Color online) (a) The achieved fidelity of the Bell state. (b) The Lyapunov field as a function of time for the generation of Bell state in two coupled quantum dots. The red lines correspond to a continuously varying control field and the blue lines correspond to the simplified control pulses.
has advantage over the highly oscillating state obtained by applying simple laser fields. Taking the experimental generation of laser fields into consideration, we find out that the control field can be simplified into a train of laser pulses, and the effectiveness of the control is still maintained. The Lyapunov control method is a robust control method whose convergence and stability properties have been extensively studied and has proven to be highly effective. It is interesting to study how other control methods compare with the Lyapunov control method outlined in our paper. Our results in the present paper may be realized experimentally in the near future. Acknowledgements This project was supported by the National Natural Science Foundation of China (Grant No. 11274274). References
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