In this section, we will use the Lindblad form master equation approach to calculate the dynamics of reduced density matrix for some systems. We first start out with a simplified model to see how the two kinds of dissipations (field damping and spontaneous emission into free space) damp the populations of the two dot states and the surface plasmon states.
Consider the two QDs couple to only one surface plasmon mode k which is incident from the left end of the first small wire. The schematic diagram is the same as Fig. 4.1. The total system now is S ⊕ S0, and the master equation can be written as
˙ρ = 1 i¯h[H, ρ]
+ Γk(akρa†k−1
2a†kakρ −1
2ρa†kak)
+ X
i=1,2
γi(σ−iρσ+i− 1
2σ+iσ−iρ − 1
2ρσ+iσ−i), (4.32) where H = HS + HS0 + HSS0 with
HS = ¯hωka†kak, HS0 = X
i
¯hωeigiσei,ei,
HSS0 = ¯h[(g1,kσe1,g1ak+ g2,kσe2,g2eikdak) + h.c.]. (4.33)
Here, all operators and parameters are identical to those we used in previous section. Since there is only one excitation in the system, we expand the density operator of the total system S ⊕ S0 with the basis:
{|g1, g2, 1ki, |g1, e2, 0i, |e1, g2, 0i, |g1, g2, 0i}.
For convenience we label the basis kets as in ground state, and the excitation is in the surface plasmon mode k. Since we take the dissipations into account, we have to include the vacuum state
|0i = |g1, g2, 0i in our basis. Thus, the matrix representation of the density operator of the total system reads
X
Eq. (4.35) can be simplified as Now we can calculate all elements of the matrices on both sides of Eq. (4.36).
These matrices can be flatted and rearranged as
Thus, the entire problem turns out to be a system of coupled differential equations. All we need is to diagonalize the intermediate matrix A, and obtain its eigenvalues and eigenvectors to do the linear transformation. In this way, we can decouple the coupled differential equations and obtain the solutions ρnm(t)(n, m = k, 2, 1, 0) with given initial conditions. In the den-sity matrix of the total system S ⊕ S0, the diagonal elements ρnn(t) are the probabilities in |ni and the off-diagonal elements ρnm(t)(n 6= m) are the co-herences between |ni and |mi. Now we set the two dots are both initially in
0 2 4 6 8 10
Figure 4.2: Population dynamics without dissipations for each diagonal ele-ment.
0 2 4 6 8 10
the ground state with identical two-level spacing which is resonant with the surface plasmon mode k incident from the left end of the first wire, and the two dissipations have the same decay rate (i.e. Γk = γ1 = γ2 = γ0). In last chapter, we set the Purcell factor P = 20, for which the coupling strength between QDs and surface plasmons is about 3γ0. We further assume that the couplings of the two dots to the surface plasmon mode k are the same.
If we first ignore the dissipations (Fig. 4.2), it is similar to that of two iden-tical dots are placed inside a high Q cavity with single mode. Therefore, the populations are independent of inter-dot distance d and reveal the feature of Rabi Oscillations in cavity Q.E.D: going back and forth between the surface plasmon mode k and the two dots [38]. With dissipations, the populations are damped by the two channels individually as shown in Fig. 4.3 (a), (b), (c). Since we assume that the coupling constant g is the same for two dots (i.e. g1 = g2 = g), panels (b) and (c) of Fig. 4.2 and 4.3 demonstrate that the two dots ’see’ the same surface plasmon mode k . Note that in plotting the figure, the unit of time t is normalized to the inverse of free-space decay rate γ0.
One might argue that it is not sufficient to consider only a single-mode since the QDs are coupled to infinite propagating modes. However, from our discussions in Chapter 2, we realize that the energy spacing of QDs can be tuned such that only the lowest n-mode is effective. In addition, the lengths
of the wires considered here are finite. This means the dispersion relations of the surface plasmons are discrete. Therefore, if the QD exciton energy happens to be close to one of the discrete points of the dispersion relations, it is plausible to assume a single-mode model. The difference to the original cavity QED case is that the photon is assumed to be injected from one side of the wire. Thus, one should also take into account the mode −k to denote the reflecting surface plasmon from the other side.
Let us now consider two QDs resonantly coupled to the surface plasmon mode k and its reflecting mode −k. The Hamiltonian H can be written as
H = HS + HS0 + HSS0
and the corresponding Lindblad form master equation reads,
˙ρ = 1 The physical picture is similar to our discussions in Chapter 3: the surface plasmon with wavevector k, is injected from the left end of the first wire. It
would be either scattered or absorbed by the two QDs with certain possibil-ities. If the surface plasmon is trapped between the two dots, it is possible to create the entanglement between this two QDs. We now use the basis
|k−i → |g1, g2, 1−ki,
|k+i → |g1, g2, 1ki,
|2i → |g1, e2, 0i,
|1i → |e1, g2, 0i,
|0i → |g1, g2, 0i. (4.39)
as a complete set to expand Eq. (4.38), and assuming that, at the initial time t = 0, only the state |g1, g2, 1ki is populated. With these, the population dynamics for each basis state can then be calculated.
In Figs. 4.4, 4.5 and 4.6, we show the population dynamics for three different inter-dot distance kd = π2, π4 and 2π (or π), respectively. Notes that not only the coupling strengths are the same (g1 = g2 = g), but also the decay rates for dissipations are assumed to be identical. A very interesting point in Fig. 4.4 (a) is that the excitation never goes to the |g1, g2, 1−ki state.
Now we can go further to study the entanglement dynamics of the two dots by introducing the ”concurrence” [48] as a criterion to quantify the entanglement. For a general state ρ of two qubits, the spin-flipped state is
0 2 4 6 8 10
0 2 4 6 8 10
0 2 4 6 8 10
written as
ρ0 = (σy ⊗ σy)ρ∗(σy⊗ σy). (4.40)
The concurrence is a positive value between 1 and 0, defined as
C(ρ) = max{0,p
λ1−p
λ2−p
λ3−p
λ4}, (4.41)
where σy is the y component of Pauli matrices, and {λ1, λ2, λ3, λ4} are eigen-values of ρρ0 in decreasing order. If all eigenvalues of ρρ0 are all negative, then the concurrence is zero, which means the state is not entangled at all.
For maximally entangled state, the concurrence is unity.
We can therefore use this criterion to quantify the entanglement. First of all, we need to have two qubits, which means we have to trace out the surface plasmons (the system S’):
T rS0ρ = h1k|ρ|1ki + h1−k|ρ|1−ki = ρS.
Substituting this ρS into Eqs. (4.40) and (4.41), we calculate the concurrence for kd = (2n+1)π2 (n = 0, 1, 2...), (4n+1)π4 (n = 0, 1, 2...), even multiple of π and odd multiple of π for the cases without (with) dissipations shown in Fig. 4.7 (4.8). As see in Fig. 4.7 (a), for kd = (2n+1)π2 (n = 0, 1, 2...), we have a periodically maximal entanglement, which is different from our results in Chapter 3. This is because, in Chapter 3, we studied the stationary state which is an average of many measurements. We assume that once there is
0 2 4 6 8 10
Figure 4.7: The concurrence dynamics without dissipations for kd = (a)
(2n+1)π
2 (n = 0, 1, 2...) (b) (4n+1)π4 (n = 0, 1, 2...) (c) even multiple of π and 63
0 2 4 6 8 10
Figure 4.8: The concurrence dynamics with dissipations (Γ−k = Γk = γ1 = γ2 = γ0) for kd = (a) (2n+1)π2 (n = 0, 1, 2...) (b) (4n+1)π4 (n = 0, 1, 2...) (c)
64
no detection of any outgoing surface plasmons at the two ends of wire, the total state would be projected into the state of two qubits. Here, however, we include also the probabilities of surface plasmons by using the density matrix ρ. Therefore, for the cases of kd = even multiple of π and odd multiple of π, no maximal entanglement can be created. In addition, since we only take into account two modes here (k and −k), some differences are expected if we include more modes. One also notes, in Fig. 4.8, the concurrences decay with time due to dissipations. If one can further reduce the dissipations,higher entanglement can be achieved between the two dots.
In real experiment [15], the samples are prepared by spinning QDs onto a glass substrate with a PMMA layer coverage above. Then, dry silver wires are deposited on top of it. The coupling strength between the QDs and surface plasmons would not be identical for each dot. Therefore, it is desirable to investigate how the concurrence changes with different coupling strengths, i.e. varying g1,˜k and g2,˜k in Eq. (4.37). For simplification, we turn off the dissipations and show the concurrences for different coupling strength ratio of the first dot to the second one (Fig. 4.9).
A surprising result is that if g1/g2 is a ratio between two odd integers, the concurrence for kd = (2n+1)π2 (n = 0, 1, 2...) becomes unity at some points in time. To prove this, we first use Laplace transformation to analytically solve Eq. (4.38). After tracing out the system S0 and obtain the state of the
0 5 10 15 20
Figure 4.9: The concurrence dynamics for kd = (2n+1)π2 (n = 0, 1, 2...) without dissipations for the ratios g /g =(a) 1 (b) 1 (c) 1 and (d) 3.
66
two-dot excitons (qubits), we can derive an analytical form of the condition for C(ρ) = 1:
e−i
√2(g1g2+1)t(−1 + e2i√2t)(−1 + e2i
√2g1g2t) = ±4.
This equation can be further simplified as
Sin(√ 2g1
g2
t)Sin(√
2t) = ±1.
One immediately finds that for the requirement of Sin(√
2t) = ±1, the con-ditions are
t = 2ξ + 1 2√
2 π (ξ = 0, 1, 2, 3...).
With the second requirement for Sin(√ 2gg1 to achieve maximum entanglement at some points in time (t = 2ξ+12√2π).
Instead of setting the initial state is in |g1, g2, 1ki, here, we would like to study two special cases for different initial state. First, we consider that if the state is prepared in a pure state of the two QDs initially:
ρ(0) = |ψ(0)ihψ(0)| = 1
√2(|e1, g2, 0i + |g1, e2, 0i) 1
√2(he1, g2, 0| + hg1, e2, 0|).
We find that, for kd =odd multiple of π, the state will stay in this triplet state without evolving with time, and the concurrence is always unity as shown
in Fig. 4.10 (a). This is because the triplet state is a eigenstate of the total Hamiltonian [eq. (4.37)] with eigenvalue ¯hωeg. So, it is straightforward that an eigenstate will not evolve. But this only holds for two QDs with the same energy spacing ¯hωeg. Similarly, if the initial state is prepared in the singlet state ρ(0) = |ψ(0)ihψ(0)| = √12(|e1, g2, 0i−|g1, e2, 0i)√12(he1, g2, 0|−hg1, e2, 0|), for kd =even multiple of π, the state will not evolve as well with the same reason, and the concurrence is also always unity as shown in Fig. 4.10 (b). Second, if the initial state is prepared in the mixed state ρ(0) =
1
2(|e1, g2, 0ihe1, g2, 0| + |g1, e2, 0ihg1, e2, 0|). As shown in Fig. 4.11, the con-currences for different kd are calculated. Surprisingly, for kd = (2n + 1)π2 (n = 0, 1, 2...), the concurrence is always zero. The condition for this is written as
cos2(1 + eikd
eikd2 ) − cos h2(1 + eikd eikd2 ) = 0.
One can easily simplify it and obtain 1 + eikd
eikd2 = ± i eikd− 1 eikd2 .
With this, one identifies that when kd = (2n + 1)π2 (n = 0, 1, 2...), the concurrence always vanishes as seen in Fig. 4.11(a).
0 2 4 6 8 10 t H1êg
0L
0.2 0.4 0.6 0.8 1
C
HaL
0 2 4 6 8 10
t H1êg
0L 0.2
0.4 0.6 0.8 1
C
HbL
sol_2kDd_exact_pure.nb 1
Figure 4.10: The concurrence dynamics for (a) kd=odd multiple of π with
|ψ(0)i being the triplet state and (b) kd=even multiple of π with |ψ(0)i being the singlet state.
0 5 10 15 20
Figure 4.11: The concurrence dynamics without dissipations. The initial state is in the mixed state for kd = (a) (2n+1)π2 (n = 0, 1, 2...) (b) multiple of π (c) (4n+1)π (n = 0, 1, 2...) and (d) (3n+1)π (n = 0, 1, 2...).
70
4.4 Conclusion
In this chapter, we keep the main configuration in chapter 3, but alter-nate the mediator from the infinite long wire to two small wires which are evanescently coupled to the same dielectric waveguide. In this way, one could not only minimize the Ohmic losses resulting from propagating through the metal wire, but also achieve the remote entanglement between the two QDs.
In section 4.1, we introduce the open quantum theory to show how a pure composite density matrix of two systems goes to a mixed reduced density matrix in the presence of interactions between two systems. In the second section, we derive the Lindblad form master equation, which is the main approach we used to study the time dependent behaviors of the system. In the last section of this chapter, we first consider the two QDs coupled to only one resonant surface plasmon mode and apply the master equation to calculate the population dynamics for each basis state. We show that it is legitimate to only take one surface plasmon mode into account because one can tune the energy spacing of the QDs close to the discrete points in the dispersion relations of surface plasmons. We therefore take one surface plasmon mode k which is resonant with the dots plus its reflected mode −k to investigate the entanglement dynamics without dissipations. We find that if the inter-dot distance kd = π2, maximal entanglement can be achieved
at some points in time when g1/g2 equals the ratios of odd integers. We then study two special cases for the initial state prepared in pure and mixed state. It is found that for pure state, the triplet and singlet states don’t evolve with time and the maximal entanglement is hold for kd=odd multiple of π and even multiple of π individually. For mixed state, we prove that the concurrence is alwasy zero when kd = (2n+1)π2 (n = 0, 1, 2...).
Chapter 5
Summary and outlooks
In this thesis, we make use of the physical properties of surface plasmons to study a series of problems essentially based on the strong interactions between QDs and surface plasmons. In the first chapter, we introduce some backgrounds of the surface plasmons and the motivations. In the second chapter, we apply the Fermi’s golden rule to calculate the decay rate of a QD exciton into the surface plasmon modes. We find that the decay rate is greatly enhanced due to the strong coupling between surface plasmon and the QD.
The unreasonable infinite enhancement tells us that it is not legitimate to use Markovian treatment around the band-edge . We thereby deal with the problem with a non-Markovian way, and obtain the oscillatory behaviors of decay dynamics. In the third chapter, we consider a surface plasmon incident from the left end of a long wire to study the scattering resulting from the
interactions with two QDs. We find that if there is no out-going surface plasmon detected, the entire state collapses into the entangled state of the two QDs. We also obtain two conditions for achieving maximal entanglement.
In the latter part of chapter 3, we propose a way to store the entangled state and a experimental procedure to verify that if the entangled state has been prepared or not. In the last chapter, we keep the main configuration in chapter 3, but use two small wires to replace the original infinite long one to minimize the ohmic losses during propagation. In stead of applying the ”projection” concept we used in chapter 3, we use the density matrix approach to obtain the population dynamics of each basis state and introduce the Lindblad form master equation to include the dissipations. After tracing out the surface plasmon modes, we obtain the reduced density matrix of the two QDs, which is used to calculate the concurrence dynamics. We find that when the inter-dot distance kd = (2n+1)π2 (n=0,1,2,3...), the maximal entanglement can be achieved. We also investigate that when the ratio of coupling strength of the two QDs equals a ration of two odd integers, the concurrences recover to unity at some points in time for kd = π2. In addition, for a triplet (singlet) initial state, the concurrence is always unity for kd = odd (even) multiple of π. For an initially mixed state, we prove that under the condition of kd = (2n+1)π2 , the concurrence always vanishes. With the advantage of the strong coupling between QDs and surface plasmons, we
ٛٛٛٛ|g1 >
J
|g2 >J
|g3 >ٛٛٛٛ |e1 > ٛٛٛٛ|e2 > ٛٛٛٛ|e3 >
Figure 5.1: The schematic diagram for a one-dimensional array to simulate Bose-Hubbard model.
propose a future work on the simulation of quantum phase transition [27, 28].
Consider a one-dimensional array, each site in this array contains a QD which is put close to a small metal wire (See Fig. 5.1) and is thus coupled to the surface plasmons with coupling strength g. Each site is also coupled to one another with coupling strength J. So, once the surface-plasmonic polariton is created, it can transport back and forth from one site to the next. The Hamiltonian of each cell can be described by a atom-field Hamiltonian plus one hopping term as [27, 49]
H =X
i
Hiaf −X
i,j
Ji,ja†kiakj −X
i
µiNi, (5.1)
with
Haf =X
k
¯hωka†kak+ ¯hωegσee+X
k
¯hgk(σ+ak+ σ−a†k). (5.2)
Where, Haf denotes the atom-field Hamiltonian with gkdenotes the coupling strength between QD and surface plasmon. The second term in Eq. (5.1) is the hopping term with Jij = J denotes the coupling strength for nearest neighbors and J = 0 otherwise. a†ki (aki) is the creation (annihilation) opera-tor for k−mode surface plasmon at site i, σee = |eihe| with ωeg is the energy spacing of each dot. ωk is the frequency of k−mode surface plasmon, and σ+(−)= |eihg| (|gihe|) denotes the atomic creation (annihination) operators;
Ni is the total number of photonic and atomic excitations, and µi is the chemical potential at site i in the grand canonical ensemble.
In this way, we can regard this system as an analogy [27, 28] to a conven-tional one-dimensional lattice in condensed matter physics and investigate the Mott insulator-to-superfluid phase transition in our system.
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Publication list :
1. “Aharonov-Bohm Effect in Concentric Quantum Double Rings”,
Guang-Yin Chen, Yueh-Nan Chen, and Der-San Chuu, Solid State
Communications 143, 515 (2007).2. “Proposal for detection of non-Markovian decay via current noise”, Yueh-Nan Chen and Guang-Yin Chen, Phys. Rev. B 77, 035312 (2008).
3. “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire” , Guang-Yin Chen, Yueh-Nan Chen, Der-San Chuu, Opt.
Lett. 33, 2212 (2008)
4. “Quantum-dot exciton dynamics with a surface plasmon: Band-edge quantum optics”, Y. N. Chen, G. Y. Chen*, D. S. Chuu, and T. Brandes, Phys. Rev. A 79, 033815 (2009).
5. “Detecting non-Markovian plasmonic band gaps in quantum dots using electron transport”,Yueh-Nan Chen, Guang-Yin Chen, Ying-Yen Liao, Neil Lambert and Franco Nori, Phys. Rev. B 79, 245312 (2009).
6. “Coherent single surface-plasmon transport in a nanowire coupled to
6. “Coherent single surface-plasmon transport in a nanowire coupled to