Surface plasmons, propagating electromagnetic waves on the surface of metal nanowires in our model, must be damped due to Ohmic losses or the leakages during transmission (see Fig. 4.1). For two QDs, if they are initially in the ground state, each of them is possible to be excited by the surface plas-mons. But meanwhile, they are coupled to the vacuum as well. Therefore, besides decaying into surface plasmons modes, they may also decay into the free space. Since now we consider small nanowires with finite length, the Ohmic losses could be minimized. And, from our previous discussions in chapter 2, the pheonmenon of large Purcell factors due to the strong cou-pling between dots and surface plasmons should still hold. Thus, we can take these two decay channels : field dampings and spontaneous emissions into free space, as dissipations in our model. Instead of using the quantum jump effective Hamiltonian, we introduce in this section the Lindblad form master equation approach [47], in which the two dissipations are both included.
We start out with a general Hamiltonian, H = HS + HR + HSR, where HS and HR are Hamiltonian for S and R respectively, HSR is the interaction between system S and reservoir R. The density matrix corresponding to the total system S ⊕ R reads ρSR = ρS ⊗ ρR, while the reduced density matrix of the system is written as ρS = T rRρSR.
The Schr¨odinger equation of ρSR is
˙ρSR = 1
i¯h[H, ρSR], (4.1)
we can transform this Schr¨odinger equation into the interaction picture and get
˙˜ρSR= 1
i¯h[ ˜HSR(t), ˜ρSR], (4.2)
with ˜ρSR= ei/¯h(HS+HR)tρSR(t)e−i/¯h(HS+HR)t, and ˜HSR(t) = ei/¯h(HS+HR)tHSR(t)e−i/¯h(HS+HR)t. Setting the starting point of interaction is t = 0 and integrating Eq. (4.2),
we directly obtain
Substituting this back to Eq. (4.2) for ˜ρSR(t) inside the commutator gives
˙˜ρSR= 1 interested in, after tracing out R, Eq. (4.4) becomes
˙˜ρS = 1 Since one could always write ˜HSR as a sum of products of operators si of system S and operators Ri of reservoir R,
H˜SR(t) = ¯hX
i
˜
si(t) ˜Ri(t), (4.6)
we assume that the mean value of the observable ˜Ri in state ρR is zero ( i.e.
T r[ρRR˜i] = 0 ). We can then eliminate the leading term i¯1hT rR{[ ˜HSR(t), ˜ρSR(0)]}
with the cyclic property of trace T r[ABC] = T r[BCA] = T r[CAB]. Finally, we have
If the interaction between the system and reservoir is very weak and the reservoir is relatively large, one can expect the reservoir is virtually unaffected (stay in initial state) during the interaction. Thus, the density matrix of the total system can be expanded as
˜
ρSR(t) = ˜ρS(t)˜ρR(0) + O(HSR), (4.8)
The Born approximation can be made here to neglect the higher order terms in Eq. (4.7) and give
We can now substitute Eq. (4.6) into Eq. (4.9) and obtain
˙˜ρS = − X
Now, we can use this master equation, i.e. Eq. (4.10), to discuss the two dissipations taking place in our model separately. First, we focus on the field damping dissipation and ignore two QDs for present discussion. Considering the surface plasmon modes as a system, and the modes which damp the surface plasmon fields as a reservoir. The Hamiltonian can be written as
HS = X
where ωk is the energy of the surface plasmons, a†k(ak) denotes the creation (annihilation) operators for each k mode; b†j and bj represent the modes of reservoir with frequencies ω0j; κj,k denotes the coupling constant between the surface plasmons and reservoir. In our model, these j modes play the role of transmission losses from Ohmic losses and the leakages between dielectric waveguide and nanowires. From Eqs. (4.6) and (4.12), we can specify ˜si and R˜i respectively as
˜
Substitute Eq. (4.13) into Eq. (4.10), we obtain
where we take the reservoir S to be a thermal equilibrium mixture of states,
˜
oscillator with frequency ωj at temperature T. Here, kB is the Boltzmann’s constant. We can make a change of variable τ ≡ t − t0, Eq. (4.14) then For a large reservoir containing infinite modes, we can also change the
sum-mation in Eq. (4.15) to an integration by introducing the density of state
g(ω), that is,P
From Eq. (4.17), we can easily see that if τ is large enough, the oscillat-ing exponential would average other ”slow-varyoscillat-ing” functions, g(ω0), κ(ω0), n(ω0, T ) to zero, which means, comparing to the evolution time of ˜ρS, the correlations of reservoir survive only within a very short time scale τ . We can therefore make an approximation to replace ˜ρS(t − τ ) by ˜ρS(t). This is called Markovian approximation, which states that the evolution of ˜ρS(t) de-pends only on its present state and is independent of its past history. After making this Markovian approximation, Eq. (4.16) turns out to be the master equation in Born-Markovian approximation,
Since the reservoir correlations, Eq. (4.17), vanish in the limit of large τ , we
can therefore extend the τ integration to infinity and obtain where, P is the Cauchy principal value. α and β are then written as
α = πg(ωk)|κ(ωk)|2+ i∆k,
Eq. (4.23) is still in the interaction picture, we can transform it back to Schr¨odinger picture, and it reads
˙ρS = X
Here, the frequency shift ∆k is the so-called Lamb shift in quantum electro-dynamics, which is generally very small and can be conventionally neglected.
Furthermore, we assume that the total system is at temperature T=0, then the mean photon number n is zero. The final master equation in Born-Markovian approximation can be written as
˙ρS = 1 Eq. (4.25) is the Lindblad form master equation with Lindblad operator ak which governs the field damping of the surface plasmons due to Ohmic losses and leakages. Γk in Eq. (4.25) is identified as the decay rate of each k mode into this field-damping dissipation channel.
Our next step is to derive the Lindblad form master equation for the dis-sipation due to the QD excitons decaying into free space. We can now ignore the surface plasmons and start out with the Hamiltonian which describes the interaction between the two dots and vacuum,
HS0 = X i running from 1 to 2. In the Hamiltonian, HR0 describes the vacuum as harmonic oscillators with frequencies $j for each j mode. And HS0R0 is the interaction between the two dots and the vacuum, σ+i(−i) = |eiihgi|(|giihei|), and ηi,j is the coupling constant.
The master equation for the reduced density matrix for the dots can now be easily obtained since the calculation is exactly similar to how we derived Eq. (4.25). Thus, we could have it only by replacing ak and a†k by σ−i and σ+i respectively
˙ρS0 = 1
i¯h[HS0, ρS0] +X
i
γi(σ−iρS0σ+i− 1
2σ+iσ−iρS0− 1
2ρS0σ+iσ−i), (4.27)
where γi is exactly the decay rate γ0 for the dot excitons into free space, which can be exactly evaluated as γi = γ0 = 4π²1
0
4ω3eigi℘2i
3¯hc3 with ℘i = ehgi|ˆq|eii denoting the dipole moment of the i-th dot.
Eq. (4.27) is the Lindblad form master equation for the reduced density matrix of the two QDs. It describes the dissipation of spontaneous emission into free space resulting from the coupling to vacuum.
Now, we would like to move back to our model Hamiltonian: H = HS+ HS0 + HSS0 + HR + HR0 + HSR+ HS0R0, which describes the two QDs couple to multi-mode surface plasmons (see Fig. 4.1), and the two dissipations discussed before. It can be written as a combination of Eqs. (4.12) and
(4.26) plus the do-surface plasmons interaction HSS0, which is
where g1(2),kis the coupling strength between surface plasmon modes and the first (second) QD, and d is the inter-dot distance. The equation of motion for this total system can be written as
˙ρ = 1
i¯h[H, ρ], (4.29)
we can exactly expand the H and rewrite Eq. (4.29) as
˙ρ = 1
i¯h{[HS+ HR+ HSR, ρ] + [HS0+ HR0 + HS0R0, ρ] + [HSS0, ρ]}, (4.30) from Eq. (4.30), we identify that the first commutator corresponds to our discussions for deriving Eq. (4.25), and the second commutator corresponds to Eq. (4.27). After tracing out the reservoirs R and R0, the remaining terms in the commutator is HS + HS0 + HSS0, and the equation of motion for the
reduced density matrix of a composite system χ = S ⊕ S0 can be easily obtained :
˙ρχ = 1
i¯h[Hχ, ρχ]
+ X
k
Γk(akρχa†k−1
2a†kakρχ− 1
2ρχa†kak)
+ X
i=1,2
γi(σ−iρχσ+i− 1
2σ+iσ−iρχ− 1
2ρχσ+iσ−i), (4.31)
where ρχ = T rR,R0ρ, and Hχ= HS+ HS0 + HSS0.
Eq. (4.31) contains the two QDs, the surface plasmon modes, the inter-actions between them, and two kinds of dissipations such as field damping and spontaneous emission into free space. It is exactly the Lindblad form master equation we need to calculate the reduced density matrix of the two dots and to investigate the entanglement generation and its dynamics in the next section by tracing out the surface plasmon modes (system S0).