solve the the equations of motion for the reduced density matrix of the system. However, regardless the choice of approach, one key challenge is the computational e↵ort required in computing such systems. We will briefly illustrate this below for the master equation approach derived in previous section.
As we observe in Eq. (2.13), the integral term consists of a system unitary operator term US(t, t1). For a time-independent system, we can simplify this term into exp [ iHS(t t1)], which can be typically recasted in terms of the eigenvalues of HS after projecting the entire equation into its energybasis. Therefore, the overall term involves only a single integral which can be easily solved numerically. For time-dependent systems, however, the unitary operator is instead given by exp [ iT Rt
t1HS(s)ds], where T is the time ordering operator.
By inspection, we can easily observe that one has to repeat the integrals for each density matrix element for the unitary operator for each time step, and as such this will eventually be a daunting task for systems with long driving time (which is of our interest in subsequent chapters) or of large Hilbert space. We therefore will have to resort to further approximations to our master equations, as mentioned in the concluding remarks of the previous section.
2.4 Time-Dependent Master Equation in Lindblad Form
We have reviewed and derived the master equation in Eq. (2.13) using nothing more than the standard time-dependent perturbation theory in quantum mechanics. However, as men- tioned in the previous section, in many time-dependent driven systems, such general form of master equation may make the problem intractable, or at least require an extremely long computation time to be solvable practically. Hence, to simplify the problem at hand, as mentioned earlier, we perform Markovian approximation and rotating wave approximation to reduce the master equation into a more solvable form, otherwise known as the Lindblad master equation.
In this section we first consider a quantum system described by a time-dependent Hamil- tonianH(s), wheres=vtis the rescaled time, v is the driving speed of the parameters and
t is the real time. To include the e↵ects of dissipation in the adiabatic driving, we consider the master equation in Lindblad form:
v d
ds⇢(s) = i[H(s),⇢(s)] +
⇢
A(s)⇢(s)A†(s) 1 2
⇥A†(s)A(s)⇢(s) +⇢(s)A†(s)A(s)⇤ (2.15) where A(s) = P
mnAmn(s)|m(s)i hn(s)| and also that A†(s) = P
mnA⇤nm(s)|n(s)i hm(s)|. Here, |m(s)i and |n(s)i are instantaneous eigenstates of H(s) at given s. The subject of interest here is the probability that the system is in thejth energy level,⇢jj(s), where⇢jj(s) = hj(s)|⇢(s)|j(s)i. Here, is the coupling strength between the system and the environment.
To proceed further, we cast the entire master equation in the time-dependent basis of the system Hamltionian and consider the e↵ect of dissipation and dephasing individually.
2.4.1 Dissipative Lindblad Equation
In a dissipative system, the energy levels of the system will undergo transition due to its interaction with the environment, and this can be represented by the choice of Lindblad operator whereby the indices m 6=n. Such choice of Lindblad operator will always induce a transition between the m and n levels and mimic the e↵ect of dissipation to the environ- ment. Furthermore, such form of Lindblad operator will not be able to commute with the Hamiltonian. To proceed, we project the instantaneous eigenstate of H(s) into the master equation and we obtain the following:
For the diagonal terms, we have
˙
⇢jj(s) + v
X
m6=j
|Amj(s)|2⇢jj(s) =Cjm(s) +
v[Dmm(s) +Djn(s) +Dnm(s)] (2.16) where |Amj(s)|2 =Amj(s)A⇤mj(s) and v = dsdt.
Cjm(s) = X
m6=j
(⇢jm(s)hm(s)˙ |j(s)i+⇢mj(s)hj(s)|m(s)˙ i) (2.17)
2.4. TIME-DEPENDENT MASTER EQUATION IN LINDBLAD FORM
Dmm(s) = X
m6=j
|Ajm(s)|2⇢mm(s) (2.18)
Djn(s) = 1 2
X
mn,m6=j,n6=j,m6=n
Amn(s)A⇤mj(s)⇢nj(s) +Amj(s)A⇤mn(s)⇢jn(s) (2.19)
Dnm(s) = X
mn,m6=j,n6=j,m6=n
Ajn(s)A⇤jm(s)⇢nm(s) (2.20) where Cjm(s) is the term due to the coherence of the instantaneous eigenstates which exist even in closed systems, while Dmm(s), Djn(s) and Dnm(s) are terms that arise due to the dissipation. In particular, Dmm is related to the instantaneous population of the states at instant s.
At this stage, to solve for the diagonal terms, we will now multiply both sides by a factor of exp [
Rs s0
v
P
m6=j|Amj(s0)|2ds0] = exp [v↵mj(s)], where we keep the v term in order to track the order, we then have
⇢jj(s) =e v↵mj(s)⇢jj(s0) +e v↵mj(s) Zs s0
ev↵mj(s0)[Cjm(s) +
v[Dmm(s0) +Djn(s0) +Dnm(s0)]ds0 (2.21) For the o↵ diagonal terms, we have
˙
⇢ij(s) ⇢ij(s)(⌦
j(s)|j(s)˙ ↵ ⌦
i(s)|˙i(s)↵ ) + i
vgij(s)⇢ij(s) +
vF1(s)⇢ij(s)
= X
m6=i6=j
⇢im(s)⌦
m(s)|j˙(s)↵
+⇢mj(s)⌦˙i(s)|m(s)↵
+ ⇢(j,i)
⌦i(s)˙ |j(s)↵
+ v
⇥Aij(s)A⇤ji(s)⇢ji(s) +K1,diag(s) +K1,o↵(s) +K2(s)⇤
(2.22)
with
⇢(j,i) =⇢jj(s) ⇢ii(s) (2.23)
F1(s) = |Aji(s)|2 +|Aij(s)|2+ X
n6=i6=j
|Ani(s)|2+|Anj(s)|2 (2.24)
K1,diag(s) = X
m6=i6=j
Aim(s)A⇤jm(s)⇢mm(s) (2.25)
K1,o↵(s) = 1 2
X
m6=im6=j
(Ajm(s)A⇤ji(s)⇢mj(s) +Aij(s)A⇤im(s)⇢im(s)) (2.26)
K2(s) = X
mmn6=i6=n m6=j n6=i6=j
Aim(s)A⇤jn(s)⇢mn(s) 1
2(Anm(s)A⇤ni(s)⇢mj(s) +Amj(s)A⇤mn(s)⇢in(s)) (2.27)
Once again, F1(s),K1,diag(s),K1,o↵(s) andK2(s) are terms that arise due to dissipation.
In this case, K1,diag(s) arises due to the population of the other levels other than the i-th or j-th levels, whereas K1,o↵(s) is related to the instantaneous o↵ diagonal terms between i-th, j-th and the other states. K2(s) on the other hand is related to the instantaneous o↵
diagonal terms of the other states other than i-th andj-th states.
To proceed, we multiply both sides by exp [ i ij(s) + vi✓ij(s) + vf1(s)], where we define
ij(s) = i Zs s0
⌦i(s0)|i(s˙ 0)↵ ⌦
j(s0)|j(s˙ 0)↵
ds0 (2.28)
as the geometric phase di↵erence and
✓ij(s) = Zs s0
gij(s0)ds0 (2.29)
as the dynamical phase di↵erence between the ith andjth state, as well as
f1(s) = Zs s0
F1(s0)ds0 (2.30)
2.4. TIME-DEPENDENT MASTER EQUATION IN LINDBLAD FORM
We then obtain
⇢ij(s) =e [ i ij(s)+vi✓ij(s)+vf1(s)]⇢ij(s0) +e [ i ij(s)+vi✓ij(s)+vf1(s)]
Zs s0
e i ij(s0)+i✓ij(s0)+vf1(s0)⇥
✓ X
m6=i6=j
⇢im(s0)⌦
m(s0)|j(s˙ 0)↵
+⇢mj(s0)⌦˙i(s0)|m(s0)↵
+ ⇢(j,i)
⌦˙i(s0)|j(s0)↵
+ v[Aij(s0)A⇤ji(s0)⇢ji(s0) +K1,diag(s0) +K1,o↵(s0) +K2(s0)]
◆ ds0
(2.31)
At this stage, the solutions of both diagonal and o↵ diagonal elements are both formal solutions, and in order to obtain a physical interpretion of the results, we have to investigate the limiting cases whereby the driving speed is either fast or slow (close to the adiabatic regime). We will perform such further approximations in the subsequent chapters.
2.4.2 Dephasing Lindblad Equation
In this section, we will consider the situation whereby the environment does not induce population transfer to the system, and this is analogous to the pure dephasing case considered earlier on, except that now we are working in the time dependent basis. In this case, the Lindblad master equation can be simplified to the following:
v d
ds⇢(s) = i[H(s),⇢(s)] +
⇢
A(s)⇢(s)A(s) 1 2
⇥A2(s)⇢(s) +⇢(s)A2(s)⇤
(2.32) whereby A(s) =P
mAm(s)|m(s)i hm(s)|. Since the Lindblad operators only have diagonal contributions, we use a single index to denote the elements in the Lindblad operators, unlike the dissipation case described in the dissipative Lindblad equation.
Once again, we project the master equation into instantaneous eigenstate ofH(s),|m(s)i and hj(s)|, we find that the diagonal and the o↵diagonal elements will be respectively given by:
⇢jj(s) = ⇢jj(s0)
Z s1
s0
Cmj(s)ds (2.33)
⇢jk(s) = e i⇠(s)⇢jk(s0) +e i⇠(s) Z s
s0
ei⇠(s0)B1(s0)ds0 (2.34) with
B1(s) = X
m6=j
⇢mk(s)⌦j˙(s)|m(s)↵
+X
n6=k
⇢jn(s)D
n(s)|k(s)˙ E
(2.35) and we defined ⇠(s) = jk(s) + 1v✓jk(s) i↵jk(s) and = v for brevity of notation. Here,
↵jk(s) =Rs
s0[Aj(s0) Ak(s0)]ds0.
At this junction, we will make one important remark. As we observed in both diagonal and o↵-diagonal expressions in dissipation and dephasing cases, the dissipation case has a significantly more complicated form due to the multiple possible choices of the Lindblad operators, and therefore typically more difficult to solve. We will then restrict ourselves to two levels systems whenever dissipation is present. Nonetheless, we can still arrive at various meaningful insights as we will see in the subsequent chapters.