To derive an expression for slow population transfer under dephasing, we will continue our derivation from Section 2.4.2, where the diagonal and o↵diagonal terms of the density matrix are given by, as seen in Chapter 2:
⇢jj(s) = ⇢jj(s0)
Z s1
s0
Ckj(s)ds (5.22)
⇢jk(s) =e i⇠(s)⇢jk(s0) +e i⇠(s) Z s
s0
ei⇠(s0) 1(s0)ds0 (5.23) with
1(s) = X
m6=j
⇢mk(s)⌦j(s)˙ |m(s)↵
+X
n6=k
⇢jn(s)D
n(s)|k(s)˙ E
(5.24) and we again 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. To proceed, we observe that for ⌧ 1, we can perform integration by parts and expand with respect to v, and the o↵diagonal term, up to first order with respect to v is given by:
⇢jk(s) = e i⇠(s)⇢jk(s0) +e i⇠(s)
vei⇠(s)
igjk(s0) + jk(s0) (5.25)
⇥ ⇣
1(s0) + ⇢(i,j)(s0)D
j(s0)|k(s˙ 0)E⌘ s0=s
s0=s0
where jk(s) = [Aj(s) Ak(s)].
Substituting this into the diagonal element, we obtain
⇢jj(s) =⇢jj(s0) +P1(s) +P2(s) (5.26) with
P1(s) = X
k6=j
v e i⇠(s)
igjk(s0) + jk(s0)
Dj(s0)|k(s˙ 0)E s0=s
s0=s0
⇢jk(s0) +c.c. (5.27)
5.3. POPULATION TRANSFER UNDER DEPHASING
and
P2(s) = Zs s0
2 64
v ⇢(i,j)(s0) D
j(s0)|k(s˙ 0)E2
igjk(s0) + jk(s0) +c.c.
3
75ds0 (5.28)
+ Zs s0
2 4 vD
j(s0)|k(s˙ 0)E
igjk(s0) + jk(s0) 1(s0) +c.c.
3 5ds0
To verify the validity of the derivation, we compare our results with our earlier work [19].
In this case, we will substitute the zeroth order result for ⇢jk(s) into P2(s) and as a result, after the integration of the second term, P2(s) will be of at least O(v2) and hence we can drop that particular term. We finally recognise that we will fully recover the results from Ref. [19] if we let e i⇠(s) ! 0 as v ! 0, which is the limit that we have considered in the work. As a result, the population at j-th level will be given by:
⇢jj(s) =⇢jj(s0) +vX
k6=j
Dj(s0)|k(s˙ 0)E igjk(s0) + jk(s0)
s=s0
⇢jk(s0) +c.c.
2X
k6=j
[⇢jj(s0) ⇢kk(s0)]
Z s1
s0
v jk(s) D
j(s0)|k(s˙ 0)E 2 2jk(s) +gjk2 (s) ds,
(5.29)
Here we note that in comparison to the dissipative case, the contribution due to the population di↵erence is of first order with respect to v, and also, the initial state coherence has a non-zero contribution to the non-adiabatic correction to the population transfer. This is significantly di↵erent from the dissipative case where the final population at s1 is purely dictated by the Lindblad operators, and the choice of initial state has little e↵ect on the final state of the system.
5.3.1 Example
To illustrate the theory we have derived in the previous section3, we consider a three-level system described by the HamiltonianH(s) =g0Sx+sSz, where Sx and Sz are spin-1 opera- tors. In this case, for a constant driving speedv ofs, the non-adiabatic transition probability on the second excited state is given by
⇢11=⇢11(s) ⇢11(s0) = 2v[⇢11(s0) ⇢22(s0)]D1 vD2, (5.30) with
D1 = 1 4 g02
2 66 4
(2 2g20) cot 1(g0)
g03 +
3tan 1
✓ p 2
g02+4
◆ p 2
g02+ 4 + 2 g04+g20
3 77
5 (5.31)
and
D2 = 4p
2g0{Re [⇢12(s0)] g1 2Im [⇢12(s0)]}(g1+ 2s0 1) g13+ (4 + 2g12)p
g12+ (2g1+ 1) (2s0 1) , (5.32) where ⇢12(s0) is the initial o↵-diagonal density matrix element describing the coherence between the two excited states and g1 ⌘
q
g02+ (1 2s0)2.
Figure 5.3 shows the agreement between theory and numerics for this case, for a wide range of the dephasing rate and for various initial states. Expectedly, the results only diverge as v approaches a large value, since our derivation is based on the fact that v has to be sufficiently slow such that the correction beyond the first order of v can be ignored.
We also explicitly compare the percentage di↵erence between the theory and numerics in Fig. 5.4 and find that over a wide range of v, the percentage di↵erence between our theory and numerical results di↵ers by less than 1%. In the fast driving limit, the two results expectedly diverge, with the percentage di↵erence at around 4% for the mixed state and more than 10% for the coherent state. Comparing the two cases shown in the main panel of Fig. 5.3, we see the impact of initial state coherence on non-adiabatic transition probabilities, albeit dephasing. The initial state coherence in fact will significantly increase the transition probability, especially as the degree of coupling increases.
3We refer the readers to Ref. [19] for illustrations on two levels systems using di↵erent adiabatic protocols.
5.3. POPULATION TRANSFER UNDER DEPHASING
We further observe that under dephasing, interestingly, the final transition probability also displays a non-monotonic behaviour with respect to the coupling , and the peak is at about the regime of regardless of the type of initial state preparation. Physically, this can be understood as the suppression of the oscillatory motion of the population as it is being driven across the avoided crossing. We further notice that in the limit of large , the final transition probability essentially reaches zero and this is a manifestation of the quantum Zeno e↵ect. The system in monitored too strongly in this case, hence any non-adiabatic transitions will no longer be possible.
Figure 5.3: Transition probability of ⇢11 as a function of . The coherent state is initially prepared in | (s0)i = 0.8|E1i + 0.1|E2i+ 0.1|E3i, and the mixed state is initially in ⇢(s0) = 0.8|E1i hE1|+ 0.1|E2i hE2|+ 0.1|E3i hE3|. Here we set g0 = 1 and v = 10 3. Inset shows the transition probability when the state is prepared in a pure state| i=|E1i.
Figure 5.4: Percentage di↵erence of the transition probability of ⇢11 between the numerical results and our theory as a function of . The coherent state is initially prepared in | (s0)i = 0.4|E1i+ 0.3|E2i+ 0.3|E3i, and the mixed state is initially in⇢(s0) = 0.4|E1i hE1|+ 0.3|E2i hE2|+ 0.3|E3i hE3|. Here we set g0 = 1 and v = 10 3. Inset shows the transition probability when the state is prepared in a pure state | i=|E1i.