4.3 Modeling of molecular charge qubits
4.3.2 Electron–vibration coupling strength
As mentioned before, the overall coupling strength between the qubit and the vibra-tional modes is determined byγ viaγi = (π/2)(g2i/ωiηi). Here, this is calculated by redetermining the energy splittings while the nuclear coordinates are displaced ac-cording to the mode with maximal intensity in the IR spectrum, for whichηimax = 1 (see Appendix E). It is assumed that this mode has a maximal electronic coupling strength. Consequently, a maximal shift of the energy splitting is expected when the nuclear coordinates are displaced. For a vibrational mode with frequency (i.e., eigenvalue)ωi, the displacement coordinates are given by the eigenvectors associated to that mode. They are obtained by using the AM1 method (see Appendix B).
A molecular system with an energy splitting ∆N (~ ≡1) has an IR mode with maximal intensity at frequency ωimax, i.e., the mode with ηimax = 1 in Appendix E. This mode has a force constant k as follows from the curvature at the potential minimum. The calculated nuclear displacement coordinates, rj, indicate that each atom is displaced in its normal mode direction by one unit of the oscillator length
38 Design of organic π-conjugated molecular charge qubits
(see Sec. B2). By summing over all atomsi, a potential energyEpot = (1/2)kP
ir2i due to the nuclear displacement is introduced. This displaced nuclear configuration is then used to recalculate the electronic states, yielding an energy splitting E(dis). Typically, displacements generate an energy difference 0 for the electron in one of the two wells (i.e., a static bias) rather than modifying the tunnel coupling between the two wells. The energy splitting for an according two-level system is thenE(dis) = p∆2N +20, whereas in equilibrium E(equil) = ∆N. The difference between the two is thenδE =E(dis)−E(equil).
The displacements introduced correspond to the Epot/ωimax =υ-th excited state of the considered mode leading to 0 ' p
(δE)2+ ∆2N − ∆N. In turn, a single excitation of this mode corresponds to displacements smaller by a factor √
υ. It is assumed that the electronic qubit states couple bilinearly to vibrations, i.e., ∝ (1/2)σzgj(aj+a†j) in Eq. (4.3.3). Hence, all nuclei are maximally displaced relative to a single excitation in the modeωimax. This results in an energy difference 0/√
υ which is identified with the coupling constantgimax.
This result can be directly replaced in γi = (π/2)(g2i/ωiηi), for which ηimax = 1 due to the fact that only the IR mode with maximal intensity is considered. Thus,
γimax = π 2
g2imax ωimax
. (4.3.5)
This indicates that the electron–vibration coupling constants gi determine also the damping strength of the vibrational bath. This is given by the decay of the bath autocorrelation in Eq. (2.2.11), as shown in Appendix E.
Results for the systems with N = 0,1, and 2, obtained using the described methodology, are shown in Table 4.2. It is worth to note, however, that the coupling
k υ-th excited E(dis) 0 gimax γimax
System [eV/˚A2] state [cm−1] [cm−1] [cm−1] [cm−1]
∆0 193.34 33rd 2325.27 926.61 161.30 20.14
∆1 204.81 34th 3196.35 1392.27 238.77 43.18
∆2 219.64 35th 5868.50 3639.54 615.19 278.28
Table 4.2: Electron–vibration coupling strength for the ∆N systems with N = 0,1, and 2.
strength obtained for the system ∆2 is quite large. This might be a consequence of the several modes at energies around ~∆2 (see Table E3 and Fig. E5), which already results in a system with strong coupling to vibrational modes and will then exhibit strong damping (see results in Sec. 5.1). Due to their smaller molecular
4.3. Modeling of molecular charge qubits 39
size, displacing the atoms from their equilibrium position according to the most intense mode, as before, in the systems with N = 3 and 4 induces a very strong electronic rearrangement, which gives E(dis) < E(equil), resulting in very small γ values. This is physically meaningless due to the fact that a smaller molecule will have a stronger electron–vibration coupling, which is consistent with the results in Table 4.2 for the largest systems. It is important to note that more elaborate ways to calculate the electronic level widths for different environments are available [127, 128]. The structural deformation due to the pseudo-Jahn-Teller effect [121] is negligible sinceγimax ∆N for all calculated systems, in contrast to the cases where a strong coherent electron–vibrational coupling leads to qualitative modifications of the electron dynamics [129, 130].
40 Design of organic π-conjugated molecular charge qubits
CHAPTER 5
Dynamics of π-conjugated molecular charge qubits
In this chapter, the real-time dynamics of the different molecular charge qubits in Table 4.1 is investigated in terms of the time-dependent population difference
P(t) = hσzit= tr [ρ(t)σz] (5.0.1) of the left and right wells. The reduced density matrix ρ(t) is calculated by means of the interative QUAPI scheme presented in Sec. 2.2.1. An initial preparation ρ(0) = |LihL| in the left well is assumed. For all cases, ~∆N > kBT at room temperature and thermal effects are negligible for the following considerations. All results reported below are for T = 300 K.
The long-lived oscillatory correlations of the ∆4 system resulting from the large number of peaks at frequencies ωi < ∆4 lead to very long memory times τ =Kδt in the autocorrelation function (see Fig. E10 in Appendix E). This makes difficult to find an optimal parameter combination ofK and δt, and therefore, no results are shown for the ∆4 system in this chapter. However, results obtained by choosing a compromise between a large enough memory time (given by a large K) and a small enough Trotter error (given by a small δt step) are presented in Appendix F for comparison purposes.
41
42 Dynamics of π-conjugated molecular charge qubits
5.1 Dynamics of undriven molecular charge qubits
Here, it is assumed that(t) = 0 in Eq. (4.3.2). The resulting dynamics is shown in Fig. 5.1 for all cases withN = 0-3. Because, in general, the strength of the electron–
vibration coupling increases by decreasing the size of a molecule, coupling strengths
-1 0
1
P(t)
-1 0 1
0 t∆0
100 200
γ [cm-1]
100
50 300
(a)
-1 0
1
P(t)
-1 0 1
0 t∆1
100 200
γ [cm-1]
100
50 300
(b)
-1 0
1
P(t)
-1 0 1
0 t∆2
100 200
100
50
γ [cm-1] 300
(c)
-1 0
1
P(t)
-1 0 1
0 t∆3
100 200
100
50
γ [cm-1] 300
(d)
Figure 5.1: Time-dependent population differenceP(t) as a function of the damping strength γ for the heterostructures ∆N with N = 0-3 (a)-(d).
5.1. Dynamics of undriven molecular charge qubits 43
up to 300 cm−1 have been investigated. Coherent oscillations of the population difference are found. Figure 5.2 compares the dynamics of all systems when the electron–vibration coupling is fixed at γ = 250 cm−1. The corresponding dynamics of the ∆4 system is shown in Fig. F1.
0 20 40
-1 -0.5 0 0.5 1
P(t)
∆0
∆1
∆2
∆3
t∆N 60
Figure 5.2: Time-dependent population difference P(t) for a fixed value γ = 250 cm−1 of the electron–vibration coupling for the heterostructures ∆N with N = 0-3.
By a fit of the population difference to the expression P(t) = e−Γtcos (ωt), it is possible to extract the coherence time T1,N = Γ−1 and the quality factor QN = ω/Γ =T1,N/∆N. The latter gives the number of operations that can be implemented in a physical qubit within the decoherence time [131]. The resulting values forT1,N and forQN as a function of the damping strengthγfor all molecular systems ∆N are shown in Fig. 5.3. Coherence times and quality factors for particular γ values are given in Table 5.1. It is found that the coherence times and, consequently, the quality factors decrease with increasing coupling. This can be understood when considering the autocorrelation function in Eq. (2.2.11) of the vibrational bath (see Appendix E): in the limit of small γ values, the autocorrelation function is slowly damped, and a long-lasting coherent decay ofP(t) is observed. In the opposite limit of largeγ values, the bath-induced memory effects are quickly lost and therefore,P(t) decays faster. Within the investigatedγ values, it always hold that ∆N γ, which explains the absence of a completely incoherent decay of P(t) in Fig. 5.1. It is important to note that~∆N is not in resonance with any vibrational mode in any of the five cases, which (together with the lacking point symmetry) excludes possible strong pseudo-Jahn-Teller effects [121]. Put differently, no significant polaron formation occurs. In such a case, a breakdown of the Franck-Condon (or, equivalently, the Born-Oppenheimer) approximation is expected. Due to the large number of peaks
44 Dynamics of π-conjugated molecular charge qubits
101
Coherence time [ns]
103
102
0 100 200 300
∆0
∆1
∆2
∆3
γ
[cm-1]0 100 200 300
∆0
∆1
∆2
∆3
γ
[cm-1]100
Quality factor
102
101 103
Figure 5.3: Coherence timesT1,N(top) and quality factorsQN (bottom) as a function of the damping strengthγ for the heterostructures ∆N with N = 0-3.
around 1500 cm−1, a significant spectral overlap occurs in all heterostructures (see Appendix E). This leads to an effective broad single mode with a large spectral weight. This is the reason for the system ∆2, in particular, to exhibit the smallest quality factors. On the contrary, this effective mode has a negligible influence on the dynamics of the ∆4 system resulting in quality factors an order of magnitude
5.1. Dynamics of undriven molecular charge qubits 45
γ
System 5 cm−1 20 cm−1 45 cm−1 100 cm−1 250 cm−1
∆0 535.394 222.275 201.258 159.552 53.986
(79) (31) (27) (20) (6)
∆1 528.736 147.498 72.650 42.911 31.258
(101) (28) (14) (7) (5)
∆2 503.075 124.848 55.739 30.982 17.641
(111) (27) (11) (6) (3)
∆3 345.562 94.453 42.308 24.068 17.846
(133) (36) (16) (10) (8)
Table 5.1: Coherence timesT1,N (in ns) and quality factors QN (in parentheses) for all heterostructures with N = 0-3 for specific γ values as indicated.
larger (see Appendix F).
Notice that for small γ values, the quality factors increase for larger N, i.e., when the dynamics is influenced by N main vibrational bands below ~∆N. The situation, however, is more complicated when increasing the damping strength γ.
In the intermediate regime of γ values, the system ∆0 exhibits the largest quality factors, with a maximum around 100 cm−1. For this particular system, the energy difference between the electronic transition (given by ∆0) and the first vibrational peak (see Table E1) is 98 cm−1. A width γ of the vibrational transition close to this value will bring the electronic and vibrational transitions into resonance and therefore, sustained coherence is expected. This results in large coherence times and quality factors as observed in Fig. 5.3. A similar argument is valid for the systems
∆1 and ∆3. In these cases, the corresponding energy differences are 89 cm−1 for
∆1 (taken with respect to the first vibrational peak at lower energy as compared to
~∆1, see Table E2), and 96 cm−1 for ∆3 (taken with respect to the next vibrational peak at higher energy, see Table E4). For these two cases, however, no maximum in the quality factors is observed. The reason for this is the large spectral weight at the electronic transition,J(∆N), resulting from the overlap of the vibrational peaks around~∆N (at larger energies for both cases, see Figs. E3 and E7).
Thus, it is possible to conclude that the design of the molecular heterostructures allows tochemicallyengineer the coherence times over very broad time scales for such molecular quantum devices. Similar chemical engineering was shown for molecular spin systems [132], where the electron spin phase memory time was extended by changing the substituent chemical groups in the molecular structure.
46 Dynamics of π-conjugated molecular charge qubits