1. Landau-Zener Tunneling
Edward Burt, Christopher Field1
1
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
In this project, Landau-Zener Tunneling is explored. We find the adiabatic eigenstates of our
system exhibit avoided level crossings and show this gives rise to Landau-Zener tunneling between
these eigenstates. We also show that the adiabatic eigenstates exhibit interference phenomena for
integer detuning ratios. Finally we relate this system to a cascade of Mach-Zehnder interferometers.
Landau-Zener (L-Z) tunneling is a quantum mechan-
ical tunneling effect. L-Z tunneling is observed when a
system spontaneously moves from the adiabatic state |−
to the higher energy adiabatic state |+ . This occurs
when the system evolves naturally in time; not infinitely
slowly. A system where L-Z tunneling occurs multiple
times can exhibit Mach-Zehnder interferometry. This is
like an interferometer with pairs of mirrors which keep
reflecting the same light through the interferometer. A
visual way of interpreting this is Figure 1.
This project evaluates a system with a time dependent
Hamiltonian defined as:
H = −
1
2
( (t)σz + ∆σx) (1)
where σz and σx are the Pauli spin matrices defined as:
σz ≡ |0 0| − |1 1| , σx ≡ |1 0| + |0 1| (2)
(t) is a time dependent parameter with dimensions of
energy and ∆ is a constant with the dimensions of energy.
In this project we will consider the case where:
(t) = 0[2 sin(ωt) − 1] (3)
0 is a constant with dimensions of energy and ω is a
frequency. The ratio: 0
ω defines the detuning ratio, this
project looks at a range of detuning ratios for the system.
FIG. 1. A plot showing repeated L-Z tunneling gives rise to
Mach-Zehnder interference. Beginning at the dot, the first L-
Z tunneling at t1 causes the state to follow the dotted path.
The second L-Z tunneling interferes at t2. The state then
moves away from the L-Z tunneling point and returns to the
starting position. This is one period of Mach-Zehnder inter-
ferometry. This figure was taken from Oliver et. al. [2]
The diabatic states, which will be used as the basis in
this report, are defined as:
|0 =
1
0
, |1 =
0
1
(4)
The adiabatic eigenstates of this system are given by
the eigenstates of the Hamiltonian (1), these can be
shown to be:
|+ =
−E
∆
1
, |− =
+E
∆
1
(5)
where E = ( 2
+ ∆2
)
1
2 . In the limit (t) >> ∆ the
adiabatic eigenstates reduce to the diabatic eigenstates
to within a phase factor. The corresponding adiabatic
eigenvalues are given by:
λ± = ±
1
2
E (6)
Using the diabatic eigenstates as our basis we can de-
fine the state of the system as:
|Ψ(t) = c0(t)|0 + c1(t)|1 (7)
where c0 and c1 are time dependent probability ampli-
tudes of the |0 and |1 states respectively. By substi-
tuting (7) and (1) into the time dependent Schr¨odinger
equation:
H|Ψ(t) = i
∂
∂t
|Ψ(t) (8)
we find two coupled differential equations (9) and (10)
for the coefficients of the diabatic states in (7).
˙c0(t) =
i
2
(∆c1(t) + (t)c0(t)) (9)
˙c1(t) =
i
2
(∆c0(t) − (t)c1(t)) (10)
These equations can be solved numerically to find
the probabilities of being in |1 ; P1 = |c1|2
and |0 ;
P0 = |c0|2
. Solutions are found for specific values of
the constants ∆, 0 and ω.
The transition probability of a system which starts in
|0 and ends in |1 can be derived as long as the system
starts and ends away from an avoided crossing point.
P0, 1 = 1 − exp
−∆2
π
2 ν
(11)

2. 2
FIG. 2. A plot of the adiabatic eigenvalues as a function of
time. This plot shows multiple avoided level crossings of the
adiabatic eigenvalues for ∆
0
= 0.1 and ω
0
= 0.001
This transition probability is given in (11), where ν
is defined as the first time derivative of the difference
between the two adiabatic eigenvalues (12), in the limit
∆ → 0 evaluated at t∗
.
ν =
d
dt
(λ+ − λ−) (12)
t∗
is defined as the time where (t) = 0. Using this we
find ν =
√
3ω 0 for our system. The derivation of the
equation P0, 1, (11), can be found in [4].
P0, 1 is equivalent to the upper state probability P1 that
comes from the numerical solutions to (9) and (10). This
gives us a method for comparing the numerical solutions
we find to the analytical solutions from (11).
Avoided level crossing, meaning the adiabatic eigen-
values do not cross paths, can be seen in Figure 2. This
implies that the system should stay in the lower state
|− if it began there. However, this only holds if (t) (3)
changes slowly with time. The adiabatic theorem states
that if (t) changes infinitely slowly with time, then the
state should not change [3]. In a real system (t) does
not vary infinitely slowly. In this case L-Z tunneling can
cause the system to tunnel from the lower state |− to
the upper state |+ , with a finite probability. In this re-
port, the system examined has (t) >> ∆. In this limit
the adiabatic eigenstates are approximately equal to the
diabatic eigenstates respectively:
|+ ≈ |0 , |− ≈ |1 (13)
In this project we used Matlab to calculate the numer-
ical solutions and plot our figures. We used ODE45 as
our differential equation solver with tolerances of 10−12
.
The first part of this project was to show that our system
exhibited avoided level crossing. After this we found the
coupled differential equations (9) and (10) and solved nu-
merically to find the probabilities P1 and P0. Using (11)
we compared our numerical answer against the analytical
one to check consistency and show that the simulation of
the system was accurate. Finally we looked at the system
over a range of detuning values to replicate the interfer-
ence phenomena found in [2].
FIG. 3. A plot showing the probability of occupying the
diabatic eigenstates as a function of time. The system starts
from the lower diabatic state |0 with ∆ = 1 , = 1 , ∆
0
= 0.05
and ω
0
= 0.001. This figure shows that the probabilities
oscillate around a late time solution after an initial period.
FIG. 4. A plot of the transition probability from |0 to |1 as
a function of time for ∆ = 1 , = 1 , ∆
0
= 0.05 and ω
0
= 0.01.
This plot shows the numerical solution oscillates around a late
time value which matches the analytical solution.
The probability of tunneling from |0 to |1 and vice
versa is given by P1 and P0 respectively and can be seen
in Figure 3. This shows the L-Z tunneling effect. The re-
sults of numerically solving for |c1|2
, which is equivalent
to the transition probability P0,1, can be seen in Fig-
ure 4. An analytical solution was also found as outlined
previously (11). Plotting the two results shows that the
numerical solution tends toward the analytical solution
at late times, see Figure 4.
A complete explanation of the mean transition prob-
abilities should include the behaviour over time. Gener-
ally, the probability of transition oscillates over time and
decreases in amplitude over oscillations. The rate of de-
crease in amplitude appears to be related to the detuning
ratio. A larger detuning ratio gives a faster decay in am-
plitude of the oscillations. A specific example can be seen
in Figure 5. The decrease in transition probability for
integer detuning ratios with respect to time can explain
the decrease in mean probability seen in Figure 7. It also
highlights why a long period of time was required to give
a reliable mean transition probability. An interference-
like pattern can be seen in Figure 6. Resonant peaks are
observed at integer values of the detuning ratio. This
matches the results seen in Figure 1C of Oliver et. al.
[2].

3. 3
FIG. 5. This plot shows the probability of being in the
adiabatic state |+ as a function time. In this plot the de-
tuning ratio was 0
ω
= 8 with = 1, 0 = 1 and ∆
ω
= 0.01.
The points were determined over the range of times t = 0 to
t = 300000π
2ω
.
FIG. 6. This plot shows the probability of being in the adi-
abatic state |+ as a function of the detuning ratio 0
ω
for
= 1, 0 = 1 and ∆
ω
= 0.01. The points were determined
over the range of times t = 0 to t = 10000π
2ω
. with the ex-
ception of the point at 0
ω
= 8 which was run for t = 0 to
t = 60000π
2ω
. The reason for this was a good average required
a large number of oscillations. This value was run over more
points as it only just managed a single oscillation over the
expanded range. Expanding the range to at least one oscilla-
tion gave us a more accurate mean showing the constructive
interference at integer detuning ratios.
To draw analogy between the two plots, Figure 7 ex-
amines the first (I) interference fringe in Figure 1C of
Oliver. The amplitude of the |+ state probability is
shown as the colour in Oliver. The underlying physics of
the interference fringes is beyond the scope of this report.
However the interference is explained nicely in the review
letter by Shevchenko, Ashhab and Nori [1]. Qualitatively,
integer values of the detuning ratio causes constructive
interference. Due to averaging, the peak heights vary
inconsistently in Figure 6. Hence the decrease in peak
amplitude observed in Oliver can not be seen [2]. This is
addressed in Figure 7.
To show complete consistency with the results seen in
a superconducting flux qubit in Oliver, the probability
peaks were examined in more detail. Figure 7, shows the
decreasing mean probability of the |+ state with respect
to detuning. This corresponds to the result seen in Fig-
ure 1C of Oliver, where the colour map shows decreasing
FIG. 7. This plot shows the probability of being in the
adiabatic state |+ as a function of the ratio 0
ω
with = 1,
0 = 1 and ∆
ω
= 0.01. The points were selected to be integer
detuning ratios only. The mean probability was determined
over the range of times t = 0 to t = 200000π
2ω
. This gave enough
oscillations of each resonant value to give a good average and
shows the decrease in the constructive interference amplitude.
The green line of best fit shows the trend to be a negative
linear correlation.
amplitude with increasing detuning ratio [2]. This ex-
periment could be extended by running our simulation
of the system over more points and for a longer time.
This would show any longer term patterns but requires
a large amount of computing power. Another extension
would be to look at the system when we include deco-
herence effects, this would help capture the full quantum
nature of our system and could be linked to other re-
sults found in [2]. One further interesting area to look
at would be when (t) ≤ ∆, other values of ω
∆ and how
these parameters affect the peak probability drop off we
found in 7.
In conclusion, this report has shown that the theo-
retical results predicted from L-Z tunneling match the
experimental results found by Oliver [2] very closely. We
have shown that our system exhibits avoided level cross-
ing of the adiabatic eigenstates, which is characteristic
of L-Z tunneling. This tunneling probability was shown
to match the analytical solution at late time. Finally we
showed an interference pattern in the tunneling proba-
bility, matching the results found in [2].
[1] S.N. Shevchenko et.al, Landau-Zener-Stckelberg Interfer-
ometry, Physics Reports 492, pp 1-30, (2010).
[2] W. D. Oliver et.al, Mach-Zehnder Interferometry in a
Strongly Driven Superconducting Qubit, Science, 310, pp
1653-1657, (2005).
[3] M. Born and V. A. Fock, ”Beweis des Adiabatensatzes”,
Zeitschrift fr Physik A, 51 (34), pp 165180 (1928).
[4] Quantum Engineering, A. M. Zagoskin, Cambridge Uni-
versity Press, Chapter 1.5.2, (2011).