The Two-Level System

In this chapter some general properties of a two-level system will be discussed. It is again emphasized, that the two-level system appears in many areas of quantum mechanics.

Examples

resonant quantum tunneling, spin transitions at the level crossings in AFM wheels, chemical covalent bonding, chemical reaction rates, cold atoms, photons, qubits for quantum computing IX.3.a. Energies and Eigenfunctions

Hamiltonian in the Energy Representation

Just for reminding this possibility to express operators, and the Hamiltonian in particular, in a basis independent way, the energy representation of the two-level Hamiltonian is given:

1 1 E 0 0 E Hˆ

1

0 +

= Comments

1) 0 , 1 are effective eigenstates, they do not have to be related in an obvious way to the

"true" eigenstates of the "true" Hamiltonian

2) are the true energy eigenvalues, i.e., correspond exactly to that of the "true"

Hamiltonian E ,0 E₁

3) as with the matrix elements in the two-level Hamiltonian also the energies may depend on parameters of the "true" Hamiltonian, in our case e.g. the magnetic field

Matrix and Operator Representation in a "General" Basis

As basis we may chose the states 0 , 1 , then as presented already in the above,

[ ]

x

[ ]

y Z

0 1 1 0 1

*

0 ˆ Reb ˆ Imb ˆ

1ˆ 2 b 2

Hˆ b ε −ε σ + σ − σ

ε − +

= ε

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ε

= ε

0 Hˆ

0 = 0

ε , ε₁ = 1Hˆ 1 , b= 0Hˆ 1

( )

²0

2 0 1 1

0

0 2

1

E ε 2+ε − ε −ε +Δ

=

Def.: Δ₀ =2b

( )

²0

2 0 1 1

0

1 2

1

E ε 2+ε + ε −ε +Δ

=

It should be noted that the quantity Δ₀, which affects the energy spectrum in n important way, is directly related to the non-diagonal matrix element. Hence, the mixing of the states

implied by b ≠0 may be detected in the energy spectrum! This may be seen e.g. by looking at the energy difference Δ= E₁−E₀ (energy gap) between the two states, which is

(

ε₁ −ε₀

)

² +Δ²₀

= Δ

Hence, Δ₀ is given by the minimal energy gap between the two eigen states.

As regards the eigen functions, and obvious, general Ansatz would be 0 =a0 +b1 and 1

a 0 b

1 = − . Since a and b are complex, this would suggest 4 free parameters, but normalization requires a² + b² = 1, leaving 3 free parameters. Furthermore, the global phase of the wave function is irrelevant, such that finally any possible wave function in our two dimensional Hilbert space can be described uniquely and unambiguously by only 2 parameters. A "clever" Ansatz for the eigenfunctions could be

(

^{0 e 1}

)

e

0 = ^iΦ α −β ^−iϕ

(

^{e 0 1}

)

e

1 = ^iΦ β ^iϕ +α Comments

1) here α, β are chosen to be real withα,β≥0 and α² +β² =1 2) the global phase Φ is irrelevant for matrix elements of operators

3) only 2 free parameters are left, which may also be chosen with α=cos(θ) as the two angles θ, ϕ. Hence, there is a one-to-one correspondence of any state in the Hilbert space with a point on a sphere. This sphere is called the Bloch sphere.

Two Representations of the Energy Level Diagrams

-2 -1 0 1 2

-2 -1 0 1 2

Δ₀ ε₀

ε₁

E₀ E₁

energy

B

|1〉

|0〉

ε1

ε0

E₀ E1

Δ0

|1〉

|0〉 ε

ε 1 0

E₀ E₁

|1〉

|0〉

ε₁ ε0

E₀ E₁

- 5 - Vorl. #19 (2. Feb. 2011) Comments

1) the non-diagonal element b describes an interaction between the two state, i.e., a physical process which allows the system to get from 0 to 1 , and vice versa

2) the magnitude of b is directly related to the energy gap at the level crossing, Δ₀ =2b 2) E₀ ≤ε₀,ε₁, that is, the ground-state energy is always lowered by a b >0

3) b >0 implies a mixing i.e., superposition of the states 4) most importantly, 0 , 1 are not eigenstates any more IX.3.b. Time Dependence I: Quantum Oscillations

In this chapter we will assume that the two-level Hamiltonian is not explicitly time dependent.

The time dependence is generally described by the time-dependent Schrödinger equation t

, Hˆ t t ,

i ψ = ψ

∂ h ∂

For a time-independent Hamiltonian it can be formally solved by

⇒ ,t e ,0 e ^{E t}c_n n

i

n t

Hˆ i

h n

h −

− ^{ψ =}

∑

=

ψ ,

where on the right-hand-side the wave function is written in energy representation.

In the following we will address this question: If the system had been in the state ψ,0 at time t = 0, what is the probability for finding (measuring) the system to be in the state

0

ψ, at a later time t?

The answer is obviously 0 2

, t , ) t (

P = ψ ψ

Molecular Nanomagnets ©Oliver Waldmann

Hence, if the system is an energy eigenstate, the system continues to stay in the very same eigenstate for indefinite times. This is physically expected and is just another, equivalent way to express that the energy is a constant of motion.

Comments

1) as a conclusion, in order to obtain non-trivial time evolutions, the system has to be prepared in a non-eigenstate

2) the considered situation is of course that of a free quantum system, without any interactions to any sort of environment

System at Time t = 0 is NOT in an Eigenstate

Now we consider the situation that the system is at time t = 0 not in an energy eigenstate. For our two-level system it shall be written as

1 0 0

, =α +β

ψ .

The time evolution of the state and the probability to find it again in the initial state later on is then obtained as

⇒ ⎟⎟⎠

Thus, if at t = 0 the system is not in an energy eigenstate, it will oscillate with a certain amplitude between the two energy eigenstates. This phenomenon is called (coherent)

- 2 - Vorl. #20 (4. Feb. 2011) quantum oscillation. The oscillation amplitude is maximal for the totally symmetric or anti-symmetric state ^{ψ,0 = 1}₂

(

^{0 ± 1}

)

^.

Comment

The total energy E of the system hence cannot be any of the energy eigenvalues, in particular it must be larger than the ground state energy E₀: E= Hˆ >E₀

Decoherence

The interaction or coupling of the quantum system to any sorts of environment will in general

"destroy" quantum coherence, disrupting the above oscillation. The environment could be something like a measurement, as a result of which the wave function will collapse into one of the eigenfunctions of the measured observable. Often the environment is a thermal bath. In this case the quantum oscillation will be damped with a typical time scale, the decoherence time. The resulting time evolution bears some analogy to that of a harmonic oscillator.

Comments

1) if the coupling to the environment is very strong, no quantum oscillations will occur, and the system or process is said to be incoherent (over-damped oscillator)

2) with increasing environmental coupling not only the oscillation amplitude gets more and more damped, but also the tunneling rate or tunneling splitting Δ as it appears in the cos term gets smaller, one may say that the tunneling gets thwarted by the interaction to the environment

3) the process of decoherence cannot be described in a Hilbert space, i.e., pure state picture, it may be treated with the density matrix formalism.

IX.3.c. Time Dependence II: Landau-Zener-Stückelberg Transitions

In contrast to the previous section, in which the Hamiltonian was assumed to be time-independent, we allow now for an explicit time-dependence. A closed-form solution is then in general not possible. A solution is however known for a specific situation, which, fortunately, is encountered often in experiment.

Description of the Problem

A system shall be described by a two-level Hamiltonian, in which the diagonal elements vary linearly with time, i.e., where ε_0/1 =m₂¹gμ_BB(t)=m₂¹vt with a constant sweep rate v. At time t → -∞ corresponding to B → -∞ the system should be in the state 0 , which is then also eigenstate, 0 = 0 . The level

crossing is now crossed by sweeping the field B → ∞ with constant velocity.

-2 -1 0 1 2

-2 -1 0 1 2

Question

What is the probability P∞ for finding the system in the same state 0 = 0 at time t → ∞?

Δ₀ ε₀

ε₁

E₀ E₁

energy

B

P_∞

Solution

Although the derivation is quite intricate, the solution is remarkably simple:

) v 2 / exp(

P_∞ = −πΔ²₀ h Discussion

v → ∞: ⇒ P∞ → 1

if one passes the level crossing very quickly, the system doesn't find the time to tunnel to the other state, and hence remains in the starting state.

v → 0: ⇒ P_∞ → 0

if one travels across the level crossing very slowly, the system has all the time it needs to tunnel to the other state, and hence does so.

Whether the sweep rate is fast or slow depends on the tunneling rate Δ₀. In fact, the problem can be characterized by two time scales:

= tunneling time, i.e. time it takes to tunnel / 0

2πh Δ

= time for which the system stays in the level-crossing regime v

0/ Δ

The Landau-Zener-Stückelberg (LZS) transition rate or tunneling probability 1 - P_∞ is determined by the ratio of these time scales, i.e.,

0 0

/ v / Δ Δ h .

Example: resonant quantum tunneling and magnetization in Mn₁₂ und Fe₈

Mn12: S = 10

-1.0 -0.5 0.0 0.5 1.0

-100 -80 -60 -40 -20 0 20 40

T < 300 mK: tunneling regime Fe8

E/|D|

B (T)

During a sweep of the magnetic field from, let's say negative to positive, the system at each level crossing undergoes a LZS transition with a probability of 1 - P (where P is of course related to the particular tunneling splitting at the particular level crossing), i.e., tunnels to the other state or other side of the energy barrier. The example of Fe8 shown to the right sows explicitly that with increasing sweep rate of the field the probability to tunnel, i.e., to change the state gets smaller.

-10 -8 -6 -4 -2 0 2 4 6 8 10 E

M

-10 -8 -6 -4 -2 0 2 4 6 8 10 E

M -12-10 -8 -6 -4 -2 0 2 4 6 8 10 12

E

M

- 4 - Vorl. #20 (4. Feb. 2011) Measuring the Tunnel Splitting

The LZS relation for the tunneling probability 1-P allows a means to determine experimentally from the M(B) curves the extremely small tunneling splitting at the level crossings in single-molecule magnets. The sweep rate is obviously known. However, also the probability 1-P can be directly inferred from the height of the magnetization step at the level crossing. Hence knowing v and 1-P, the tunneling splitting Δ0 can be determined.

Although it varies by orders of magnitudes for different molecules, the tunneling splitting is generally very small, a value of Δ ≈ 10^-6 K is typical.

Im Dokument Molekulare Nanomagnete: Quantenphysik zum Anfassen Molecular Nanomagnets (Seite 94-100)