What Are Quantum Beats in Atomic Physics?

[jonas_nilsson]

Hi!

I'm doing a basic course in atomic physics, and right now I'm looking at quantum beats. I've found several sources for basic descriptions, but there are some things that confuse me. I'm going to present some of the material in short here, and hope for comments that might make things more clear!

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In Haken&Wolf's "the physics of atoms and quanta" p.387 , the description basically goes as follows:

An electron is excited to a state that is a superposition of two stationary states:
$$\psi(r,0) = \alpha_1 \phi (r) + \alpha _2 \phi _2 (r)$$

in time the electron will make a decay transition to the ground state (wavefunction $$\phi _0$$). The occupation probability of the ground state decreases exponentially with the decay constant $$2\Gamma$$.

Here's what I oppose:
"The total wavefunction therefore takes the form
$$\psi(r,0) = \alpha_1 exp(-i E_1 t /\hbar - \Gamma t) \phi_1 (r) + \alpha _2 exp(-i E_2 t/ \hbar - \Gamma t) \phi _2 (r) + \alpha _0(t) \phi _0 (r)$$"

This must be a statistical mixed state appropriate for a big bunch of atoms, or? I mean after a decay, the wavefunction should collapse to the ground state wavefunction.

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I slightly different approach, that makes more sense to me, is given in Wolfgang Demtröder's "Laser spectroscopy" p. 661ff:

As above, a superposition of states is assumed at t=0, here just a bit more general with an unspecified number of states. At time t the wavefunction has evolved into

$$\psi (t) = \sum c_k \psi _k (0) exp((-i \omega _{km} + \gamma _k/2)t)$$
with
$$\omega _{km} = (E_k - E_m)/\hbar$$
Here the psi_k are stationary states (the excited ones) and m denotes a lower state to which we observe decay.

Looking at the simple special case used in Haken&Wolf with just a superposition of two states (1,2) and decay to 0, we get:
$$\psi (t) = \left[ c_1 \psi_1(0) exp(-i E_1t/\hbar) + c_2 \psi_2(0) exp(-i E_2t/\hbar) \right] exp(i E_0t/\hbar - \gamma/2)$$

Here's what I don't get: how does the last exponent with E_0 and gamma end up there? How do you motivate that? And how is the norm preserved?

Then, Demtröder looks at the matrix elements of the dipole operator to get the transition probabilities to the ground state. The presented end-results (the emission intensity beating) in the two books are similiar.

 

[eljose79]

Hi!

First of all, it's great to hear that you're studying atomic physics and exploring quantum beats. It's a fascinating topic that has many practical applications in spectroscopy and quantum information processing. I'll do my best to address your questions and provide some clarity on the material you've presented.

In both Haken&Wolf and Demtröder's explanations, they are describing the time evolution of a quantum system that is initially in a superposition of states. This means that the system is in a state that is a combination of two or more stationary states, each with their own energy level and wavefunction. In this case, the electron is initially in a superposition of two excited states, \psi(r,0) = \alpha_1 \phi (r) + \alpha _2 \phi _2 (r).

As time passes, the electron will make a transition to the ground state, with a decay constant of 2\Gamma. This means that the probability of the electron being in the ground state will decrease exponentially over time.

In Haken&Wolf's explanation, they are using a statistical mixed state approach to describe the system. This means that they are considering a large number of atoms, each with their own superposition of states. As the atoms decay, the wavefunction of the system as a whole will collapse to the ground state wavefunction.

In Demtröder's explanation, they are using a more general approach and considering an unspecified number of states in the superposition. They also include the decay rate, \gamma, in their time evolution equation. This is because the decay rate is related to the energy difference between the two states, as seen in the term \omega _{km} = (E_k - E_m)/\hbar. The last exponent in the equation, exp(i E_0t/\hbar - \gamma/2), is a phase factor that takes into account the energy difference between the excited states and the ground state.

To address your question about the norm being preserved, this is due to the conservation of energy in the system. The total energy of the system remains constant, even as the electron decays to the ground state. This is reflected in the time evolution equation, where the energy terms are all conserved.

Overall, both explanations are valid and provide different perspectives on the phenomenon of quantum beats. I hope this helps to clarify some of the confusion and encourages you to continue exploring this fascinating

 

[charng]

Hi there! Quantum beats are a phenomenon observed in atomic and molecular systems where the energy levels are not strictly discrete, but instead have some degree of overlap. This results in a situation where an excited state can decay to multiple lower energy states, leading to interference patterns in the emission intensity.

In the first description you provided, the wavefunction is described as a superposition of two stationary states, and the occupation probability of the ground state decreases exponentially with time. This is a simplified model that assumes a single atom in isolation, and does not take into account the effects of interactions with other atoms or the environment. In this case, the wavefunction would indeed collapse to the ground state after a decay event.

The second approach, as described by Demtröder, takes into account the effects of interactions and uses a more general superposition of states. In this case, the wavefunction does not collapse to a single state after a decay event, but rather evolves according to the time-dependent Schrödinger equation. The last exponent with E_0 and gamma arises from this time evolution, and the norm is preserved through the use of the dipole operator and transition probabilities.

Overall, both approaches are valid and provide different levels of understanding of quantum beats. The first one is simpler and more intuitive, while the second one is more comprehensive and takes into account the effects of interactions. I hope this helps clarify some of your confusion. Let me know if you have any further questions!