285. Dephasing and diffusion of quantum particles

Posted by Ariel Amir

February 25, 2009

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This is a guest post by Ariel Amir from the Weizmann Institute of Science. Dmitry.

I was kindly asked by Dmitry to describe a recent work, where diffusion emerges from the decoherence in a noisy environment. This work was done together with Y. Lahini (an experimentalist) and H. Perets (an astrophysicist), and can be found on condmat (arXiv:0902.0890). Let me start by describing what a tight-binding model (for those of you outside Condensed Matter). This will be a convenient tool for us to understand decoherence, and other interesting physics such as ‘motional narrowing’.

Let’s assume we have a bunch of N atoms, which are arranged with equal spacing in one dimension, where neighbouring atoms are far away from each other. Let’s also simplify life, and assume we have one electron in our system, which can sit on the lowest unoccupied level of each of the atoms. The electron can tunnel from one atom to its nearest neighbour (where we neglect next-nearest neighbour tunneling since it is exponential in the distance). The Hamiltonian of the system is an NXN matrix, which For N=4, for example, would read:

$H=\sum E_0c_i^{\dag}c_i+T\sum_{\langle i,j\rangle}c_i^{\dag}c_j$ (1)

where $\langle i,j\rangle$ are nearest-neighbours in the lattice, is a constant related to the properties of the tunnelling and E_0 is some arbitrary energy, which we can take to be 0 for convenience.

If we put the electron initially at some atom, we have to solve the Schrodinger equation:

$i\frac{dA_j}{dt}=T(A_{j+1}+A_{j-1})$ . (2)

When this is done for the above case, we get ballistic motion, meaning the variance of the electron wavefunction grows as the square of time (i.e., the typical size of the wavefunction grows linearly with time). This is demonstrated in the following figure, showing the evolution of the wavefunction with time:

The spread of a quantum-mechanical wavepacket in a perfectly ordered lattice showing ballistic spreading.

Figure 1: The spread of a quantum-mechanical wavepacket in a perfectly ordered lattice showing ballistic spreading.

In fact, the same mathematical description is relevant for many other physical phenomena, not just electrons on atoms: this can also happen with light, for example, which makes these tight-binding models a generic, very useful, tool.

It turns out that when the energies of all atoms (the diagonal terms of the matrix) are random (i.e., the matrix is the same but now the diagonal will have fluctuations), the behavior can be completely different, if the magnitude of the ‘disorder’ is large enough (compared to what? This depends on strongly on dimensionality. For a 1D infinite system, any tiny amount of disorder would be sufficient). P.W. Anderson showed that this can give rise to localization of the wavefunction: so the electron, instead of moving ballistically, would ‘get stuck’ after a short time, and remained confined in a small regime. This was exciting enough to get Anderson a Nobel prize, and to create a huge interest in the phenomenon that continues to this day.

The question we asked ourselves, is the following: What would happen if instead of being constant, the energy of each atom would fluctuate in time? This actually happens in many physical systems: imagine an electron in a lattice of molecules (this is called a molecular solid, and is all around us: take ice, as a concrete example). The molecules would be vibrating, since they are at finite temperature. This would effectively mean a time-dependent disorder in the above matrix.

In this case the Schrodinger equation would read:

$i \frac{dA_j} {dt}=T(A_{j+1} + A_{j-1})+\xi{}(j,t)A_j$ , (3)

where $\xi$ can now be modelled by a noisy term: it can be characterized by a typical magnitude W and correlation time $\tau$ .

Before we continue, let’s simplify things a bit. What would happen if T=0 ? (the atoms are infinitely far apart, so there is no tunneling). In this case, the electron would stay in a given place. Something still changes in time though: the phase of the electronic wavefunction. The Schrodinger equation for the amplitude is given by:

$i\frac{dA_j}{dt}=\xi_(j,t)A_j$ (4)

so if we write $A=|A|e^{i \phi}$ , we get: $\frac{d\phi}{dt}=\xi$ . This equation tells us that a noisy environment will cause the phase to be a random variable, too. This is dephasing. The typical time it takes the phase to ‘lose its memory’ is the dephasing time, which is of great interest in realistic systems in the last years, since it is one of the factors limiting the applicability of quantum information and quantum computers.

We can relate the correlation time of the phase to the parameters of the noise, and obtain, in the case $W\tau\ll\hbar$ , the relation:

$\tau_\phi=\frac{\hbar^2}{W^2 \tau}$ . (5)

This means the shorter the correlation time of the noisy environment, the longer the dephasing time. A similar physical phenomenon occurs in Nuclear Magnetic Resonance, where a spin loses its phase due to its interactions with the many fluctuating spins around it. This is called motional (or dynamical) narrowing. In our paper, we give an exact formula for the correlation-time of the phase, for any form of a noisy environment.

To summarize so far, I have introduced the so-called tight-binding model, and showed that without the tunnelling term, but with a noisy environment, we get dephasing. What happens when a small tunnelling term is present?

In our work, we showed that there are two processes in the problem, which occur on different time scales: at short time scales, we have dephasing, which causes the phases at each site (atom) to be a random variable. Without the tunneling, this would be the end of the story, since the probabilities to be at each site cannot change in time. However, we showed that the interplay of the tunnelling and the dephasing, causes diffusion of the probabilities: it is, in a sense, a compromise between the ballistic motion (when there is no noise in the environment) and Anderson localization (when the noise does not change in time).

This is a plot of the typical form of the wavefunction, obtained by numerical solution of the above equation:

The spread of a quantum-mechanical wavepacket in a noisy environment.

Figure 2: The spread of a quantum-mechanical wavepacket in a noisy environment.

As expected from our results for diffusion, this is a Gaussian profile. If you want to convince yourself that the motion is really diffusive, take a look at the following figure, demonstrating the growth of the extent of the wavefunction (the standard deviation) as a function of time: as expected, one gets a growth proportional to square-root of time.

The typcial size of a spreading wavepacket in a noisy environment, as a function of time.

Figure 3: The typicial size of a spreading wavepacket in a noisy environment, as a function of time.

We are able to calculate the diffusion constant D, for any form of the noise, and it turns out that the ‘motional narrowing’ discussed previously, finds its way into the diffusion constant: D is proportional to the dephasing time. This is the main result of our work. The following plot shows the correspondence between our theory and our numerical simulations results:

Theory Vs. Numerical simulations for the diffusion coefficient

Figure 4: Theory vs. numerical simulations for the diffusion coefficient.

For short correlation times of the noisy environment, the diffusion constant is proportional to $\tau_\phi$ – the motional narrowing regime. However, for long correlation times, the diffusion is proportional to $W^{-1}$ : remarkably, in this regime there is no dependence on the correlation time (or on the details of the noisy environment).

For more details, and various other references related to this problem, check out arXiv:0902.0890.

Condensed Matter, Journal club, Master Mason, Quantum computation

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4 Comments »

Comment by Dmitry

2009-02-25 16:29:16

Hi Ariel,

thanks for explaining dephasing! (or better say – decoherence?) Are $\tau$ ’s in the formulae (5) and (6) different? I understood that $\tau$ in the r.h.s. is correlation time for the noise, while $\tau$ in the l.h.s. is correlation time for the quantum phase, is it correct?

Cheers,
Dmitry.

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Comment by Ariel Amir

2009-02-26 11:46:41

Yes, sorry for this typo.
What I meant in eq (5) is $\tau_\phi$ on the LHS and $\tau$ on the RHS (as you point out to). Eq (6) implies D proportional to $\tau_phi$ (maybe better to omit the equation in this case). Maybe you should change this in the post, to avoid further confusion?

Thanks,

Ariel

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Comment by Dmitry

2009-02-26 11:51:49

Dear Ariel,

done! I have a question regarding Laplace operator in (2) and (3) – if this is latticized second order derivative, where is -2A_i term?

Cheers,
Dmitry.

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Comment by Ariel Amir

2009-02-26 13:22:41

Thanks a lot for correcting the blog. Regarding the -2A_j: if you look at the time derivative of A_{j+1}, for example, it would contain a term T A_j (since the Hamiltonian is hermitian). It is easy to check that the sum of probabilities \sum_j |A_j|^2 is conserved, a direct consequence of the hermiticity of the Hamiltonian. So you don’t need this term. Notice, though, that in the emerging diffusion master equation the term you mentioned exists, to account for the same conservation law in the ‘classical’ case, of diffusion.

Ariel.

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285. Dephasing and diffusion of quantum particles

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