Okay. So, finally now that we have, have this analysis of a particle in a box which
is a toy model or you know, simplified model for an atom for, for the states of
an electron in an atom, now let's see how we can use this to implement qubits. Okay,
so remember our model had the particle in the box and, and we saw the solution
equation for, for a free particle. We just confine with, with these boundary
conditions that psi of zero equal to psi of i equal to zero, and then we got this
quantized solution. So first of all, we saw, saw where does this quantization of
energy comes from. It comes from the fact that the, that the, that the, that the
stationary states of the, of the electron, the, the states of constant energy are
these the Eigenstates of H. And, they are, they, they are, they are, standing waves
in this, in this, in this interval. And they happen to have these, these quantized
energies, which, which vary as n² So they are, they are numbered, n equal to one,
you know. The, the states are numbered, n equal to one, two, three, and so on. The
energy of the n state is proportional to n² and the, the corresponding, the, the n
states has, has eight nodes in it. It has, it has these okay. So now how do we use
this to implement a qubit? Well, what you could do is make sure that our electron
has energy at most e sub three, it has energy strictly less than e sub three.
Right? And so now, if it has energy strictly less than e sub three it's, it's,
it's confined to be in one of these energy levels, one or two. Okay? So we can, we
can confine it to, to have lower energy and we can call this n equal to one state
the zero state and the n equal to two state the one state. Right? And so in
general, our electron will be in superposition of zero and one, of, and,
and so we'd have implemented a qubit. So, let's look at this more, more, more
carefully. So, so we've, we've made our total energy much smaller than E3, so that
we want it to be in superposi tion of just these two Eigenstates of H in which we
call the zero and one state. And then, in general of course, we, we are in a
superposition of the two, and so we'd be in some superposition alpha times the zero
state plus beta times the one state. The zero state is psi one which is, which is,
which is of course, it's a continuous state, a quantum state, right? And it's
given by, by this particular superposition. It's, it's, it's wave
function is this normalization times sine pi x over l. And similarly the, the, the,
the, the one qubit state is, is, is given by this normalization times sine two pi x
over l. And, and now as we let this, this, this state evolved in time, of course,
the, this, this first part of the superposition is pretty precess at the
rate proportional to e1 and the second one will precess four times as fast because
its at a rate proportional to e2. And, if we factor out this precession due to e1 so
we could look at the qubit in a rotating you know, frame, then still, you have a
relative rotation which is proportionate to e2 minus e1. Right? So if you call this
delta e Which is e2 minus e1, for the hydrogen atom, this is about ten electron
volts and if you look at the corresponding frequency which is this divided by h, its
about, the frequency is about 2.5 times ten to the fifteen Hertz. Okay, so this,
this frequency turns out to be very close to the frequency of optical light. So this
is why you can actually control these atomic qubits by, optically by you know,
through, through interaction with light pulses. Okay so, so this gives us our
implementation of a qubit atomically as well as suggest how one actually
implements gates and those are done by, by, by very precise light pulses directed
at this qubit.