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Okay. So, finally now that we have, have
this analysis of a particle in a box which

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is a toy model or you know, simplified
model for an atom for, for the states of

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an electron in an atom, now let's see how
we can use this to implement qubits. Okay,

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so remember our model had the particle in
the box and, and we saw the solution

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equation for, for a free particle. We just
confine with, with these boundary

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conditions that psi of zero equal to psi
of i equal to zero, and then we got this

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quantized solution. So first of all, we
saw, saw where does this quantization of

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energy comes from. It comes from the fact
that the, that the, that the, that the

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stationary states of the, of the electron,
the, the states of constant energy are

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these the Eigenstates of H. And, they are,
they, they are, they are, standing waves

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in this, in this, in this interval. And
they happen to have these, these quantized

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energies, which, which vary as n² So they
are, they are numbered, n equal to one,

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you know. The, the states are numbered, n
equal to one, two, three, and so on. The

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energy of the n state is proportional to
n² and the, the corresponding, the, the n

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states has, has eight nodes in it. It has,
it has these okay. So now how do we use

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this to implement a qubit? Well, what you
could do is make sure that our electron

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has energy at most e sub three, it has
energy strictly less than e sub three.

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00:02:03,002 --> 00:02:09,085
Right? And so now, if it has energy
strictly less than e sub three it's, it's,

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00:02:10,009 --> 00:02:16,069
it's confined to be in one of these energy
levels, one or two. Okay? So we can, we

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can confine it to, to have lower energy
and we can call this n equal to one state

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the zero state and the n equal to two
state the one state. Right? And so in

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general, our electron will be in
superposition of zero and one, of, and,

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and so we'd have implemented a qubit. So,
let's look at this more, more, more

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00:02:45,093 --> 00:02:52,094
carefully. So, so we've, we've made our
total energy much smaller than E3, so that

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00:02:52,094 --> 00:02:59,039
we want it to be in superposi tion of just
these two Eigenstates of H in which we

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00:02:59,039 --> 00:03:04,093
call the zero and one state. And then, in
general of course, we, we are in a

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00:03:04,093 --> 00:03:10,026
superposition of the two, and so we'd be
in some superposition alpha times the zero

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00:03:10,026 --> 00:03:16,070
state plus beta times the one state. The
zero state is psi one which is, which is,

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which is of course, it's a continuous
state, a quantum state, right? And it's

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00:03:23,039 --> 00:03:29,042
given by, by this particular
superposition. It's, it's, it's wave

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00:03:29,042 --> 00:03:35,072
function is this normalization times sine
pi x over l. And similarly the, the, the,

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00:03:35,072 --> 00:03:43,004
the, the one qubit state is, is, is given
by this normalization times sine two pi x

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00:03:43,004 --> 00:03:51,058
over l. And, and now as we let this, this,
this state evolved in time, of course,

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00:03:51,058 --> 00:03:57,059
the, this, this first part of the
superposition is pretty precess at the

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rate proportional to e1 and the second one
will precess four times as fast because

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its at a rate proportional to e2. And, if
we factor out this precession due to e1 so

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00:04:13,009 --> 00:04:20,027
we could look at the qubit in a rotating
you know, frame, then still, you have a

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00:04:20,027 --> 00:04:32,016
relative rotation which is proportionate
to e2 minus e1. Right? So if you call this

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00:04:32,016 --> 00:04:40,070
delta e Which is e2 minus e1, for the
hydrogen atom, this is about ten electron

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00:04:40,070 --> 00:04:49,093
volts and if you look at the corresponding
frequency which is this divided by h, its

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00:04:49,093 --> 00:04:58,052
about, the frequency is about 2.5 times
ten to the fifteen Hertz. Okay, so this,

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00:04:58,052 --> 00:05:06,080
this frequency turns out to be very close
to the frequency of optical light. So this

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00:05:06,080 --> 00:05:15,020
is why you can actually control these
atomic qubits by, optically by you know,

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00:05:15,020 --> 00:05:23,008
through, through interaction with light
pulses. Okay so, so this gives us our

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00:05:23,008 --> 00:05:31,004
implementation of a qubit atomically as
well as suggest how one actually

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00:05:31,004 --> 00:05:39,006
implements gates and those are done by,
by, by very precise light pulses directed

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00:05:39,006 --> 00:05:43,003
at this qubit.