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welcome back everybody to our course introduction to quantum optics in this lecture today we want to start discussing the different field states of the electromagnetic field that can occur when we just look at a single mode of the radiation field and in the last class we derived the operators for the electric field for the magnetic field for the vector potential and today we want to see how different field states actually give rise to different properties of the light field in this single mode so let's get started so remember in the last class we had introduced the electric field operator as a sum over positive frequencies and negative frequencies and in this lecture today we just want to consider a single node of the radiation field so this simplifies things dramatically before we had to sum over all the modes characterized by their different wave vectors K but today we're just gonna pick a single node and look at field States of that single mode so let's pick a certain mode labeled by some wave vector okay but in the subsequent course here we're not gonna write out this K anymore explicitly so when we you think of that really think of just one mode where we singled out on K vector with one polarization but now we're just gonna stick with that and we're just gonna drop that index from now so this means we just have a single mode so we don't have to sum over all the modes and the single mode is characterized by this electric field operator that's by this operator for the electric field of our light field so we see we have the destruction operator the creation operator for creating and destroying photons and we have here this prefactor now in order to save us some work in writing down this we're going to kind of simplify this field operator by working in natural units in units where we don't have to write out explicitly this pre-factor anymore so all the electric fields that we're going to write down in the subsequent lectures unless stated otherwise are going to be considered in this kind of natural unit square root of H bar Omega divided by two epsilon zero V where V is the volume of our box for which we quantize the modes and we're actually going to see that this kind of feel strength is a very characteristic feel strength for the vacuum field even when you have no photons present in the system so very often therefore this is also called kind of the vacuum field strength okay so we measure now all of the electric field in units of this vacuum field strength and then we can just write the electric field operator in the following way we don't have to write down this pre factor anymore it's just a times e to the minus I Chi plus a dagger he to the I Chi where Chi was this phase factor Omega t minus kr minus PI over two so now let's look at distinctive states of the electromagnetic field that we can have and the first states we're actually going to start with in this lecture the so called states the eigenstates of our harmonic oscillator where we have a defined photon number n photons in the mode of the electromagnetic field and we can also say that we have the nth degree of excitation of the harmonic oscillator attached to that mode those are kind of the two equivalent viewpoints we can take so we have n photons in the radiation mode and this is an eigenstate of the number operator so now let's for example calculate what the action of the electromagnetic field Hamiltonian onto such a fox State is you remember what the outcome of this would be did you get it right well the electromagnetic field Hamiltonian for a single mode radiation field that's just H bar Omega times a dagger A plus 1/2 now applied to state n and that's just h-bar Omega n plus 1/2 applied to state n and since the N is an eigenstate of this number operator n with eigen value small n then that's just H bar Omega n plus 1/2 times n with the eigen energy of this state given by h-bar Omega n plus 1/2 okay that was simple enough let's look at a few other things that's for example calculate calculate the fluctuations in the photon number for such a Fock state well first of all we have to remember how to calculate fluctuations if you want to calculate the variance of an observable in quantum mechanics over a given step size how would you do that well one way to do it is to just calculate psi of let's say an observable a minus the average value of a squared this gives us the variance of this operator a sorry I should write operator here anymore the variance of a of this kind of observable a so now let's apply this to our number operator so we have the same thing here the number operator is n so we now want to calculate the fluctuation so that would be just the expectation value over the state we want to calculate the fluctuations in this is the Fock state for the state we're looking at here the operator is n the average value of n is given here the expectation value of n and now we have to calculate this well let's just do that let's just multiply this out this square that's just n square minus 2 and hat plus N squared expectation value okay so now these are just scalars we can just pull them in front of the expectation value so we have basically in expectation value of the operator N squared minus two times expectation value of n hat squared plus expectation value of n hat squared and this just gives us N and hat squared minus and hat squared so we see that the fluctuations are given by the expectation value of the operator squared minus the expectation value of the operator squared so what would that be in case of the number states well this is just kind of gives us N squared so let's do this now explicitly this gives us n n squared and minus this also gives us n so this gives us here another N squared and since we can pull this just in front of the expectation realize just a scalar now we just see that this is just the same thing so it's just N squared minus N squared and therefore 0 and that's of course completely natural the expectation value of the fluctuations of a state of the number of photons that we have that Fox ID is just 0 because we have a precisely defined number of photons in that mode so the fluctuations vanish there are 0 fluctuations in the photon number for this Fox statement ok that was simple enough let's calculate something else let's calculate for example the expectation value of the electric field so at a certain phase and/or Chi so the electric field operator for the single mode radiation field that's just 1/2 AE to the minus I Chi plus a dagger e to the I Chi we calculate that over the expectation value over our Fox State n so we basically get 1/2 and he had U to the minus I Chi and plus 1/2 and dagger e to the I Chi and and this again just scalars we can just pull them out and then we have the destruction operator on n gives us squared n times n minus 1 and the creation operator gives on n gives us growth n plus 1 times n plus 1 but n plus 1 is orthogonal to n and n minus 1 is also a thornell to n so both of these terms actually vanish so this one 0 and this one is also 0 so we have the strange result that the expectation value of the electromagnetic field is actually 0 so there seems to be no electromagnetic field present in the system and now this is a bit weird because see the state we're talking about could contain a lot of photons could contain let's say 10 to the 21 photons could be a huge amount of photons in that mode but still the expectation value of the electromagnetic field of the electric field would be 0 so so why is that how can that be well before we interpret that let's calculate something else let's calculate the fluctuations of the electric field ok in order to calculate the fluctuations remember as we just derived in the last slide this is just expectation value of the operator squared so the electric field operator squared minus the expectation value of the electric field operator squared ok so this one we know is just zero this is what we just calculated but the first term we don't know what it gives us so we just basically have to put in our electric field operator and do the calculation this gives us 1/4 and a hat e to the minus I Chi plus a dagger e to the I Chi n squared and now we just have to square that term so we'll see what we get 1/2 1/4 here and a hat squared e to the minus 2 I Chi for the other term we get a dagger squared e to the 2i Chi and then we get the cross terms where the phase factors cancel plus a hat a dagger plus a dagger a applied to him now these first terms applied to n will create two photons this term will create two photons and therefore the state will be orthogonal to end so it's going to give us zero this term is going to destroy two photons so this gives us n minus two here and it's also going to be orthogonal to n so that also is going to kind of be zero so we only have to think of these last two terms here a a dagger and a dagger a now what's a a dagger remember I could just rewrite that as a dagger A plus one due to the commutation relations between a a dagger and being one and so this now is the number operator and this is the number operator n so basically I just get 1/4 in to n plus 1 applied to n and therefore I get just 1/4 and 2 n plus 1 and so and over to 1/2 and plus 1/2 the fluctuations of this state seem to increase with the number of photons so it's a kind of very strange result we had the expectation value of the field operator being zero but the fluctuations the variants that we encounter here seems to be growing linearly in the photon number so the fluctuations are becoming larger and larger but still the expectation values zero how can we reconcile those two views well one way to view that if we want to think of the field state in terms of classical oscillating electromagnetic fields electromagnetic waves sinusoidal oscillating electromagnetic waves with the frequency Omega then we can think of it in the following way we can actually think of this Fock state as a superposition of all kind of different sinusoidal phase with kind of random phases equally distributed between 0 and 2pi so here I've just picked twenty kind of random sinusoids with random phases between zero and 2pi and when you add them all up when you calculate the expectation value of the electromagnetic field you see that for almost every positive value you find an equal negative value such that the expectation value will average out to zero however the fluctuations as you can see here those are going to still be very very large in the system so now let's add a few more waves to this now I've done the same thing I've just picked hundred sinusoids with random phases and you can indeed see even better what I said before that for every kind of positive value here of the electric field we also encounter a negative value of the electric field for kind of a sinusoid with a different phase such that the expectation value of the electric field will be zero but still a kind of the fluctuations are going to be very very large here so these fluctuations here this Delta e that we see here that's just going to be square root of 1/2 n plus 1/2 so this is going to grow we're like the square of the photon number kind of for large photon numbers and it's going to become larger and larger now you actually also see something interesting that even for N equals zero which we call the vacuum state where there are no photons in the system even for this vacuum State we have fluctuations of the order of one half of this natural kind of unit of the electric field that we introduced in the beginning of this lecture so that's what we call this kind of feel strength of vacuum field fluctuation strengths because the characterize is naturally the strength the standard deviation of the vacuum field fluctuations so this is a very important result it tells us that even the vacuum field even when there are no photons in the Preston no photons in the system the system exhibits fluctuations the electric field fluctuates so if you go into a system with no photons and you measure the electric field at some point in time you'll find a non-zero value only when you average over many of those measurement results you'll get the expectation value of zero electromagnetic field but for kind of a single measurement of the electric field and we'll look how we can kind of do that in the subsequent lectures you will actually find a finite result even for a system the vacuum itself when there are no photons present in the system and these vacuum field fluctuations they're actually very very important they're going to be able to we're going to see that they actually can introduce the dynamics into the atom even when there are no photons in the system they actually the cause of spontaneous emission in which we can in fact view as being triggered by these vacuum fluctuations all right this is all I wanted to tell you today in the lecture about Fox States they actually very strange States they're actually very non classical state they don't give us the classical oscillating electromagnetic field that we used to in Maxwell's equations but rather give us a strange result of a vanishing electric field however with large large fluctuations so clearly these Fock States are not the classical States not the classical fuel States and in the next class we want to actually look what kind of States give us the best approximation to classical fuel states that we know from actual solutions thanks a lot for watching today and see you in the next class