In lesson one, we have seen the notion of number states, which are the eigenstates of the Hamiltonian describing radiation in a single mode. We will apply this notion to the case when the number of photons is one. I've repeated here the expression of the Hamiltonian of radiation in mode l, expressed as a function of the creation and annihilation operators a_L, and a_dagger L, whose commutator has a value one. The number states |n_l>, are eigenstates of the Hamiltonian with the eigenvalues hbar omega_L, times n_L plus one half. They are also eigenstate of the number of the photons observable N_l equals a-dagger l a_l. A one photon state is a number state with n_l equals 1. It means that if we measure the number of photons in such a state, with a perfect detector, we will find with certainty the result one. But what would we obtain if we could measure the average value of the electric field in the one photon state. The answer is zero because a applied to the one photon state yields an orthogonal state so the average of a in the one photon state is null. Similarly, the average of a_dagger in state one is null. A null result would also be obtained as the average value of any quadrature component. Do you remember the observables Q_l and P_l that we introduced in lesson one? They are analogous to the position and momentum of a harmonic oscillator within multiplying constants. The average value is also null in a one photon state because the average of a or a-dagger is null in a number state. Does it mean that the field is null, in a one photon state? As we did in lesson one, about vacuum, we can answer that question by evaluating the average of the square of the electric field. When we square the electric field operator, we get a term a_dagger squared, which gives the state n = 3, orthogonal to a state n =1. So it's contribution is null. Similarly, the term a squared has a null average. We are thus left with the average of a_dagger a +a, a-dagger. Using the commutator a, a_dagger = 1, we obtain 3 times the square of the 1 photon amplitude. We can then think of the electric field as a fluctuating quantity whose fluctuations have magnitude of E1 times the square root of three. A similar calculation with the quadrature components yields delta Q and delta P equals root of three times hbar over two. If we write then the Heisenberg product, delta Q delta P, we find three times the minimum limit. The one photon state is not a minimum dispersion state. This is true for any number state except for vacuum. At this point, I do not want to hide the problem, linked to the fact that E1 seems to depend on the arbitrary volume of quantization. In a real experiment, it is possible, at least in principle, to measure the average of E squared. And this measurement will give a well defined result. So the result of the calculation should not depend on the arbitrary value of the volume of quantization. The solution of this paradox lies in the fact that real one photon states come in wave-packets of finite extension. For instance, when a single atom in an excited state emits one photon, the emitted radiation is not stationary. It has a time dependence. It is null before the emission starts and decays to zero exponentially after the start of the emission. You will learn how to describe such an evolution in a future lesson. It involves several modes of the radiation associated with several different frequencies. I'm sure that you understand what I mean if you think of the fact that the Fourier transform of a time varying function Involves several frequencies. Well, in a sense we have a similar situation here. So in principle, to study one photon state, we should wait until we have studied the formalism of multi-mode radiation states. I want, however, to introduce you the intriguing properties of such states as soon as possible, that is, today. I will do it using a simplified model of a single mode one-photon state. This model is not entirely rigorous, but trust me, it captures the quantum features I want you to understand at this stage. The model is the following. Firstly, we consider a traveling plane wave that has a limited transverse section S determined by the experimental setup. For instance, we can think of collecting almost all the light emitted by a single atom using a parabolic mirror with the atom at its focus. If S is much larger than the square of the wavelengths of the light, diffraction is negligible and it is not a bad approximation to think of a plane wave with a constant transverse section. For instance, yellow-orange visible light has a wavelength of 0.6 micrometers and a beam with a section of one squared centimeter has negligible diffraction over hundreds of meters of propagation. We also assume constant amplitude in the transverse section. This last assumption is not very realistic but it simplifies significantly the calculation without modifying the physics. So we will used the so called top_hat model. Secondly, we think of a traveling plane wave with a limited temporal duration, T, which evokes, for instance, a finite lifetime of the emitting excited level. Those of you familiar with Fourier transform will immediately object that we cannot have a monochromatic wave with a limited duration. This is very true, but if we think of an amplitude rising slowly to its steady-state value and falling slowly to zero after a duration T, then the frequency spread can be negligible compared to the radiation frequency, provided that the rise and fall times are long compared to the period of the oscillating field. This is possible if the duration T is very large compared to the period of oscillation. Let us take again the example of a single atom emitting visible light at the frequency of say 5 times 10 to the 14th hertz. The duration of the emission of the one photon wave-packet is always larger than a few nano seconds, more than one million of oscillation periods. There is plenty of room to switch on and switch off slowly the amplitude and it is not a too bad approximation to think of a monochromatic wave with a constant amplitude for a finite duration, T. So that is to say a wave packet with a length cT. Where c is the velocity of light. Note that if we take, for instance, T equals 10 nanoseconds, light travels three meters only. Much less than the distance after which diffraction becomes significant, hundreds of meters in the example given above. So, our model is consistent. We will thus take a mode with a quantization volume S_l c T_l to which we attribute a physical meaning. The number of photons in that mode is well defined and it can be measured in principle with the detector D wider than S, integrating the signal over a period of time longer than the wave packet duration T. You are now going to learn how to write explicitly photon detection signals in quantum optics, and you will be able to calculate these signals for one photon wave packets.
