Let us start with a short presentation of the Heisenberg formalism of quantum optics of various authors. I have no doubt that many of you were frustrated when seeing these expression of the double photo detection signal at time t. Why not be interested in the probability of a double photo detection at two different times. It turns out that it is not easy to express the rate w2 of double photo detection at two different times with the formalism we have used up to now, the Schrödinger formalism. In that formalism, the operators do not depend on time, and the time dependence is in the state vector. The quantities related to measurements expressed as quantum averages of the form shown here, then involve one value of time only. In contrast, in the Heisenberg formalism, the state vectors do not evolve, and keep their value at the initial time, while the operators do evolve with time. It becomes then possible to have an expression involving two different times of detection as shown here. This is the expression of the rate of joint photodetections at (r1, t1) and (r2, t2). It is a very useful expression, provided that you know how to use the Heisenberg formalism. I'm now going to show you how to pass from the Schrödinger formalism to the Heisenberg formalism, so that you will be able to determine yourself the expression of the electric field operators. You can expect a good surprise, the time evolution you will find is the most natural one can think of. Can you make a guess and write it on your notepad? I give you a hint. The Heisenberg formalism leads to an expression of the electric field operator that resembles closely its expression in the classical formalism. In the Schrödinger representation, state vectors evolve according to the Schrödinger equation, which is a first order linear differential equation. As a consequence, the state vector at time t can be deduced from the state vector at time t0, by application of the linear operator, U(t, t0) called the evolution operator. One can show that this operator is unitary which is not a surprise, since the norm of the state vector has to remain constant, equal to one. I give here without demonstration the differential equation obeyed by U. You can find the demonstration of this equation and more about the evolution operator in the book of Cohen-Tannoudji Diu and Laloe. In the case of a conservative system, the Hamiltonian H_hat does not depend on time, and the evolution operator takes a simple form of a complex exponential of time with the operator H in the exponent. You can check that when you apply it to an eigenstate Psi_n of the Hamiltonian H corresponding to an energy E_n, you obtain the usual evolution described by exponential of minus i E_n / hbar. Let us now use the evolution operator to express psi(t) as a function of psi(t0) in the quantum average at time t of an operator A. We consider here an operator that does not depend on time, as most operators in the Schrödinger point of view. Collecting together the terms U_dagger A U in the expression of the average of A, we obtain an operator that evolves with time, A(t). This is the Heisenberg form of A. Then the average of A at time t takes a form in which the operator evolves with time while the state vector retains its initial value. This is the Heisenberg expression of the quantum average of A. In the Heisenberg formalism, the state of the system does not evolve, it keeps its initial value. In contrast, the operators do evolve as a function of time. You can easily extend the reasoning above to a product of operators. Introducing between A and B the product U, U_dagger whose value is 1. It is not a difficult calculation, do not hesitate to do it. To complete our survival kit in the Heisenberg quantum world, we need to have an evolution equation for the operators, as we had an evolution equation for the state vectors in Schrodinger territories. You can find that evolution equation by taking the time derivative of the Heisenberg form of the operator and using the evolution equation of U. Remember that in our calculation, we only consider operators whose Schrödinger form does not depend on time. This is what you should find at the end of your calculation. The time derivative of A(t) is proportional to the commutator of A and H. In the not-too-frequent case when the Schrödinger form of the operator depends on time, there is an extra term shown here for completeness. Let us apply what we have just learned, to the electric field operator, whose Schrödinger form is written here. Remember also the Hamiltonian of radiation in the absence of charges. Starting from the Heisenberg evolution equation for the annilihation operator, and using the commutation relation for a and a_dagger, you can easily find this simple result. The solution of this differential equation is an oscillating exponential, so that the Heisenberg form of the electric field E(r,t) writes as a sum of terms proportional to exponential of i k_ell r - omega_ell t times the annilihation operator a_ell, at time 0. Of course, one should not forget the hermitian conjugate since the field is a real observable. Had you guessed correctly that form? It is identical to the classical expression, with each classical number alpha_ell at time zero replaced by the operator a_hat_ell at time zero. We have just encountered one of the important feature of the Heisenberg formalism. Expressions and equations resemble closely the corresponding classical expressions and equations. Personally, I do not find this feature always an advantage, because it may prevent one from fully appreciating the difference between the quantum behavior and the classical behavior. I use the Heisenberg formalism as a convenient tool necessary for instance to calculate the joint photodetection signals at two different times. But I tend to privilege the Schrödinger formalism when there is no reason to use the Heisenberg formalism. But you should know that not all quantum opticians share my position, and you should not be surprised to find an extensive use of Heisenberg formalism in some books or scientific papers. This is another reason why I want you to know the essential of that formalism which leads sometimes I must admit to develop simpler calculations. We can now write the expressions of the single and double photodetection signals using the Heisenberg formalism. It is not a big change in the case of w1 although the calculations are sometimes slightly simpler. w1 can equivalently be expressed as the average of E- and E+ at the same time taken in the radiation state at t0. It is now possible to express w2 for two different detection times t1 and t2. This formula was established by Roy Glauber it is one of the most important in quantum optics and you should remember it. It is not easy to demonstrate so I'll just ask you to accept it. Note the order of the operators which must be respected since they do not commute. The more compact form of the second line respects that order, since the order of the operators must be inverted when taking an hermitian conjugate. In both expression, the E- and E+ operators, are arranged in normal order, that is with the a operators, on the right hand side, and a_dagger, on the left hand side. A very important consequence is the null value of the photodetection signals in the vacuum.
