[MUSIC] Hello everybody. In the lecture, you was shown how to quantize the electromagnetic field by quantizing plane waves. In the homework, you have seen that the method also applies to the quantization of a standing wave mode, describing the electromagnetic field confined inside of cavity. Indeed, the quantization method introduced in the first lesson for the electromagnetic field is quite general and applies to any classical oscillating system. As an example,let us consider the case of mechanical oscillators. A paradigm of quantized mechanical oscillator consists in a single trapped ion. As seen on the left picture, the fluorescence of a single ion is easily seen with a CCD camera. Indeed, you can even see the fluorescence of a single ion with your own eyes. Since more than 30 years, experimentalists like Dave Wineland and Rainer Blatt have developed exquisite techniques for trapping and observing single ions. By using laser-cooling method, one can cool ions down to the ground state of their quantized motion. When reaching this degree of control of motion a quantum description becomes necessary. Not surprisingly the quantum state of motion of a massive particle trapped in a harmonic potential is described by using the formalism of a quantized harmonic oscillator. Let us now consider the case of two trapped ions. Their classical motion is described as the super position of two harmonic modes. The first mode is called the center of mass mode. It corresponds to an in-phase oscillation of the two ions. The second mode is the relative vibration mode also called the stretch mode. It corresponds to an oscillation of the relative distance between the two ions. It usually oscillates at the higher frequency. This mode can be quantized as two independent unique oscillators. This approach generalizes to the quantum description of the motion of N trapped ions, whose classical motion can be described in term of the superposition of N harmonic modes, each of them being characterized by its own oscillation frequency. Remarkably, each of these modes can be quantized individually as a single harmonic oscillator. Also each mode involves many interacting particles. Going one step further the quantum state of motion of a solid state crystal is also quantized as a collection of harmonic oscillators. Using the 1D picture, the crystal can be seen as an infinite chain of atoms or ions linked by strings. Phonons are the elementary excitation of vibration modes of this chain. Mechanical oscillator quantization also plays an essential role in the fast developing field of optomechanics. It basically consists in controlling and manipulating the quantum state of motion of a mechanical oscillator by using laser light. As an example let us consider a micrometer sized oscillator whose scale is at an intermediate scale between that of a few dropped ions, and that of a macroscopic crystal. The picture on the right represents a ten micrometer square Indium Phosphide suspended membrane. The thickness of the membrane is in the range of 100 nanometers. It presents drone modes with very high quality factor. Each of these modes is quantized as a single harmonic oscillator. In addition, the membrane is a good optical mirror for finding a mean to apply a force by using the radiation pressure of reflected light. Here again, as in the case of trapped ions. One can control the emotion of the mirror down to the ground state by using laser cooling techniques. Optomechanics is indeed a fast developing field of research with a fascinating perspective of observing quantum feature at the scale of a nearly microscopic object whose mass is orders of magnitudal wave from the mass of single atoms or ions. Going back to the case of the electromagnetic field, let us finally consider the quantization of a simple electric oscillator, the LC oscillator. But let us first answer a simple question. Why to quantize electric oscillators? Indeed, quantized LC oscillators play an essential role in the field of circuit quantum electrodynamics, called circuit-QED. In this field one manipulates the quantum state of artificial two layer atoms made of Josephson junction. These artificial atoms have become popular object as cubit for performing quantum information tasks or quantum simulations. Superconducting cubits are coupled to each other in a controlled way, by using superconducting circuits made of superconducting inductors and capacitors as shown on the picture. Such circuits have extremely low losses and present extremely high finesse electromagnetic resonances which have to be quantized in order to get to consistent quantum description of the system. Quantizing an electric circuit involving the motion of a huge number of electrons and capacitors and inductors may be seen as an extremely challenging task. Indeed, it is not. We will now show that the simple quantization formalism introduced for fill quantization applies as well to electric circuit quantization. Let us consider the LC circuit depicted on the left of the slide. It's classical dynamical state is described by the voltage difference V between the plates of the capacitor. And the current I, circulating in the wires connecting the capacitor to the inductor. Equivalently, the state of the circuit can be as well described by using another pair of variables. The charge q of the capacitor and the magnetic flux Phi is the inductor. As you will see, this pair of variables is well adapted to quantization of the circuit. Let us now express the total electromagnetic energy stored in the circuit as a function of these two variables. The total energy will be the sum of two terms. The first one HL equals to one half times phi square over L. It corresponds to the magnetic energy stored in the inductor. The second term, HC equals one half times q square over C. It corresponds to the electric energy stored in the capacitor. Let us finally denote by H of q and phi, the total electromagnetic energy of the circuit, expressed as a function of the variables q and phi. We will now check that q and phi are canonically conjugate variables. For that purpose, we apply the simple recipe presented in the lesson. Given the Hamiltonian on the left let us write the Hamilton equation of motion. By deriving H with respect to phi one gets q dot equals phi over L, equals the current I. And we know that q dot is equal to the current one phi the usual equation, linking the magnetic flux to the current in the inductor L. The second Hamilton equation reads phi dot equals minus dH over dq equals minus q over C. This equation expresses that the voltage at the capacitor equals the voltage at the inductor. These two equations are the proper equation of motion of the circuit. As a result we know know that q and phi are canonically conjugate variables. As an exercise, you can check for the current I and the voltage V do not form the pair of canonically conjugate variables. Let us combine to the Hamiltonian H of q and phi, q and phi being canonically conjugate variables, this quadratic Hamiltonian in q and phi is that of a harmonic oscillator. We can consider that the first term which is proportional to phi square plays the role of the kinetic energy, whereas the second term which is proportional to q square plays the role of a potential energy. We are now ready to proceed with quantization. We just have to turn q and phi into operators verifying the commutation relation q phi equals ih bar. To charge q and the flux phi play the role of the position and momentum as defined for mechanical oscillator. As usual, in the case of mechanical oscillator we now define the relation and creation operators a and a dagger as a function of the the motionless charge and flux operator q over q zero. And phi over phi zero. You can finally take such a Hamiltonian with h bar omega times a dagger a plus one-half. In this expression omega equals one over square root of LC. Omega over 2 phi is the usual oscillation frequency of the LC circuit. Of course, you can say that the commutator of a and a dagger is equal to one. Finally we are seeing that the quantum description of the LC oscillator amounts to the quantization of two well-defined canonically conjugated variables, q and phi. When expressing the energy of the circuit as a function of these variables, one recognizes formulas of the Hamiltonian of how many oscillator. In the quantum formulation, q and phi are operators with a commutator equal to ih bar. No matter that the valuables are microscopic valuables involving the motion of a microscopic number of electrons contributing to the current I and interacting with electromagnetic shield. So macroscopic point of view adopted here is the good one for quantizing an electric oscillator. It is infinitely simpler than the microscopic point of view which would consist in quantizing the motion of pull electrons and the coupling to the quantized electromagnetic field. Such a procedure would be a nightmare involving a huge and practical number of variables. Finally, the take home message is the following, any classical oscillating phenomena can be quantized in the same way. First identify the proper classical independent modes of oscillation of the system and then quantize those modes as independent harmonic oscillators. That's exactly the method we applied when quantizing the electromagnetic field. [MUSIC]