Good morning. This morning I will try to introduce you to cavity quantum electrodynamics and to circuit QED which are 2 related fields in quantum information which illustrate the increasingly tighter link between atomic and molecular physics and quantum optics on the one side and condensed matter physics on the other one. I will start by describing experiments we have performed in Paris with microwave photons bouncing between 2 mirrors and interacting with atoms crossing the gap between the mirrors one by one. And then I will describe similar experiments in circuit QED in which the Rydberg atoms are replaced by artificial structures, so-called Josephson qubits, coupled to radio frequency photons. These experiments have started as a fundamental test of quantum measurement theory. And they are now turning into a promising field for application in quantum information science. So let’s start with cavity QED, you have an atom interacting with Fabry-Perot cavity, cavity made of 2 mirrors facing each other. And this is an ideal system which can be modelled as a 2 level atom or qubit, coupled to a harmonic oscillator which is a field mode which is resonant or quasi resonant with a transition between the 2 levels. The coupling is measured by a quantity called the vacuum Rabi frequency which describes a rate at which at resonance the 2 level system and the oscillator exchange energy. This rate is rather fast for the microwave domain because we are not using just any atom, we are using giant circular Rydberg atoms whose size is of the order of 0.1 micrometre, which means that superposition state between these Rydberg atoms has a huge dipole coupling, very strongly to microwaves. The relaxation of this system which describes the rate at which they are coupled to the environment is very long. So it’s another system to study the exchange of quantum information between light and matter. Here is a more realistic view of the system. You see the 2 mirrors which are copper mirrors speckled with a thin layer of niobium and the atom interacts one at a time between these mirrors. The mirrors are very good. You have more than 1 billion bounces of the microwave photons between the mirrors and they can stay inside the cavity as I said for more than a tenth of a second. The atoms are just crossing the cavity one by one. They are coupled to the cavity field and then they are detected, and the information we get is from the atoms. We use state selective field ionisation which is described on this slide. You see here the Coulomb potential of the core of the atom. And the 2 levels are relevant for our experiment. If you apply an electric field, static electric field you lower the barrier on one side and the electron escapes. And, of course, the upper state escapes in a smaller field than the lower state. So usually we apply a ramp of electric field and we get the ionisation of the 2 levels at slightly different times which discriminate between the 2 states. And we get a bit of information, atom detected in e or g, for each atom that we detect in this way. Here I describe the most basic process that we are studying, which is at resonance the exchange of energy between the atom and the cavity. You see here the atom entering the cavity in the upper state. The cavity is empty so you have zero photon inside. After a while the atom goes to the lower state and they meet one photon. And in fact we have a 2 level system. The system oscillates between e,0> and g,1>. After a longer time it comes back to state e,0>. And this is called the vacuum Rabi oscillation at frequency omega over 2 Pi. So if you just compute the state of the atom field system you get the superposition of e,0> and g,1> with probability amplitudes which oscillate at the Rabi frequency. If you stop the interaction for Omega t equal to Pi over 2, which is Pi over 2 pulse, then you get a maximum entanglement between the atom and the field. The atom leaves the cavity and the 2 system atom and field are non-locally entangled. So you can perform experiments on the atom and in this way you project the field into the corresponding state. This can also be considered as a quantum gate between the atom and the field. In fact you can use this entanglement to entangle 2 atoms. Here is what happens after the atom has crossed the cavity for Pi over 2 pulse so you get a maximum atom field entanglement. Then you send a second atom through the system and you see the second atom enter the system here. If there is no photon in the system he stays in the lower state of the transition. But if there were one photon in the system it absorbs the photon and gets excited. So the field in the cavity is mapped on the second atom. For that you just have to have a Pi pulse for the second atom interacting with the field. And you end up with 2 entangled atoms. And you can play various variants of this experiment to perform quantum gates between atoms crossing the cavity one at a time. Up to now I have considered the resonant interaction between atoms in the cavity field. Let’s now turn to non-resonant interactions which provides a way to detect photons without destroying the photons. In fact, if you think about detection of light most of the time it’s destructive. The basic effect is a photoelectric effect. A photon impinges on the atom. An electron is ejected and you detect the ejected electron. This is the way we detect in most photodetectors and it is also the way our eye is working. In fact, the photons get on to our retina and then they are destroyed. So the question we ask ourselves is, is it possible to detect the photon in a gentler way, to have an interaction which does not absorb the photon but which leaves an imprint on the atom. And cavity QED provides a way to do that by using light shifts. So on this slide I remind you briefly what the light shifts are. You have at atom interacting non-resonantly with a field. So it cannot absorb or emit photons. But the energy level of the atoms are slightly modified by the field. And you see this shift is proportional to the square of the coupling, that is the square of the Rabi frequency, to the field energy that is a number of photons. And it is inversely proportionate to the detuning delta between the atom and the field. And another important feature is that the shifts go in opposite direction for the 2 states e and g. So you see that as the atom is crossing the cavity the transition frequency between the 2 states increases. And if you have a superposition of states it accumulates a phase shift which is proportional to the photon number. And the interaction is so strong that this phase shift can reach a value Pi per photon which means that the superposition emerges from the cavity with opposite phases depending upon whether there is zero or 1 photon in the cavity. If you can measure this phase shift you realise in fact a non-destructive quantum non-demolition measurement of the photon. So how do you measure a phase shift in atomic physics? What you use is what is known as Ramsey interferometer. You see here again the cavity with the energy level shifted. What you do is that you apply a microwave pulse which mixes a state e and g before the atom enters the cavity and another pulse after. And this technique of separating the oscillitory pulses is called Ramsey interferometer. And this is a technique which is used in caesium atomic clocks for instance. So in the end you detect the atom and you obtain, in the probability to detect the atom in one state or the other you have fringes, you have an oscillatory behaviour. It is an indifference effect because you don’t know when you detect the atom whether it underwent the transition in the first or in the second zone. You have an amplitude for each and these amplitude interfere. And the important point here is that the fringes that you get have a phase phi of n, which depends upon the photon number, which is proportional to the photon number. For instance, if you plot the probability to detect the atom in one state as a function of the phase of the interferometer that you control by changing the frequency of the Ramsey zones, you get a set of fringes if you have zero photon in the cavity, and displaced fringes if you have 1 photon, and twice displaced if you have 2 photons and so on. Now, if you fix the phase of the interferometer at the point where the slope of the fringes is high what you see is that the probability for finding the atom in state e or in state g takes different values, depending upon n. So if you are able to measure this probability by sending many atoms in the cavity before the field decays you will pin down the photon number inside the cavity. But for that you need indeed many atoms crossing the cavity one by one before the photons get lost which explains why you need a very high Q cavity for this experiment. Now, the process by which you detect the field as atoms go through the cavity one by one, is in fact a Bayesian process. You acquire progressively information about the field. And each piece of information allows you to update the photon number probability distribution. We studied this in detail in a paper that we published with our resident colleagues, Luiz Davidovich and Nicim Zagury, back in 1992. And you see here a simulation of the process when you start at the bottom of this figure with the Poisson distribution, with a coherent field which has a Poisson distribution of the photon number. You see after the first atom has been detected that some photon number has been decimated because when you detect the atom in one state it means that the probability, that the photon number is such that you could not detect the atom in that state this probability decreases, it’s a kind of Bayesian argument. And as you detect atom you keep decimating different photon numbers until in the end only one photon number is left. So we simulated this and it took us many years before we could do the experiment. At last 15 years later we had a very good cavity and you see here the real experiment. At the beginning we don’t know how many photons are in the cavity. So we start with a flat distribution and then, as the atoms are detected, the probability evolves until we are left with a single value here, 5 photons in the cavity. So it’s an ideal example of quantum measurement in which the system collapses into an eigenstate of the measured observable which is here the photon number. There is a very interesting situation if you have a Pi shift of photon in this case the set of fringes that you obtain for even photon numbers and those which you obtain for odd photon numbers are phase opposite. So if you set the system at the top of a fringe a single atom will give you, in a single shot, the parity of the photon number. And if the photon number is very small, if you are sure that you have no more than one photon in the cavity, a single atom is able to count the photon number without the zero 1. This is the first trace we ever got with this kind of experiment. You see here a very small thermal field which is evolving between zero and 1 photon. And you have a typical telegraphic signal which shows that photons pop in the cavity and then disappear. And if you look at the duration of each photon you see a very large dispersion. Some photons live long, some photons only live very short. This is just because the probability law for the lifetime of a photon is exponential. So we studied these kind of signals. And, in fact, they realised a quantum gate. If the field is zero or 1, it’s a qubit, and if the atom is e or g, it’s another qubit; these 2 qubits are coupled by conditional dynamics. Atom is found in one state, if zero is zero, photon is the other state if there is 1 photon. And you can play with these gates, entangle the atom in the field in a dispersive way which is quite different from the resonant entanglements that I described before. You can also observe quantum jumps of the field. If you keep measuring, you see, as long as you have a given photon number, the fringes have a well-defined phase. But when a photon is lost due to relaxation in the walls of the cavity, the phase of the fringe suddenly jumps. And you see this as a jump in the signal. And you see here for instance how a field containing 5 photons is decaying back down to vacuum step by step. Usually when you think about the decrease of a field in electromagnetism you have a picture and exponential decay. But the exponential decay is just an average over many, many photon events. If you continuously count the photons, you don’t see an exponential decay. What you see is a staircase decay down to the vacuum. Another interesting point is the relationship between the photon number and the phase which illustrates the principle of complementarity. You see, the initial state, if you start from a coherent state, has a Poisson distribution of the photon number, just because the phase is well-defined so the photon number cannot be well-defined. But as the QND process evolves the field collapses into a Fock state so you go from a situation where in phase space you have a well-defined phase. This is just the Q function of the field which shows that in phase space you have a vector pointing in a well-defined direction down to a situation where the phase is completely random. And you have this circular distribution for the phase. So the question we ask ourselves is, how does this process occur? We know that acquiring information, the photon number destroys information about the phase, but how is it happening? And what is the phase distribution after the first detected atom? So you see here a simulation. We start from a coherent field which has a well-defined phase. After the first atom this evolves into a superposition of 2 fields with opposite phases. If you have a bi-phase shift of photons then for the second atom, you apply Pi over 2 phase shift of photon. And each of these components is split into 2 – again – and then you have 8 components, The important point is that for the first atom you get a superposition of 2 fields with opposite phases which will go like a cat state, the Schrödinger cat state, for reasons which will become clear in a moment. So let’s describe the experiment in which we observe this cat state. Again we make use of the Ramsey interferometer. We start by preparing in the cavity a coherent field which is just injecting, with a classical source, a field with a well-defined phase in the cavity and we send 1 atom across a cavity. And before the atom enters we prepare it by microwave pulse in superposition of the 2 states e and g. Then the atom crosses the cavity and it changes the phase of the cavity field. The atom behaves as a single atom index of a fraction which changes temporarily the frequency of the field. So the field accumulates 2 phases, 2 opposite phases depending on whether, in which state the atom is. And here you have a typical Schrödinger cat situation. You have a 2 level atom which is entangled with a cavity field which may contain many photons. If you detect it at this point you would collapse the field into a field which will have 1 phase or the other. The trick is to let the atom cross the second pulse which mixes the levels again. And then you maintain the cat's ambiguity because when you detect there is no way of knowing in which state the atom was when it crossed the cavity. So when you detect finally the atom in e or in g, the field is projected into a cat state superposition. It’s interesting to look at this process in the vocabulary of quantum logic. And I have drawn here the circuit, the logical circuit to describe this experiment. The red line corresponds to the cavity field and the blue line to the atom. So you see, the first thing you do from the left to the right is to make a unitary transformation on the field. Injecting a field in the cavity is displacing the field in the phase spaces, is a unitary operation. Then you let it interact with the atom. The atom has undergone a Pi over 2 pulse here. Then you have the Pi phase shift of photon which is a conditional gate. You displace a field by Pi whether the atom is in e or g. And then you apply a second Pi over 2 pulse and then you detect. And when you detect you prepare Schrödinger cat. If you find the atom in state e you prepare the Even cat because the photon number is even. And this is superposition beta> plus minus beta>. If you detect the atom in the other state you get the Odd cat, beta>minus minus beta>. And if you don’t detect the atom or if you miss the detection you get a statistical mixture of the 2 which amounts to a statistical mixture of the field in one of the 2 states. So you see that this preparation procedure is random. You don’t know whether you will find an Even or an Odd cat before you detect it. But this is only half of the experiment. Then you have to detect the Schrödinger cat. And for that we complete the circuit by the last line. So you skip sending atoms across a cavity. What you do is first to displace the field again by a controlled amount and then you have another quantum gate which performs a QND measurement of the displaced field. So finally you reconstruct the field by displacing it in phase space and measuring on many copies the photon number distribution in the displaced field. Then you can use a computer to reconstruct the field and you have, of course, to repeat the experiment for many displacements. And you get this kind of result. This is a Wigner function of an Even cat. And so you recognise the 2 Gaussian peaks corresponding to the 2 coherence superposition. And in between you have fringes in the Wigner functions which are a signature of the coherent nature of the superposition. This corresponds to about 12 photons. Here you have the same but the Odd cat, if you detect the atom in the other state and finally you see what happens if you don’t distinguish between e and g. You don’t get the fringes anymore, you get the statistical mixture. And in fact what decoherence is about, it’s a phenomenon which transforms the coherent cat into a statistical mixture due to the coupling with the environment. And we indeed were able to study decoherence by taking snap shots of this Wigner function for increasing delays. And you will see here on this movie what happens. You see that the fringes get blurred. As soon as you lose a photon in the environment you get some information leaking outside hich tell you whether the cat was Even or Odd. And then you lose track of the coherence within the 2 cat components. So in the last minutes I would like just to show you, to describe to you the corresponding experiment in the circuit QED. So you see here again the sketch of cavity QED. In circuit QED you replace your atom by macroscopic device. In fact it’s a circuit in which one or several Josephson junctions are embedded. And this circuit is coupled in various ways to a cavity. It’s a strip line resonator or a close 3D cavity. And this is a superconducting system. The coupling is much stronger than in cavity QED, just because of the size of the dipole is much larger with the atoms we have 0.1 micron. Here we can go from 100 micron to 1 millimetre in size. So you have a huge coupling which means that the processes go much faster. And you have to keep the system at very low temperature, much lower than the one which are required in cavity QED. I will just fly over this topic because a huge number of groups now working with circuit QED. I just named here a few of these groups. And on the next slide I give you 2 examples of circuit QED systems. This is a kind of device, John Martinis used in Santa Barbara. You recognise on the left in this black area the qubit is imprinted, the superconducting circuit which is imprinted here, it’s called a phase qubit. So in this qubit the phase difference between the 2 parts of the Josephson junction and the charge difference act as conjugate quantum mechanical variables. And the phase can be considered as a kind of position of the quantum system which evolves into a qubit potential well which near the minimum has a shape which is quasi parabolic. And you can define quantum states zero and 1. The important point is that the state 2 is the transition between 1 to 2 is non-degenerate with the transition between zero and 1 because of the anharmonicity. Which means that if you have a radio frequency field which is resonant between zero and 1, you can forget about the other state, as we do in cavity QED. And this qubit is coupled to a coplanar resonator which is a long coaxial line, which is closed at the 2 ends by capacitance and which plays a role, a Fabry-Perot role in our experiment. Here you have the other kind of set-up of the Yale group, the group of Rob Schoelkopf. In the latest version you have a qubit called a Transmon qubit which is suspended inside an aluminium 3D cavity. And what I want also to say is that when you use these artificial atoms because of the stronger coupling you have 1,000 times faster Rabi frequency. But, of course, decoherence is also faster. So in fact you scale down the time scale of the experiment by a factor of 2 to 3 orders of magnitude, which allow you to make much more operations. But you have to go very fast. The number of operations you can go during the relaxation time is of the same order of magnitude that in cavity QED. You see here, for example, results from the Yale group. These are Schrödinger cats they have observed. They look very similar to the cavity QED cats. On the left you have a 2 legged cat, superposition of 2 coherence states. They also studied superposition of 3 coherence states in the middle and of 4 coherence states on the right. You see here the cavity structure. You have in fact 2 cavities, 2 aluminium cavities. The cavity number 1 is the one which stores the photons. And the cavity number 2 is a detection cavity. In fact, the Transmon is bridging, interacting with the 2 cavities at once. It interacts through the cavity QED process with cavity number 1. And it modifies the dispersive properties of cavity 2 in different ways whether the qubit is in state e or g. So you use the coupling to cavity 2 as a detection. The way the Schrödinger cats are prepared is according to a circuit which is very similar to the circuit we use in cavity QED. You see here the red line corresponds to the storage cavity, the black line to the qubit, and the blue line on the bottom to the detecting cavity. So you see at first the qubit undergoes a Pi over 2 phase shift. You fill the field in the cavity and at the end of this process you get an atom in the superposition state interacting with the coherent field. Then you apply the conditional biphase shift and you get entanglement exactly the same kind of cat, beta> e> plus minus beta> g> that we get in cavity QED. At this point the 2 methods diverge a little bit. What you do next is that you shift the amplitude in the cavity by the amount beta. So beta goes to 2 beta and minus beta goes to zero. And now you have a different kind of cat. You have a cat which is the preparation of a large field and a vacuum. Now, why do you do that? The reason is the following, what you can do at this point is a conditional pulse applied on the atom. The atom will go from e to g provided the cavity is in vacuum. If the cavity contains photons the light shift puts the cavity out of resonance with the atom and prevents the system from evolving. So you have a conditional gate. And in the end what you see is that the g part of the wave function goes to e, whereas the e part does not move. And, lo and behold, the 2 systems are disentangled. Now, you have cat 2 beta zero in the presence of an atom in level e. You have suppressed the entanglement and you have a deterministic preparation of a Schrödinger cat. You don’t rely anymore on a random process. So if you want to go back to the phase cat what you have to do is to shift again by minus beta. And now you get the same cat as before with the atom in level e. At this point what you have to do is to reconstruct and you have exactly the same process as in cavity QED to reconstruct. You see that the only difference that you don’t need to send other qubits. The same qubit you just prepared, the cat, can be reset to detect it at a later time. And in this way you get the kind of cat that I showed you to. This is another example, a cat with 7 photons on average. And they have been able by looking at the fringes between the 2 coherent states to prepare cats with up to about 100 photons in the cavity. The more photons the tighter the fringes are. So I won’t say more about this. Just want to conclude by saying that by entangling superconducting qubits, by coupling them to rf photons in coplanar wave guide or in 3D cavities one can demonstrate several steps of quantum information procedures. You can, of course, build quantum gates. You can perform quantum teleportation experiments. You can also demonstrate simple algorithms like factorising 15, for example, and finding, after a long process, that 15 is 5 times 3, and perform simple error correction schemes. But these experiments are still far from large-scale fault-tolerant quantum computing. One may say that this system is competing with the ion trap kind of computer and facing different difficulties. One of the difficulties here that the qubits are not identical to each other, they are manmade, they are not given by nature. So one of the problems is to get well-controlled sized qubits. So I am concluding at this point. But I want to come back to our kind of cavity QED. I just want to say that we are still working in quantum information using our Rydberg atoms. And we have done in the last year 3 kind of experiments. I don’t have time to describe them. But I just list them here, quantum feedback experiments. Quantum feedback, the principle is that you have a non-classical state in the cavity, for instance, a Fock state, and you observe when a quantum jump occurs and then you correct for this quantum jump. So that you are able to keep on average a Fock state for an indefinite time in the cavity. And similar techniques might apply to Schrödinger cat by just observing continuous disparity of the cat state. Another kind of experiment is the quantum Zeno experiment. By continuously observing a quantum system you can freeze its evolution or you can force the evolution to stay within the boundaries of sub-space, of Hilbert space, and in this way you can tailor quantum states, you can prepare all kind of non-classical states. And finally you can use the fine structure of the Schrödinger cat state to perform very precise metrology, for instance of electric fields. And by looking at the fringes in an electric-sensitive Schrödinger cat we have been able to build an electrometer which measures electric field with the sensitivity of 30 microvolt per centimetre which is equivalent of the field of the single electron at a distance of 700 micrometres which might have interesting applications. So I am just finishing here and I want in this last slide to acknowledge my colleagues in Paris, especially Michel Brune and Jean-Michel Raimond and all the other postdocs and students who have made the experiments possible. And I thank you for your attention.