Quantum optics Eyal Freiberg

Question no.1 Derive the condition for lasing: The equations for the two levels are: Where now the radiation density has two independent components. The thermal black body density and the external density (the Latter did not exist in Einstein's treatment) Let’s write the eq. for in terms of and knowing that We obtain:

Question no.1 The General solution of this equation is in the form: And if we plug it into the rate eq. we obtain: Is determined from the initial condition thus: The final solution is: Also, the population of is immediately given as

Question no.1 Steady state implies that there is no change in population. Then we can deduce from the rate equation: We know that in equilibrium Therefore: comparing to plank’s law

Question no.1 We see that The condition for lasing is the so-called population inversion, It is clear that this cannot be achieved with a two level system as the maximum value can be reached is when

Question no.2 high intensity, coherence and directionality. The name stands for Light Amplification by Stimulated Emission of Radiation We can think of it as the Bose condensate for light: one photon that bounces back and forth in a cavity with two highly reflecting mirror, stimulates two photons into the same state as the original photon. So we get a huge amount of coherent radiation very quickly in this way. If one of the mirrors can also transmit then this output gives us the laser light. Its properties are: high intensity, coherence and directionality.

Question no.2 The modes allowed are those whose wavelength satisfies: The corresponding frequency is So we need to calculate Suppose that we have N different waves in a cavity. We then have The intensity is The peaks occur when

Question no.2 The width of each pulse is given by (distance from the central maximum at t = 0 to the first Minimum ). The extra phases come from the fact that the atoms in the walls of the cavity emit photons at random times. The intensity is now And thus: The average of cosine is zero since the phases are random.

Question no.2 The pulses can be made short by making sure that the extra phase of as many modes as possible are the same (known as mode locking) and this would also lead to an increase in intensity which would then be proportional to

Interactions of Light with Matter II In the semi-classical model atoms are quantized, but light is not. as an additional part of the Hamiltonian. The evolution of the system is obtained by solving the Schrodinger equation with the total Hamiltonian. This is frequently impossible to solve analytically and we have to resort to approximations or numeric. The effect of oscillating field is usually taken as a perturbation of the basic non-interacting atomic Hamiltonian. This leads to the time-dependent perturbation theory where the most useful result is Fermi's golden rule. This tells us the probability to obtain a transition from one level to another under a time-dependent perturbation.

Interactions of Light with Matter II and represent the two atomic levels. The and are the corresponding energies of the two states. They are the eigenvalues of the atomic Hamiltonian with and being the eigenvectors. Now, when this atom interacts with a field the Hamiltonian contains the transition elements for jumping from 1 to 2 and vice versa. We have the creation and annihilation operators and The Schrodinger equation is

Interactions of Light with Matter II By substituting in the wave-function We obtain two equations: This is a system of coupled equations which we solve for c2 by differentiating the second equation and substituting in the first equation into it.

Interactions of Light with Matter II We obtain: Trial solution leads to which has two roots: Therefore But thus the solution is therefore

Interactions of Light with Matter II If on –resonance we have For we require And so we obtain the time for a flop to be Given that we find

Field Quantization From the Schrodinger equation it follows that (note that this is only true if the Hamiltonian is time independent. Therefore and so is the time-evolution operator. The usual expectation value of an observable is given by

Field Quantization But this can be rewritten using our time-evolution operator as and so is the time dependent operator in the Heisenberg picture. Let's now derive the evolution equation for the above operator. Differentiate the equation obtaining

Interaction of Light With Matter III If the fluorescent light from a coherently - excited atom is collected and the mean intensity and the mean (intensity squared) is measured, you will find for a light beam from a single atom that < This has been measured (by Kimble, Mandel and others) and is called photon anti-bunching.

Interaction of Light With Matter III This is surprising because it says that r.m.s (root mean square) intensity is smaller than the average intensity. It is not true for thermal radiation nor for coherent states photon statistics. This relationship is impossible for a classical random process. Therefore photon anti-bunching is a signature of a purely quantum effect and quantum state of radiation. If spontaneous emission is sufficiently large and the coherent state amplitude small, this will lead to observation of anti-bunching. This is therefore ultimately an effect of quantum vacuum.

Interaction of Light With Matter III The variance of the photon number distribution is . Using commutation relations, this can be written as This can be written as The second order intensity correlation function (for zero delay time) is defined as

Interaction of Light With Matter III This quantity is basically the probability of detecting two simultaneous photons, normalized by the probability of detecting two photons at once for a random photon source. Here and after we assume stationary counting statistics. Then we have Then we see that sub-Poisson photon statistics, one definition of photon antibunching, is given by g(2)(0) < 1. We can equivalently express anti-bunching by Q < 0 where the Mandel Q Parameter is defined as

Interaction of Light With Matter III If the field had a classical stochastic process underlying it, say a positive definite probability distribution for photon number, the variance would have to be greater than or equal to the mean. Sub-Poissonian fields violate this, and hence are nonclassical in the sense that there can be no underlying positive definite probability distribution for photon number (or intensity).

Interaction III: 6th Question A coherent state is the minimum uncertainty state in position and momentum of a harmonic oscillator. This is a natural Hamiltonian from the physical perspective as it says that when a photon is lost in the field it is absorbed by the atom as vice versa. Going to interaction picture the Schrodinger equation reduces to

Interaction of Light With Matter III We assume that there are n photons in the field. Then due to energy conservation only the following superposition is possible The Schrodinger equation becomes Multiplying and we obtain

Interaction of Light With Matter III By taking the derivative of the second equation and substituting in the first: The solution is Therefore But at t = 0, , so that The probability is therefore

Interaction of Light With Matter III If the field is in the coherent state then the amplitude for the ground state at time t is Thus the probability is

Interaction of Light With Matter III If , then and implies , hence

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