- σ z b = +1 | B ˆ b | σ z b = +1 = σ z b = +1 | B ˆ b | σ b z = +1 where the spin state of a is irrelevant.. - similarly, the case θ a = π/2, θ b = π/2 is identical to θ a = 0, θ b = 0 (one actually recovers the same result for any θ a = θ b. - In the two cases (a) and (d), where θ a = θ b , i.e. - when they measure along the same axis, Alice and Bob are sure to find the same result.. - (a) Concerning the findings of Alice and of the spy, we have:. - (b) Concerning the findings of Bob and of the spy:. - (c) The probability that Alice and Bob find the same result has actually been calculated in questions 1.4(b,c), we simply have. - (d) Amazingly enough, the two expectation values are the same. - 1/2, on the average ¯ p = 3/4 if the values θ s = 0 and θ s = π/2 are chosen with equal probabilities.. - Section 13.3: The Quantum Cryptography Procedure. - Necessarily, if θ a = θ b , the results of Alice and Bob must be the same.. - If a single measurement done along the same axis θ a = θ b gives different results for Alice and Bob, a spy is certainly operating (at least in an ideal experiment). - If θ a = θ b , on the average half of the results are the same, half have opposite signs.. - The only chance for the spy to remain invisible is that Alice and Bob always find the same results when they choose the same axis. - For each pair of spins, there is a probability 1/2 that they choose the same axis, and there is in this case a probability 1/4 that they do not find the same result if a spy is operating (question 2.4(d. - Quite surprisingly, as mentioned above, the spy does not gain any- thing in finding out which x and z axes Alice and Bob have agreed on in step 1 of the procedure.. - In experiment number 1, however, measurements 1, 7 and 11 along the x axis do give the same results and are consistent with the assumption that there is no spy around. - However, the number N = 3 is quite small in the present case.. - Among the (1 − F )N remaining measurements, Alice selects a se- quence of events where the axes are the same and which reproduces her mes- sage. - In the present case, Alice tells Bob to look at the results # 8 and # 12, where Bob can read. - We consider here a two-level atom interacting with a single mode of the elec- tromagnetic field. - When this mode is treated quantum mechanically, specific features occur in the atomic dynamics, such as damping and revivals of the Rabi oscillations.. - 14.1 Quantization of a Mode of the Electromagnetic Field. - m ω 2 x where x is the position and p the momentum of the oscillator. - hmω, the equations of motion of the oscillator are. - Through- out this chapter, we consider a single mode of the electromagnetic field, of the form. - u x e(t) sin kz B ( r , t. - and the total energy U (t) of the field in the cavity:. - sin 2 kz d 3 r. - show that the equations for dχ/dt, dΠ/dt and U (t) in terms of χ, Π and ω are formally identical to equations (14.1) and (14.2).. - The quantization of the mode of the electromagnetic field under con- sideration is performed in the same way as that of an ordinary harmonic oscil- lator. - The Hamiltonian of the field in the cavity is. - The energy of the field is quantized: E n = (n + 1/2. - The quantum states of the field in the cavity are linear combinations of the set. - The state | 0 , of energy E 0. - hω/2, is called the “vacuum”, and the state | n of energy E n = E 0 +n¯ hω is called the “n photon state”. - corresponds to an elementary excitation of the field, of energy ¯ hω.. - One introduces the “creation” and “annihilation” operators of a photon as ˆ. - (a) Express ˆ H C in terms of ˆ a † and ˆ a. - 14.2 The Coupling of the Field with an Atom 133 The observables corresponding to the electric and magnetic fields at a point r are defined as:. - ˆ a + ˆ a † sin kz B ˆ ( r. - The interpretation of the theory in terms of states and observables is the same as in ordinary quantum mechanics.. - where α is any complex number, is called a “quasi-classical” state of the field.. - (a) Show that | α is a normalized eigenvector of the annihilation operator ˆ. - Calculate the expectation value n of the number of photons in that state.. - (b) Show that if, at time t = 0, the state of the field is | ψ(0. - (c) Calculate the expectation values E ( r ) t and B ( r ) t at time t in a quasi-classical state for which α is real.. - Compare the result with the expectation value of ˆ H C in the same quasi-classical state.. - 14.2 The Coupling of the Field with an Atom. - Consider an atom at point r 0 in the cavity. - The motion of the center of mass of the atom in space is treated classically. - In the basis. - f , and the atomic Hamiltonian can be written as: ˆ H A. - n forms a basis of the Hilbert space of the { atom+photons } states.. - In the remaining parts of the problem we assume that the frequency of the cavity is exactly tuned to the Bohr frequency of the atom, i.e. - Draw schematically the positions of the first 5 energy levels of ˆ H 0 . - The Hamiltonian of the electric dipole coupling between the atom and the field can be written as:. - (b) To which physical processes do ˆ aˆ σ + and ˆ a † σ ˆ − correspond?. - 0 V sin kz 0 Ω n = Ω 0 √ n + 1 . - 14.3 Interaction of the Atom with an “Empty” Cavity. - In the following, one assumes that the atom crosses the cavity along a line where sin kz 0 = 1.. - An atom in the excited state | e is sent into the cavity prepared in the vacuum state | 0 . - At time t = 0 , when the atom enters the cavity, the state of the system is | e, n = 0. - What is the state of the system at a later time t?. - 14.4 Interaction of an Atom with a Quasi-Classical State 135 14.3.2. - What is the probability P f (T ) to find the atom in the state f at time T when the atom leaves the cavity? Show that P f (T ) is a periodic function of T (T is varied by changing the velocity of the atom).. - The experiment has been performed on rubidium atoms for a couple of states (f, e) such that d C.m and ω/2π Hz. - The volume of the cavity is m 3 (we recall that 0 = 1/(36π10 9 ) S.I.).. - One observes a damped oscillation, the damping being due to imperfections of the experi- mental setup.. - (We recall that the Fourier transform of a damped sinusoid in time exhibits a peak at the frequency of this sinusoid, whose width is proportional to the inverse of the characteristic damping time.). - (a) Probability P f (T ) of detecting the atom in the ground state after it crosses a cavity containing zero photons. - (b) Fourier transform of this probability, as defined in the text. - 14.4 Interaction of an Atom with a Quasi-Classical State. - The atom, initially in the state | e , is now sent into a cavity where a quasi- classical state | α of the field has been prepared. - At time t = 0 the atom enters the cavity and the state of the system is | e. - Calculate the probability P f (T, n) to find, at time T , the atom in the state | f and the field in the state | n + 1 , for n ≥ 0. - What is the probability to find the atom in the state | f and the field in the state | 0. - Write the probability P f (T ) to find the atom in the state | f , inde- pendently of the state of the field, as an infinite sum of oscillating functions.. - 14.2 are plotted an experimental measurement of P f (T ) and the real part of its Fourier transform J(ν). - The cavity used for this mea- surement is the same as in Fig. - 14.1, but the field has been prepared in a quasi-classical state before the atom is sent in.. - determine an approximate value for the mean number of photons | α | 2 in the cavity.. - (a) Probability P f (T ) of measuring the atom in the ground state af- ter the atom has passed through a cavity containing a quasi-classical state of the electromagnetic field. - 14.5 Large Numbers of Photons: Damping and Revivals. - Consider a quasi-classical state | α of the field corresponding to a large mean number of photons. - In this case, the probability π(n) to find n photons can be cast, in good approximation, in the form:. - This Gaussian limit of the Poisson distribution can be obtained by using the Stirling formula n. - 2πn and expanding ln π(n) in the vicinity of n = n 0. - For such a quasi-classical state, one tries to evaluate the probability P f (T) of detecting the atom in the state f after its interaction with the field.. - 14.6 Solutions 137. - one linearizes the dependence of Ω n on n in the vicinity of n 0 : Ω n Ω n 0 + Ω 0 n − n 0. - (b) Does this damping time depend on the mean value of the number of photons n 0. - Keeping the discrete sum but using the approximation (14.5), can you explain the revival qualitatively? How does the time of the first revival depend on n 0. - 14.6 Solutions. - Section 14.1: Quantization of a Mode of the Electromagnetic Field. - B = 0 are satisfied whatever the values of the functions e(t) and b(t). - 2 e 2 (t) sin 2 kz + 1. - (c) Under the change of functions suggested in the text, we obtain:. - The state | n is an eigenstate of ˆ H C with eigenvalue (n + 1/2)¯ hω, i.e.
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