- The expectation value of the number of photons in that state is:. - (d) These fields are of the same type as the classical fields considered at the beginning of the problem, with. - Therefore the expectation values of the field operators satisfy Maxwell’s equa- tions.. - (e) The energy of the classical field can be calculated using the result of ques- tion 1.1b. - More generally, the expectation value of a physical quantity as calculated for a quantum field in the state | α , will be close to the value calculated for a classical field such that E cl ( r , t. - (a) Positions of the five first energy levels of H 0 . - (b) Positions of the five first energy levels of ˆ H = ˆ H 0 + ˆ W. - Section 14.2: The Coupling of the Field with an Atom 14.2.1. - 14.4a. - Only the ground state | f, 0 of the atom+field system is non-degenerate.. - The coupling under consideration corresponds to an electric dipole interaction of the form − D ˆ · E ˆ ( r. - where ˆ D is the observable electric dipole moment of the atom.. - The term ˆ aˆ σ + corre- sponds to the absorption of a photon by the atom, which undergoes a transi- tion from the ground state to the excited state. - The term ˆ a † σ ˆ − corresponds to the emission of a photon by the atom, which undergoes a transition from the excited state to the ground state.. - The operator ˆ W is block-diagonal in the eigenbasis of H ˆ 0. - The state | f, 0 is an eigenstate of ˆ H 0 + ˆ W with the eigenvalue 0.. - 14.4b.. - Section 14.3: Interaction of the Atom and an “Empty” Cavity 14.3.1. - The time evolution of the state vector is therefore given by:. - In general, the probability of detecting the atom in the state f , in- dependently of the field state, is given by:. - In the particular case of an initially empty cavity, only the term n = 1 con- tributes to the sum. - This result corresponds to the expected value:. - Section 14.4: Interaction of an Atom with a Quasi-Classical State 14.4.1. - At time t the state vector is. - the probability to find the atom in the state | f and the field in the state. - The probability P f (T) is simply the sum of all probabilities P f (T, n):. - (b) The ratios of the measured frequencies are very close to the theoretical predictions: ν 1 /ν 0. - (c) The ratio J(ν 1 )/J(ν 0 ) is of the order of 0.9. - Assuming the peaks have the same widths, and that these widths are small compared to the splitting ν 1 − ν 0 , this ratio correponds to the average number of photons | α | 2 in the cavity.. - If one performs a more sophisticated analysis, taking into account the widths of the peaks, one obtains | α see the reference at end of this chapter).. - However, the inaccuracy due to the overlap of the peaks is greater than for J(ν 1 )/J(ν 0. - owing to the smallness of J(ν 2. - Section 14.5: Large Numbers of Photons: Damping and Revivals 14.5.1. - for integer values of n in a neighborhood of n 0 of relative extension of the order of 1. - (14.6) We now replace the discrete sum by an integral:. - using the fact that the width of the gaussian is √ n 0 n 0 . - The sine term does not contribute to the integral (odd function) and we find:. - For n 0 1, the argument of the exponential simplifies, and we obtain:. - (b) In this approximation, the oscillations are damped out in a time T D which is independent of the number of photons n 0 . - For a given atomic transition (for fixed values of d and ω), this time T D increases like the square root of the volume of the cavity. - In the limit of an infinite cavity, i.e. - One can find the damping time by simply estimating the time for which the two frequencies at half width on either side of the maximum of π(n) are out of phase by π:. - Within the approximation (14.5) suggested in the text, equation (14.6) above corresponds to a periodic evolution of period. - The time of the first revival, measured in Fig. - 14.3, is Ω 0 T 64, in excellent agreement with this predic- tion. - Notice that T R ∼ 4 √ n 0 T D , which means that the revival time is always large compared to the damping time.. - 14.3 that the functions are only partly in phase. - 14.7 Comments. - The damping phenomenon which we have obtained above is “classical”: one would obtain it within a classical description of the interaction of the field and the atom, by considering a field whose intensity is not well defined (this would be the analog of a distribution π(n) of the number of photons). - It is a direct consequence of the quantization of the electromagnetic. - field, in the same way as the occurrence of frequencies ν 0 √ 2, ν 0. - in the evolution of P f (T ) (Sect. - The pair of levels (f, e) correspond to very excited levels of rubidium, which explains the large value of the electric dipole moment d. - cooled down to 0.8 K in order to avoid perturbations to the experiment due to the thermal black body radiation (M. - 15.1 Preliminaries: a von Neumann Detector. - There are two stages in the measurement process. - These states correspond for instance to the set of values which can be read on a digital display.. - Let | ψ be the state of the system S under consideration, and | D the state of the detector D . - Before the measurement, the state of the global system S + D is. - Let a i and | φ i be the eigenvalues and corresponding eigenstates of the observable ˆ A. - The state | ψ of the system S can be expanded as. - Using the axioms of quantum mechanics, what are the probabilities p(a i ) to find the values a i in a measurement of the quantity A on this state?. - After the interaction of S and D , the state of the global system is in general of the form. - D j (15.2) We now observe the state of the detector. - What is the probability to find the detector in the state | D j. - After this measurement, what is the state of the global system S + D ? 15.1.4. - A detector is called ideal if the choice of | D 0 and of the coupling S – D leads to coefficients γ ij which, for any state | ψ of S , verify. - 15.2 Phase States of the Harmonic Oscillator. - where θ m can take any of the 2s + 1 values θ m = 2πm. - We consider the subspace of states of the harmonic oscillator such that the number of quanta N is bounded from above by some value s. - Express the vectors | N in the basis of the phase states.. - Calculate the expectation value of the position ˆ x in a phase state, and find a justification for the name “phase state”. - where x 0 is the characteristic length of the problem. - 15.3 The Interaction between the System and the Detector. - We want to perform an “ideal” measurement of the number of excitation quanta of a harmonic oscillator. - If the two oscillators are two modes of the electro- magnetic field, it originates from the crossed Kerr effect. - What are the eigenstates and eigenvalues of the total Hamiltonian H ˆ = ˆ H S + ˆ H D + ˆ V. - We assume that the initial state of the global system S + D is fac- torized as:. - We perform a measure- ment of ˆ n in the state | Ψ (0. - What is the state | Ψ (t) of the system? Is it also a priori factorizable?. - 15.4 An “Ideal” Measurement. - The oscillator D is prepared in the state
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