- 17.3 Coupling of the Cyclotron and Axial Motions. - We now study a method for detecting the cyclotron motion. - The coupling is produced by an inhomogeneous magnetic field, and it can be described by the additional term in the Hamiltonian:. - Show that the excitation numbers of the cyclotron motion (ˆ n r ) and of the magnetron motion (ˆ n l ) are constants of the motion.. - Consider the eigensubspace E n r ,n l of ˆ n r and ˆ n l , corresponding to the eigenvalues n r and n l. - (a) Calculate the time evolution of the expectation values of the position and momentum operators ˆ z and ˆ p z assuming that the state of the electron is restricted to be in the subspace E n r ,n l . - (b) To first order in , what is the phase difference ϕ between the detected current and the stable oscillator after a time τ ? Show that the measure- ment of this phase difference provides a measurement of the excitation number of the cyclotron motion.. - What are the possible results ϕ k of the measurement? Show that. - this provides a means to determine the excitation number of the cyclotron motion.. - (b) What is the state of the electron after a measurement giving the result ϕ k. - Using the previous values of the physical parameters, show that this accuracy leads to an unambiguous determination of the cyclotron excitation number.. - (d) After a measurement giving the result ϕ k , we let the system evolve for a length of time T under the action of the Hamiltonian ˆ H c . - 17.4 A Quantum Thermometer. - In practice, the cyclotron motion is in thermal equilibrium with a thermostat at temperature T . - We perform successive measurements of the phase difference ϕ in the time intervals [0, τ. - Two recordings of this measurement are represented on figure 17.1 for two different temperatures. - (a) to what phenomenon are associated the sudden changes of the signal;. - (b) what is the fraction of the time during which the electron is in the levels n r = 0, n r = 1, n r = 2. - The probability p n for a system to be in the energy level E n is given by the Boltzmann factor p n = N exp. - Give an estimate of the two temperatures corresponding to the two recordings of Fig. - 17.2 represents more accurate measurements of the occupation probabilities of the various cyclotron levels for several temperatures of the cryostat which contains the Penning trap.. - (a) Determine the normalization factor N of the probability law p n for a one-dimensional harmonic oscillator of angular frequency ω in thermal equilibrium with a thermostat at temperature T , and calculate the average excitation number ¯ n. - Time evolution of the quantum number n r corresponding to the cyclotron motion for two temperatures T a and T b. - (b) Justify the aspect of the curves of figure 17.2 and evaluate the correspond- ing temperatures of the measurements.. - (c) What is the order of magnitude of the lowest temperature one can measure with such a device?. - 0,01 Occupation probability of the cyclotron. - Occupation probabilities of the energy states of the cyclotron motion.. - 17.5 Solutions. - Section 17.1: The Penning Trap in Classical Mechanics 17.1.1. - The potential satisfies Laplace’s equation in the vacuum. - Using the expression (17.2) for the electric field, the equation of mo- tion is:. - (b) If we search a solution of the form α 0 e iωt , we find that ω is given by the equation:. - In order to obtain the motion in the xy plane, we integrate the equation of motion of α:. - The motion in the xy plane is a superposition of two harmonic motions of angular frequencies ω r and ω l. - The frequencies of the three motions are therefore:. - ω l /2π 14 kHz ω z /2π = 64 MHz ω r /2π 150 GHz (c) The motion in the xy plane is the superposition of two circular motions, one is of radius α r and has a high frequency (ω r. - Assuming α r α l , this results in a trajectory of the type represented on the following figure. - Section 17.2: The Penning Trap in Quantum Mechanics 17.2.1. - The expansion of the Hamiltonian gives ˆ H = ˆ H xy + ˆ H z with:. - 2 M ω 2 z z ˆ 2 , where we have introduced the z component of the angular momentum ˆ L z = ˆ. - We can therefore search for an eigenbasis of ˆ H in the form of a common eigenbasis of ˆ H xy and ˆ H z. - The eigenvalues of ˆ H z are therefore of the form ¯ hω z (n z + 1/2).. - (b) The commutator [ˆ a r , a ˆ l ] corresponds to ξ = 1 and ξ. - also vanish if we take the Hermitian conjugates of the previous commutators.. - 1 results in the fact that the eigen- values of ˆ n r are the non-negative integers, and the same holds for ˆ n l . - The sum and difference of the roots of the equation ω 2 − ω c ω + ω 2 z /2 = 0 are:. - (e) The eigenvectors of ˆ H xy can therefore be labeled by the two (non- negative) integer quantum numbers n r and n l corresponding to the eigenvalues of ˆ n r and ˆ n l . - The expectation value of the position in the xy plane can be calculated by using:. - As in the classical motion, the coordinates x and y are the sums of two sinusoidal functions of angular frequencies ω r and ω l . - The average motion in the xy plane is therefore the superposition of two uniform circular motions of angular frequencies ω r and ω l . - (a) The wave function φ 0 ( r ) corresponding to n r = n l = n z = 0 can be written as the product of three functions in the variables x +iy, x − iy and z. - The function φ 0 ( r ) is therefore a product of three gaussian functions in the variables x, y, z:. - hω z The discrete nature of the energy spectrum will play a decisive role only for the cyclotron motion (corresponding to ˆ a r , ˆ a † r. - For the other components of the motion, of much lower frequencies than the cyclotron motion, one expects that the thermal fluctua- tions will populate a large number of levels. - The “quantum” character of the motions will be hidden under the thermal noise.. - Section 17.3: Coupling of the Cyclotron and Axial Motions. - In the presence of the axial-cyclotron coupling, the Hamiltonian ˆ H c is:. - the excitation numbers of the cyclotron and magnetron motions) are therefore constants of the motion.. - (b) If the system is prepared in a state belonging to the subspace E n r ,n l , it will remain in this subspace since n r and n l are constants of the motion. - 2, this basis no longer corresponds to factorized functions of the variables x ± iy and z. - The coupling between the axial and cyclotron motions induces a correlation between the axial frequency and the state of the cyclotron motion.. - (a) If the system is prepared in the subspace E n r ,n l , the axial Hamil- tonian corresponds to a harmonic oscillator of angular frequency ω z. - since the evolution equations of the quantum expectation values coincide with the classical equations for a harmonic oscillator.. - Knowing the time τ, the frequency ω z and the coupling constant , one can deduce the cyclotron excitation number n r. - A given experimental result determines unambiguously the excitation number of the cyclotron motion.. - Ψ of the system after a measurement corresponds to the projection of the state vector before the measurement | Ψ on the eigensubspace corresponding to the measured result:. - where the integer n (0) r corresponds to the result of the measurement of ϕ. - (c) For the values of the parameters given in the text, one finds φ 1 = ω z τ /2 2π × 1.28. - Since ˆ n r commutes with the Hamiltonian ˆ H c , n r is a constant of the motion.. - Any further measurement of the cyclotron excitation number will therefore give the same result n r , corresponding to the same phase shift ϕ k . - The coupling with the environment can cause transitions between different eigenstates of ˆ H c , as we shall see in the next section.. - Section 17.4: A Quantum Thermometer. - (a) The sudden jumps of the signal are associated with a change of the cyclotron excitation number due to the coupling of the trapped electron with the thermostat. - We recall that otherwise n r would be a constant of the motion.. - (b) In the case of the experimental curve a of Fig. - 1, the fractions of time spent in the n l = 0, 1 and 2 levels are 80%, 19% and 1%, respectively. - In the case of curve b, the fractions are 97% and 3% for the time spent on n l = 0 and n l = 1, respectively.. - In principle, the determination of the temperature cans also be made using p 2 /p 1 , but the accuracy is poor compared to that obtained with p 1 /p 0
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