- Section 16.2: Ramsey Fringes. - The neutron spin precesses freely between the two cavities during time T , and we obtain. - By definition, t 0 = t 1 +T and t 1 = 2t 1 − t 0 +T, therefore the transition matrix in the second cavity is. - The probability amplitude for detecting the neutron in state. - after the second cavity is obtained by (i) applying the matrix U to the vector (16.5), (ii) calculating the scalar product of the result with | n. - Therefore, the probability that the neutron spin has flipped in the two-cavity system is. - In contrast, the present result for two cavities exhibits a strong modulation of the spin flip probability, between 1 (e.g. - This modulation results from an interference process of the two quantum paths corresponding respectively to:. - a spin flip in the first cavity, and no flip in the second one,. - no flip in the first cavity and a spin flip in the second one.. - Each of these paths has a probability 1/2, so that the sum of the probability amplitudes (16.6) is fully modulated.. - This form agrees with the observed variation in ω of the experimental signal.. - For that value, a constructive interference appears whatever the neutron velocity. - (b) The angular frequency ω 0 is related to the magnetic moment of the neutron by ¯ hω 0 = 2µ n B 0 which leads to µ n J T − 1 . - (c) This experiment can be compared to a Young double slit interference experiment with polychromatic light.The central fringe (corresponding to the peak at ω = ω 0 ) remains bright, but the contrast of the interferences de- creases rapidly as one departs from the center. - The probability P. - is the product of the two probabilities: the prob- ability to find the neutron in the state | n. - when it leaves the first cavity (p = 1/2) and, knowing that it is in the state | n. - the probability to find it in the same state when it leaves the second cavity (p = 1/2). - 1/2 does not display any interference, since one has measured in which cavity the neutron spin has flipped. - Section 16.3: Detection of the Neutron Spin State 16.3.1. - Expanding in terms of the energy eigenstates, one obtains for | ψ(0). - Physically, this means that the neutron’s spin state does not change since it is an eigenstate of ˆ V , while the atom’s spin precesses around the x axis with angular frequency A. - The measurement of the z component of the atom’s spin gives. - In both cases, after measuring the z component of the atom’s spin, the neutron spin state is known: it is the same as that of the measured atom. - It is not necessary to let the neutron interact with another measuring apparatus in order to know the value of S nz. - Section 16.4: A Quantum Eraser 16.4.1. - Finally, when the neutron leaves the second cavity (step d), the state of the system is:. - The probability to find the neutron in state. - is the sum of the probabilities for finding:. - (a) the neutron in state + and the atom in state. - the square of the modulus of the coefficient of | n. - There are no interferences since the quantum path leading in the end to a spin flip of the neutron can be determined from the state of the atom.. - h/2 along the y axis is the coefficient of the term | n. - a : +y in the above expansion. - Indeed the sum of the two probabilities calculated above is 1/2 as in 4.2. - As seen in question 4.2, if the atom A is present, Bob no longer sees oscillations (in ω − ω 0 ) of the probability for detecting the neutron in the state. - In the complementary set, where Alice has found. - 2 can be interpreted as the interference of the amplitudes cor- responding to two quantum paths for the neutron spin which is initially in the state | n. - either its spin flips in the first cavity, or it flips in the second one. - After Alice has measured the atom’s spin along the y axis, she has, in some sense “restored” the initial state of the system, and this enables Bob to see some interferences. - Notice that the statement in the text does not specify in which physical quantity the interferences appear. - Notice also that the order of the measurements made by Alice and Bob has no importance, contrary to what this third statement seems to imply.. - Alice can measure along the axis w = sin η u y +cos η u z , in the (y, z) plane, for instance. - For η = π/2 and 3π/2 or, more generally, if Alice measures in the (x, y) plane, the contrast of the interferences,. - 16.6 Comments. - The experimental curve given in the text is taken from J.H. - One stores neutrons in a “bottle” for a time of the order of 100 s and applies two radiofrequency pulses at the begining and at the end of the storage. - This improves enormously the accuracy of the frequency measure- ment. - Such experiments are actually devised to measure the electric dipole moment of the neutron, of fundamental interest in relation to time-reversal. - The structure of the inter- action Hamiltonian considered in the text has been chosen in order to provide a simple description of the quantum eraser effect. - We study here the measurement of the cyclotron motion of an electron. - The particle is confined in a Penning trap and it is coupled to the thermal radiation which causes quantum transitions of the system between various energy levels.. - This trap consists in the superposition of a uniform magnetic field B = B e z (B >. - 17.1 The Penning Trap in Classical Mechanics. - Show that the classical equation of motion of the electron in the trap is:. - In order to study the component of the motion in the xy plane, we set α = x + iy.. - (b) We seek a solution of this equation of the form α(t. - Show that ω is a solution of the equation:. - Show that:. - We consider the values B = 5.3 T and ω z /(2π. - (a) Show that the most general motion of the electron in the Penning trap is the superposition of three harmonic oscillator motions.. - (c) Draw the projection on the xy plane of the classical trajectory of the trapped electron, assuming that α r α l (the positive quantities α r and α l represent the amplitudes of the motions of angular frequencies ω r and ω l. - 17.2 The Penning Trap in Quantum Mechanics. - We note ˆ r and ˆ p the position and momentum operators of the electron. - The Hamiltonian of the electron in the Penning trap is, neglecting spin effects:. - We are now interested in the motion along the z axis. - (a) the expression of the operators ˆ a z and ˆ a † z which allow to write ˆ H z in the form ˆ H z. - hω z ( ˆ N z + 1/2) with ˆ N z = ˆ a † z ˆ a z and [ˆ a z , ˆ a † z. - We now consider the motion in the xy plane under the effect of the Hamiltonian ˆ H xy . - (a) Show that [ˆ a r , ˆ a † r. - (c) Recall the eigenvalues of ˆ n r = ˆ a † r ˆ a r and ˆ n l = ˆ a † l ˆ a l (no proof is required).. - (d) Show that the Hamiltonian ˆ H xy can be written as:. - (e) Deduce from this the eigenvalues of the Hamiltonian ˆ H xy. - We note | ψ(t) the state of the system at time t and we define a r (t. - Integrate these equations and calculate the expectation value of the elec- tron’s position ( x (t), y (t)) in the xy plane. - Show that the time evolution of the expectation value of the electron po- sition r (t) is similar to the classical evolution found in Sect. - We note | φ 0 the eigenstate of ˆ H corresponding to the eigenvalues 0 for each of the operators ˆ n r , ˆ n l and ˆ N z. - (b) Using the same numerical values as in question 17.1.5, evaluate the spa- tial extension of φ 0 ( r. - The experiment is performed at temperatures T ranging between 0.1 K and 4 K. - Compare the characteristic thermal energy k B T to each of the energy quanta of the “cyclotron”, “axial” and “magnetron” motions (associ- ated respectively with ˆ n r , ˆ N z and ˆ n l. - For which of these motions does the discrete nature of the energy spectrum play an important role?
Xem thử không khả dụng, vui lòng xem tại trang nguồn hoặc xem
Tóm tắt