The Quantum Mechanics Solver 16

- Express the state | Ψ (t) in terms of the phase states.
- θ k } of the oscillator D .
- Write the state | Ψ (t 0 ) of the system..
- What is the probability to find the value θ k in a measurement of the phase of the “detector” oscillator D.
- After this measurement has been performed, what is the state of the oscillator S ? Describe qualitatively what will happen if one were to choose an interaction time t = t 0.
- 15.5 Solutions.
- Section 15.1: Preliminaries.
- Since the state of the system is | ψ.
- The state of the global system is.
- The probability p j to find the detector in the state | D j is the sum of the probabilities | γ ij | 2.
- After this measurement, the state of the global system S + D is, after the principle of wave packet reduction,.
- For an ideal detector, the probability that the detector is in the state.
- α j | 2 = p(a j ) and the state of the set system + detector, once we know the state of the detector, is | φ j.
- 15.5 Solutions 151.
- Section 15.2: Phase states of the harmonic oscillator 15.2.1.
- Given the definition of the phase states, one has:.
- The phases of the expectation values of x in two phase states | θ m and | θ n differ by an integer multiple 2(m − n)π/(s + 1) of the elementary phase 2π/(s + 1)..
- Section 15.3: The Interaction between the System and the Detector.
- N are eigenstates of the total Hamil- tonian.
- The state of the system at time t is.
- The quantities n and N are constants of the motion..
- Section 15.4: An “Ideal” Measurement 15.4.1.
- Inserting the expansion of the states | N in terms of the phase states, one obtains.
- The probability to find the result θ n by measuring the phase of the detecting oscillator D on this state is p(θ n.
- After this measurement, the state of the oscillator S is simply | n (up to an arbitrary phase factor).
- In the state (15.8), the two systems are perfectly correlated.
- After a measurement of the phase of D , the state of S would be a superposition of states with different numbers of quanta..
- We see that this procedure, which supposes a well defined interaction time interval between the system and the detector, gives the value of the probability p(n.
- In addition, after one has read the result θ n on the detector, one is sure that S is in the state | n , without having to further interact with it (reduction of the wave packet).
- 15.6 Comments 153.
- 15.6 Comments.
- In practice, the case studied here is a simplification of the concrete case where the oscillators S and D are modes of the electromagnetic field.
- In an interferom- eter, where D is a laser beam split in two parts by a semi-transparent mirror, one can let the signal oscillator S interact with one of the beams.
- Next, we shall see how interferences can actually reappear if this information is “erased” by a quantum device..
- In all what follows, the mo- tion of the neutrons in space is treated classically as a uniform linear motion..
- 16.1 Magnetic Resonance.
- The eigenstates of the z component of the neutron spin are noted | n.
- The magnetic moment of the neutron is denoted ˆ µ n = γ n S ˆ n , where γ n is the gyromagnetic ratio and ˆ S n the spin operator of the neutron..
- What are the magnetic energy levels of a neutron in the presence of the field B 0 ? Express the result in terms of ω 0.
- (16.1) Let | ψ n (t.
- be the neutron spin state at time t, and consider a neutron entering the cavity at time t 0.
- ω − ω 0 | ω 1 , and that terms proportional to (ω − ω 0 ) may be neglected in the previous equations..
- (d) Show that the spin state at time t 1 , when the neutron leaves the cavity, can be written as:.
- 16.2 Ramsey Fringes.
- The neutrons are initially in the spin state | n.
- They successively cross two identical cavities of the type described above.
- the role of the detecting atom A is specified in parts 3 and 4.
- 16.2 Ramsey Fringes 157 by 16.1, is applied in both cavities.
- At the end of this setup, one measures the number of outgoing neutrons which have flipped their spin and are in the final state.
- This is done for several values of ω in the vicinity of ω = ω 0.
- At time t 0 , a neutron enters the first cavity in the state | n.
- What is its spin state, and what is the probability to find it in the state | n.
- What is the spin state of the neutron at time t 0.
- Let t 1 be the time when the neutron leaves the second cavity: t 1 − t 0 = t 1 − t 0 .
- Write the transition matrix U(t 0 , t 1 ) in the second cavity..
- Calculate the probability P + of detecting the neutron in the state | n.
- In practice, the velocities of the neutrons have some dispersion around the mean value v.
- This results in a dispersion in the time T to get from one cavity to the other.
- A typical experimental result giving the inten- sity of the outgoing beam in the state | n.
- as a function of the frequency ν = ω/2π of the rotating field B 1 is shown in Fig.
- Intensity of the outgoing beam in the state | n.
- as a function of the frequency ω/2π for a neutron beam with some velocity dispersion.
- (b) In the above experiment, the value of the magnetic field was B T and the distance D = 1.6 m.
- Calculate the magnetic moment of the neutron.
- Evaluate the average velocity v 0 = D/T 0 and the velocity dispersion δv = v 0 τ /T 0 of the neutron beam..
- 16.1 a device which can measure the z component of the neutron spin (the principle of such a detector is presented in the next section).
- of detecting the neutron in the state | n.
- between the two cavities and in the state | n.
- between the cavities and in the state | n.
- 16.3 Detection of the Neutron Spin State.
- be the two eigenstates of the observable ˆ S az .
- After the interaction between the neutron and the atom, one measures the spin of the atom.
- Under certain conditions, as we shall see, one can deduce the spin state of the neutron after this measurement..
- Spin States of the Atom..
- y in the basis.
- We assume that the neutron–atom interaction does not affect the neutron’s trajectory.
- We represent the interaction between the neutron and the atom by a very simple model.
- This interaction is assumed to last a finite time τ during which the neutron–atom interaction Hamiltonian has the form.
- 16.4 A Quantum Eraser 159 16.3.3.
- Suppose the initial state of the system is.
- Calculate the final state of the system | ψ(τ.
- After the neutron–atom interaction described above, one measures the z com- ponent S az of the atom’s spin..
- (b) After this measurement, what prediction can one make about the value of the z component of the neutron spin? Is it necessary to let the neutron interact with another measuring apparatus in order to know S nz once the value of S az is known?.
- 16.4 A Quantum Eraser.
- We have seen above that if one measures the spin state of the neutron between the two cavities, the interference signal disappears.
- In this section, we will show that it is possible to recover an interference if the information left by the neutron on the detecting atom is “erased” by an appropriate measurement..
- A neutron, initially in the spin state | n.
- Immediately after the first cavity, there is a detecting atom of the type discussed above, prepared in the spin state | a : +y .
- By assumption, the spin state of the atom evolves only during the time interval τ when it interacts with the neutron..
- Write the spin state of the neutron–atom system when the neutron is:.
- What is the probability to find the neutron in the state | n.
- At time t 1 , Bob measures the z component of the neutron spin and Alice measures the y component of the atom’s spin.
- In your opinion, which of the following three statements are appro- priate, and for what reasons?.
- (a) When Alice performs a measurement on the atom, Bob sees at once an interference appear in the signal he is measuring on the neutron..
- (c) The experiment corresponds to an interference between two quantum paths for the neutron spin.
- By restoring the initial state of the atom, the measurement done by Alice erases the information concerning which quantum path is followed by the neutron spin, and allows interferences to reappear..
- Alice now measures the component of the atom’s spin along an ar- bitrary axis defined by the unit vector w .
- Show that the contrast of the interferences varies proportionally to | sin η.
- 16.5 Solutions.
- Section 16.1: Magnetic Resonance

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