Physics 566: Quantum Optics I Problem Set 4

Due: Thursday Oct. 3, 2019 Problem 1: Magnetic Resonance: Rabi vs. Ramsey (25 Points) The technique of measuring transition frequencies with magnetic resonance was pioneered by I. I. Rabi in the late 30's. It was modified by Ramsey (his student) about 10 years later, and now serves as the basis for atomic clocks and the SI definition of the second. All precision atomic measurements, including modern atom-interferometers and quantum logic gates in atomic systems, have at their heart a Rabi/Ramsey-type geometry. (i) Rabi resonance geometry. Consider a beam of two-level “spins” with energy splitting

passing through an "interaction zone" of length L, in which they interact with a monochromatic field oscillating at frequency w that drives transitions between and .

(a) Suppose all the spins start in the state , and have a well defined velocity v, chosen such that

, where is the bare Rabi frequency. Plot the probability to be in the excited state ,

as a function of driving frequency . What is the “linewidth” of ? Explain your plot in

terms of the Bloch-sphere. (b) Now suppose the spins have a distribution of velocities characteristic of thermal beams:

, where . Plot vs. L for D=0,

(you may need to do this numerically). At what L is it maximized - explain? Also plot as in (a), as a function of w with . What is the linewidth? Explain in terms of the Bloch-

sphere. (ii) Ramsey separated zone method As you have seen in parts (a)-(b), assuming one can make the velocity spread sufficiently small, the resonance linewidth is limited by the interaction time L/v. This is known as "transit-time broadening" and is a statement of the time-energy uncertainty principle. Unfortunately, if we

ω 0

↓ ↑

↓

ΩL / v = π Ω ↑

P↑ ω P↑

f (v) = 2v04 v

3 exp(−v 2 / v02 ) v0 = 2kBT / m P↑

P↑ L = Lmax

E cos( t)ωL0

L

B⊥ cos ωt( )

make L larger and larger other inhomogeneities, such as the amplitude of the driving field come into play. Ramsey's insight was that one can in fact "break up" the p-pulse given to the atoms into two p/2-pulses in a time t=l/v (i.e. ), separated by no interaction for a time T=L/v.

The free interaction time can then made much longer.

(c) Given a mono-energetic spins with velocity v, internal state , and field at a

detuning so that find:

, , and show that mapping of the state on the Bloch-sphere. (d) Plot as a function of w. Plot also for the case of finite spread in velocity as in

part (b). What is the linewidth? Problem 2: SU(2) Interferometers (25 points) There is a formal equivalence between a Mach-Zender-type optical interferometer and a so-called Ramsey interferometer for any two-level quantum system, which gets its name from the Ramsey separated zone method of Problem 1. We also call this an SU(2) interferometer. (a) Consider the following optical transformation: A symmetric beam splitter with transmission amplitude t and reflection amplitude r.

Ωτ = π / 2

LωLE cos( t)0ωLE cos( t)0

l

B⊥ cos ωt( ) B⊥ cos ωt( )

ψ (0) = ↓z

Δ <<Ω Ωtot = Ω2 + Δ2 ≈ Ω

ψ (τ = l / v) ψ (τ + T = (l + L) / v) ψ (2τ + T = (2 l + L) / v)

P↑(t final = 2τ +T )

a

b

We can encode a qubit in the two orthogonal paths, “a” and “b”, of a photon. We then define the standard basis

i.e., is with one photon in path-a an no photons in path-b, and vice versa for . The transformation on the basis states is

Show that the conditions for this map to be unitary are: , .

Write this map as an equivalent (up to a negligible phase) SU(2) rotation on the Bloch sphere. (b) Show that the transformation in which mode-a gets a phase shift relative to mode-b is an SU(2) rotation. What is the axis and angle of the rotation?

(c) Show how to construct an arbitrary SU(2) rotation with phase shifters and a beam splitter. Now consider a Mach-Zender interferometer The 50-50 beam splitters are thin black lines and the mirrors are thick blue lines. We assume here that the optical path lengths of the two arms of the interferometer are equal. A phase shifter f is placed in the upper arm.

↑z = 1a ,0b , ↓z = 0a ,1b

↑z ↓z

1a ,0b ⇒ t 1a ,0b + r 0a ,1b , 0a ,1b ⇒ t 0a ,1b + r 1a ,0b

t 2 + r 2 = 1 tr* + t*r = 0

φa

b

f

a

b

(d) Show that up to a negligible overall phase, the sequence

beam-splitter à mirrors à phase shift àbeam-splitter is equivalent to the sequence of SU(2) transformations

-x-rotation à -x-rotation à -z-rotation à -x-rotation (e) Given the initial state , sketch the evolution on the Bloch sphere corresponding to this sequence. (f) Show that up to an overall phase (which is negligible), the transformation from the input is

equivalent to the rotation . Using this, calculate the probability to find the state at the output port given that the state was in at the input port.

Problem 3: Inhomogeneous broadening (25 Points) Many important phenomena occur in a time short compared to the “relaxation” time for coherent phenomena (e.g. spontaneous emission lifetime). Historically, these are known as coherent transients. Here are some examples. (a) Free induction decay by inhomogeneous broadening: Consider a macroscopic ensemble of spins in a static magnetic field with a spatially inhomogeneous magnitude. The cloud is extended so that spins see an inhomogeneous distribution of B-fields in z-direction, with probability

If the spins start in a coherent superposition of and , the sample will radiate

magnetic dipole radiation at a mean frequency ( being the gyromagnetic ratio), but the signal will decay much, much faster than the radiative decay rate. The decay arises because of the inhomogeneity -- different local spins will oscillate at different local frequencies. The resulting radiation from the different components eventually getting out of phase and destructively interfering.

(i) Calculate the characteristic decay time, known as , due to inhomogeneity.

π / 2 π φ π / 2

↓z

e− iφ2σ y

1a ,0b1a ,0b

P(B) = 1

2π δB( )2e− (B−B0 )

2

2 δB( )2

↑z ↓z

�

Ω0 = γB0

�

γ

�

T2*

Take MHz, kHz. The characteristic width

is known as the inhomogeneous linewidth. (ii) If the spins all start in , qualitatively describe how to achieve an

approximate p/2-pulse to rotate all spins into the x-y plane (i.e. into a superposition state).

(b) Spin echo: Though inhomogeneous broadening will cause a decay of the ensemble averaged coherence, it is not a truly irreversible process. A procedure for recovering the coherence is known as a “spin echo”. Consider the following pulse sequence. The -pulse about the y-axis acts according to (aiii) to bring all spins onto the x-axis of the Bloch sphere. For a time , the spins dephase. The -pulse about the x-axis acts to time reverse the process. An “echo” signal will be seen at a time later. Explain this process using this Bloch sphere. Sketch the signal one would detect of the radiated fields in rf coils. (c) A spin-echo sequence is often used in a two-level Ramsey interferometer to make it more robust to inhomogeneities. Consider the following sequence

y-rotation àFree evolution for time T/2 à x-rotation à Free evolution for time T/2 à y-rotation Explain the trajectory of the Bloch sphere. To what kind of inhomogeneities is this sequence this robust? This is a direct analogy to a Mach-Zender interferometer studied in problem 2. Given what you said about the Ramsey interferometer, to what kinds of inhomogeneity is the Mach-Zender interferometer robust?

γ B0 / 2π = 1 γ δB( ) / 2π = 10

�

1/T2*

↑z

�

π /2

�

τ

�

π

�

τ

π / 2 ππ / 2

x y

Problem 4: Ramsey fringes and the measurement of T2 times (25 Points) (a) We seek to measure the coherence of between the computational basis states of a qubit . Consider a two-pulse Ramsey sequence: A “hard” pulse around x-axis with detuning D, free evolution for a time T, a second hard pulse around x-axis at the same detuning D.

During the free evolution, the coherence decays exponentially with rate . Show that, given the qubit initially in , the probability to find after the sequence is

Explain this using the evolution on the Bloch sphere. Plot this for MHz and T2=25 µs, for T=0 to 25 µs. (b) Suppose now that in addition to homogeneous decay, there is inhomogeneous decay

. Suppose that if the pulses are tuned to frequency , the probability the detuning

seen by the qubit is Gaussian distributed, , where is the mean detuning and is the spread in detunings. Calculate the probability in the same two-pulse Ramsey sequence of part (a) for µs. Comment on the result. (c) Now consider a three-pulse Hahn spin-echo Ramsey sequence: A “hard” p/2 pulse around y-axis with detuning D, free evolution for a time a “time reverse” hard pulse around x, free evolution for a time , and then a second hard pulse around y-axis at the same detuning D.

Show, , and plot for , as a

function of TB=0 to 25 µs.

0 , 1{ } π / 2

π / 2Fortschr. Phys. 51, No. 4–5 (2003) 467

∆t

t1

π/2π

τ

π/2

0.0 0.2 0.4 0.6 0.8

32

34

36

p (%

)p

(%)

p (%

)p

(%)

p (%

)p

(%)

Ramsey

∆t (µs)

33

34

35

∆t = 0.39 µs

33

34

35

∆t = 0.59 µs

33

34

35

∆t = 0.79 µs

33

34

35

∆t = 0.99 µs

0.0 0.2 0.4 0.6 0.8

33

34

35

∆t = 1.19 µs

t (µs)1

α = 2π ∆ν. t 1α

αε

x

y

z

ε

t2

∆t

π/2π/2

ε = 2π ∆ν. ( t - t ) 12

Fig. 7 Ramsey fringes measured at Ng = 0 .52 , φ = 0 and ∆ν = 41 MHz. The decay time constant of the fringesis here Tϕ ∼ 30 ns. Lower panels : echo signals obtained with the pulse sequence schematically described on the rightside, for various sequence durations ∆t. A first π/ 2 pulse brings the spin s⃗ on the −y axis. Follows a free precession byan angle α = 2π∆ν t1 during a time t1. A subsequent π pulse brings the spin in the symmetric position with respect tox axis. Follows a second free precession during time t2, which brings the spin at an angle ε = 2π∆ν(t2 − t1) with they axis. The last π/ 2 pulse results in a final z component of the spin equal to cos ε. The average switching probabilityp = (1 − ⟨cos ε⟩)/ 2 , obtained by repeating the sequence, is an oscillating function of t2 − t1. The amplitude of theoscillations is damped away from t1 = t2 (thick tick in each panel) due to fluctuations of ∆ν.

the echo signal varies as (1−⟨cos [2π∆ν(t2 − t1)]⟩)/ 2 and is therefore less sensitive to fluctuations of ∆νfrom sequence to sequence when t1 ∼ t2. In the experiment, we have recorded the switching probabilityat fixed values of ∆t, as a function of the delay t1 (left panels of Fig. 7). Up to ∆t ≃ 1 µs, fringes emergearound t1 = t2 = (∆t − τ)/ 2 (here, τ ∼ 15 ns), indicating that during pulse sequences of this duration,coherence was at least partly conserved. As expected, the period of the oscillations is twice as short in theecho experiment than in the Ramsey experiment. The observation of spin echoes at time scales much largerthan the decay time of the Ramsey fringes indicates that in this situation decoherence was essentially due tocharge fluctuations at frequencies lower than 1/ ∆t ≈ 1 MHz. No echo was seen in experiments performedat φ ̸= 0, suggesting that the relevant phase noise was at higher frequencies.

In all our time domain experiments, the oscillation period of the switching probability closely agreeswith theory, meaning a precise control of the preparation of s⃗ and of its evolution. However, the amplitudeof the oscillations is smaller than expected by a factor of three to four. This loss of contrast is likely to bedue to a relaxation of the level population during the measurement itself. In principle the current pulse,whose rise time is 50 ns, is sufficiently adiabatic not to induce transitions directly between the two levels.

ρ01 1/T21 0

P0 =121+ cos(ΔT )e−T /T2⎡⎣ ⎤⎦

Δ / 2π = 1

T2* ω

p(Δ) = e− (Δ−Δ0 )

2

2δ 2 / 2πδ 2 Δ0

δ = 1/T *2 P0

T2* = 5

TAπ TB π / 2

Fortschr. Phys. 51, No. 4–5 (2003) 467

∆t

t1

π/2π

τ

π/2

0.0 0.2 0.4 0.6 0.8

32

34

36

p (%

)p

(%)

p (%

)p

(%)

p (%

)p

(%)

Ramsey

∆t (µs)

33

34

35

∆t = 0.39 µs

33

34

35

∆t = 0.59 µs

33

34

35

∆t = 0.79 µs

33

34

35

∆t = 0.99 µs

0.0 0.2 0.4 0.6 0.8

33

34

35

∆t = 1.19 µs

t (µs)1

α = 2π ∆ν. t 1α

αε

x

y

z

ε

t2

∆t

π/2π/2

ε = 2π ∆ν. ( t - t ) 12

Fig. 7 Ramsey fringes measured at Ng = 0 .52 , φ = 0 and ∆ν = 41 MHz. The decay time constant of the fringesis here Tϕ ∼ 30 ns. Lower panels : echo signals obtained with the pulse sequence schematically described on the rightside, for various sequence durations ∆t. A first π/ 2 pulse brings the spin s⃗ on the −y axis. Follows a free precession byan angle α = 2π∆ν t1 during a time t1. A subsequent π pulse brings the spin in the symmetric position with respect tox axis. Follows a second free precession during time t2, which brings the spin at an angle ε = 2π∆ν(t2 − t1) with they axis. The last π/ 2 pulse results in a final z component of the spin equal to cos ε. The average switching probabilityp = (1 − ⟨cos ε⟩)/ 2 , obtained by repeating the sequence, is an oscillating function of t2 − t1. The amplitude of theoscillations is damped away from t1 = t2 (thick tick in each panel) due to fluctuations of ∆ν.

the echo signal varies as (1−⟨cos [2π∆ν(t2 − t1)]⟩)/ 2 and is therefore less sensitive to fluctuations of ∆νfrom sequence to sequence when t1 ∼ t2. In the experiment, we have recorded the switching probabilityat fixed values of ∆t, as a function of the delay t1 (left panels of Fig. 7). Up to ∆t ≃ 1 µs, fringes emergearound t1 = t2 = (∆t − τ)/ 2 (here, τ ∼ 15 ns), indicating that during pulse sequences of this duration,coherence was at least partly conserved. As expected, the period of the oscillations is twice as short in theecho experiment than in the Ramsey experiment. The observation of spin echoes at time scales much largerthan the decay time of the Ramsey fringes indicates that in this situation decoherence was essentially due tocharge fluctuations at frequencies lower than 1/ ∆t ≈ 1 MHz. No echo was seen in experiments performedat φ ̸= 0, suggesting that the relevant phase noise was at higher frequencies.

In all our time domain experiments, the oscillation period of the switching probability closely agreeswith theory, meaning a precise control of the preparation of s⃗ and of its evolution. However, the amplitudeof the oscillations is smaller than expected by a factor of three to four. This loss of contrast is likely to bedue to a relaxation of the level population during the measurement itself. In principle the current pulse,whose rise time is 50 ns, is sufficiently adiabatic not to induce transitions directly between the two levels.

P0 =121− cos Δ0 (TA −TB )[ ]e−δ 2 (TA−TB )2 /2e−(TA+TB )/T2( ) TA = 10µs

T

TA TB