Applied Physics 216 — Assignment #9Final Version

Professor: Donhee HamTeaching Fellow: Jundong Wu

Date: April 21st, 2017 (Problems 1 and 2); April 30th (Problems 3 and 4)Due: 10:20am + 10 min grace period, May 13th, 2017; slide your work under through the door at

Maxwell-Dworkin Room 131.

Problem 1 (200 pt): Laser oscillator linewidth

In the presence of noise (e.g., spontaneous emission), the electric field of a single-mode laser oscillator at agiven space point may be written as

E(t) = E0 cos[ωlt+ φ(t)], (1)

where ωl = 2πfl is one of the modal frequencies of the optical resonator and φ(t) is phase noise. We ignoreamplitude noise, as self-sustained oscillators correct amplitude perturbations to restore original amplitudes.With white noise perturbation, φ(t) becomes a diffusion (random walk) process with such properties as

〈φ(t)〉 = 0; (2)

〈φ(t1)φ(t2)〉 = 2Dmint1, t2, (3)

where D is phase diffusion constant. Assume that φ(t) is a Gaussian process for any given time t. We sawa similar setup in Homework #4 in connection with atomic de-phasing processes, but the physical contexthere is very different, concerning self-sustained laser oscillation whose phase undergoes diffusion due to noise.

(a) Show that the power spectral density of E(t) is given by

S(f) = E20 ·

D/(2π)2

(f − fl)2 + (D/(2π))2(4)

with its linewidth being ∆fosc = D/π. Show also∫∞0S(f)df = E2

0/2, and thus argue that a smaller D (lessnoise) means a sharper and taller power spectral density. Note that D is the single parameter that governsthe entire phase noise dynamics and that determines the laser oscillator linewidth ∆fosc.

(b) Schawlow and Townes calculated D in fundamental limit, and the resulting ∆fosc is (with Lax’s correc-tion)

∆fosc = πhfl(∆fres)

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Pout(5)

where Pout is the laser output power and ∆fres is the linewidth of the optical resonator. This derivationassumes that the optical resonator has no loss except its coupling to the external world to produce Pout (thatis, one mirror has a non-zero transmittance). Show that Eq. (5) can be converted to

∆fosc ∼hflEstored

×∆fres ∼1

Np×∆fres (6)

where Estored is the light energy stored in the resonator, Np is the corresponding number of photons, andthe omitted proportionality constant is more or less on the order of 1. Can you see ∆fosc ∆fres and howsignificant this linewidth compression can get?

(c) Consider a 633-nm He-Ne laser with mirror reflectances of 100% and 97%, a mirror separation of 30 cm,and an output power of 1 mW. Calculate ∆fosc (the laser “oscillator” linewidth) in the fundamental limitand compare it to ∆fi (the laser “transition” linewidth) and ∆fres (optical resonator linewidth).

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Problem 2 (200 pt): Laser modelocking

A total of M x-polarized standing wave modes are oscillating in a laser. Its resonator has a length d alongthe z-axis, a free spectral range fF , and a round trip time TF = f−1F . The characteristic impedance ofthe laser medium is η. The electric field of any one of these oscillating standing wave modes, indexed by l(l = l1, l2, ..., lM ), is along the x-axis and can be written as

El(z, t) = E0,l sin(klz) sin(ωlt+ φl) (7)

where ωl = 2πfF l is the angular frequency, kl is the corresponding wave number, φl is the phase, and E0,l

is the amplitude for the l-th mode. Assume that E0,l is constant E0 regardless of l; moreover, all of the Mmodes are locked in phase according to φl = ∆l for all l = l1, l2, ..., lM where ∆ is a certain constant.

(a) By summing up electric fields of all of the M modes, show that the total electric field is given by

E(z, t) =1

2E0 cos

ωc

(t− z

v

)+ φc

X(t− z

v

)− 1

2E0 cos

ωc

(t+

z

v

)+ φc

X(t+

z

v

)(8)

where ωc and φc are respectively ωl and φl with l being the one in the middle of l1, l2, ..., lM (withoutlosing essence, you can assume that M is an odd number) and the envelope function X is defined as

X(t± z

v

)≡ sin[M πfF (t± z/v) + ∆/2]

sin[πfF (t± z/v) + ∆/2]. (9)

(b) Express the l-th mode’s magnetic field Hl(z, t) (which is along the y axis) in consistency with Eq. (7)and the mode-locking scheme described above, and by summing up these magnetic fields for all of the Mmodes, calculate the total magnetic field H(z, t).

(c) From E(z, t) and H(z, t) obtained in (a) and (b), calculate intensities I+(t, z) and I−(t, z) for the forwardand backward traveling light. I+(t, z) and I−(t, z) must be identical, except their propagation in oppositedirections. Plot the amplitude envelope I+,0(t, z) of I+(t, z) as a function of t for a given z and show thatit represents a pulse train with a temporal pulse interval of TF and a temporal pulse width ∆τ of ∼ TF /M .Also show that pulse peaks—maxima of I+,0(t, z)—are exactly M times larger than the time-domain mean

of I+,0(t, z) for any given z; this will entail proving (1/π)∫ π/2−π/2 sin2(Mx)/ sin2 xdx = M , which can be done,

for instance, via mathematical induction. Overall, if a large number M of modes are locked in phase, theyform a train of tall and sharp pulses of light that go back and forth between the two mirrors of the resonator.

(d) Estimate the temporal pulse width ∆τ for He-Ne laser, dye laser, Nd:Yag laser, Nd:glass laser, andTi:sapphire laser, assuming that these lasers are mode locked. And for each of these examples, estimate howmany light oscillations (defined with ωc) occur within one pulse duration (or equivalently, how many lightwavelengths are there within one spatial pulse width). For data needed for these estimations, you can resortto the table in Assignment #8.

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Problem 3 (300 pt): Fully quantized interaction of an LC resonant mode and a 2-level atom

Figure 1: (a) LC resonator. (b) LC resonator coupled with a 2-level atom.

(a) The LC resonator of Fig. 1(a) has a single resonant mode at an angular frequency ω0 = 1/√LC. Its

classical voltage and current, V and I, are shown in Fig. 1(a). Show that the canonically conjugate voltageand current operators, V and I, should be given by

V =

√~ω0

2C(a+ a†); (10)

I = −i√

~ω0

2L(a− a†), (11)

in order to quantize the LC resonant mode so that its Hamiltonian is HLC−photons = ~ω0(a†a+ 1/2).

(b) Calculate 〈n|V |n〉 and 〈n|V 2|n〉 where |n〉 is an eigenstate of HLC−photons with an eigenenergy ~ω0(n+1/2) (n: non-negative integer).

(c) Assume a classical voltage of V (t) = Vc cos(ω0t) (Vc: real amplitude). By equating the correspondingclassical total resonator energy to n~ω0, relate Vc to the photon number n as follows:

Vc =

√2n~ω0

C. (12)

Check that this (almost) classical result is consistent with the quantum result of part (b), for large enough n.1

(d) Let a 2-level atom of electric dipolar character be placed inside the capacitor [Fig. 1(b)]. The twoatomic states are |α〉 and |β〉, which satisfy H0|α〉 = −(~ω0/2)|α〉 and H0|β〉 = (~ω0/2)|β〉 for the pureatomic Hamiltonian H0. The capacitor consists of two parallel plates separated by distance d. Derive—bygoing through all detailed steps similar to Lecture #23—the following Jaynes-Cummings Hamiltonian forthe interacting atom + LC-photons system:2

H = H0 +HLC−photons +Hint (13)

= −(~ω0/2)σz︸ ︷︷ ︸H0

+ ~ω0a†a︸ ︷︷ ︸

HLC−photons

−~g(aσ+ + a†σ−)︸ ︷︷ ︸Hint

. (14)

Here the coupling strength g is

g =µ

d

√ω0

2~C. (15)

and µ = 〈α|µx|β〉 (x is the direction perpendicular to the parallel plates of the capacitor), which we assumeto be a positive real value without loss of essence; that is, µ is the electric dipole moment of the 2-level atom.

(e) Consider the basis |x, n〉;x = α, β; n = 0, 1, 2, 3, ... (this spans an infinite-dimensional space) to treatthe atomic and LC-photonic states in a combined manner. Show that in this basis H0 and HLC−photons are

1Part (c) concerns amplitude while part (b) concerns mean value, so proper conversion is needed in checking the consistency.2We here drop the zero-point energy in HLC−photons; and we use rotating wave approximation.

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diagonal matrices whereas Hint only connects |α, n + 1〉 and |β, n〉 for any given n. Thus it may suffice toconsider the atom + LC photons system only in a two-dimensional sub-space spanned by the basis |α, n+1〉and |β, n〉 (n is arbitrary). Show that the matrix representation of H in this basis of |α, n+ 1〉 and |β, n〉 is

Hn =

~ω0(n+ 1/2) −~g√n+ 1

−~g√n+ 1 ~ω0(n+ 1/2)

, (16)

where the diagonal elements are due to H0 +HLC−photons while the off-diagonal elements arise from Hint.

(f) Show that when the atom and LC photons do not interact (set g or Hint to zero), |α, n + 1〉 and|β, n〉 are energy eigenstates (“bare states”) of Hn with a degenerate (common or non-split) eigen energyof ~ω0(n + 1/2). In contrast, when the atom and LC photons interact (g is non-zero), two new energyeigenstates, which we call |n+〉 and |n−〉, emerge, with the degeneracy lifted—that is, their eigen energiesnow split. Demonstrate the foregoing statement by directly calculating |n+〉 and |n−〉 (“dressed states”) assuperpositions of |α, n+ 1〉 and |β, n〉 and their eigen energies. Show that the eigen energy splitting is givenby 2~g

√n+ 1.

(g) Assume that the atom-photons interaction is present (g 6= 0). We initially prepare the atom-photons sys-tem in state |β, n〉. Calculate explicitly the temporal evolution of the state, and show that the state oscillatesbetween |β, n〉 and |α, n+1〉 with frequency Ωn = 2g

√n+ 1 corresponding to the energy splitting of part (f).

You should see that this dynamics is nothing but the Rabi oscillation we had studied semi-classically earlier.Calculate the semi-classical Rabi frequency Ωsemi−classical exploiting the result of part (c), and show thatthe fully quantum mechanically derived Rabi frequency Ωn converges to Ωsemi−classical for sufficiently large n.

(h) Rabi oscillation with n = 0 and Rabi frequency Ω0 = 2g is called vacuum Rabi oscillation, which marksa striking difference from the semi-classical Rabi oscillation. In the vacuum Rabi oscillation, the combinedsystem state oscillates between |β, 0〉 and |α, 1〉. During the process of |β, 0〉 → |α, 1〉, the atom spontaneouslyemits 1 photon into the vacuum LC resonant mode (n = 0) to make a downward transition; during thesubsequent process |α, 1〉 → |β, 0〉, the atom absorbs the photon back to make an upward transition; thiscycle repeats at the frequency of Ω0. To clearly observe this vacuum Rabi oscillation, Ω0 = 2g should belarger than the photon decay rate in the LC resonator. Estimate the minimum quality factor3 Q of the LCresonator required to observe the vacuum Rabi oscillation, assuming ω0/(2π) = 10 GHz, C = 100 fF, andd = 1 µm. Feel free to estimate µ using the electron charge and the Bohr radius.

3While Fig. 1 shows a lossless resonator, here in part (h) we assume a practical loss that may be modeled as a resistanceR in parallel with the LC resonator. A well known formula for the Q of the parallel LCR resonator, which you don’t have toderive here but you should be able to, is Q = ω0RC.

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Problem 4 (300 pt): Quantized interaction of an electromagnetic standing wave mode and aspin-1/2

Figure 2: Electromagnetic cavity.

Consider an electromagnetic cavity with volume V [Fig. 2]. The free-space cavity can accommodate aninfinite number of standing wave resonance modes. Of these, we consider a fundamental standing wavemode (angular frequency: ω0) along the z-direction with x-polarized electric field Ex(z, t) and y-polarizedmagnetic field By(z, t). This standing wave mode is formed from the superposition of a forward and a reversetraveling wave propagating along the ±z directions, where the forward and reverse waves are created fromthe boundary reflections at z = 0 and z = d. The non-conducting material at the left of the z = 0 boundaryhas an infinite characteristic impedance (η1 =∞), whereas the material at the right of the z = d boundaryhas a zero characteristic impedance (η2 = 0). These boundary characteristics dictate the detailed natureof the reflections, thus determining the specific spatiotemporal dependence of Ex(z, t) and By(z, t) in thefundamental standing wave mode (you should calculate Ex(z, t) and By(z, t) and their quantized operatorsin order to solve the problems below).

(a) Place a 1H (proton) spin-1/2 with a gyromagnetic ratio γ in the position where By(z, t) is maximallyvibrating (assume a background static magnetic field along the z-direction for a Zeeman splitting of ~ω0).At this position calculate Hint—the one that appears in the Jaynes-Cummings Hamiltonian—that describesthe coupling of the spin-1/2 with photons of the fundamental mode and show that the coupling strengthgspin is given by

gspin =γ

2

√µ0~ω0

V. (17)

Calculate the Rabi frequency Ωn,spin for the combined state oscillation between | ↑, n+ 1〉 and | ↓, n〉. Showthat Ωn,spin converges to the semi-classical Rabi frequency for sufficiently large n (to this end, you can followthe procedure similar to page 10 of Lecture #23).

(b) Alternatively, place a 2-level atom of electric dipolar character (atomic resonance is at ω0) in the positionwhere Ex(z, t) is maximally vibrating. At this position calculate Hint that describes the coupling of the 2-level atom with the photons of the resonance mode. Show that coupling strength ge−dipole is given by

ge−dipole =µ

~

√~ω0

ε0V(18)

where µ is the electric dipole moment value defined in Problem 3. Calculate the Rabi frequency Ωn,e−dipole.

(c) Show that Ωn,spin/Ωn,e−dipole is independent of n and that Ωn,spin/Ωn,e−dipole 1 by direct numericalcalculation (to this end, you need to evalulate µ, for which you can use the electron charge and the Bohrradius). Since this ratio is independent of n, Ω0,spin Ω0,e−dipole. Since the vacuum Rabi frequency (Ωn=0)is directly related to spontaneous emission rate, the calculation in this part (c) shows that the magneticdipole has a far less tendency to spontaneously emit than the electric dipole.

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