5 Quantum Theory of Radiation

Exercise 5.1: Transverse and longitudinal vector ﬁelds

Consider a vector ﬁeld E(r) deﬁned on a volume of size L3 with periodic boundary conditions.

(a) Show that E(r) can be written the sum E(r) = E⊥(r)+ E(r), where the transverse and longitudinal components satisfy

∂r E (r)=0, ∂r E (r)=0. ⊥ ×

(b) Show that E⊥(r) and E(r) are uniquely deﬁned, up to a uniform r-independent term.

Exercise 5.2: Transverse and longitudinal vector ﬁelds (2)

(a) Show that the curl of any vector field is a transverse vector field. (b) Show that the gradient of any scalar field is a longitudinal vector field. (c) Show that a linear combination of two longitudinal vector fields is a longitudinal vector field, and show that a linear combination of two transverse vector fields is a transverse vector field.

Exercise 5.3: Classical electromagnetism

(a) Show that the total momentum of particles and electromagnetic ﬁelds can be written as P = pα + P⊥, (1) α with q p = m v + α A (r ) (2) α α α c ⊥ α and 1 3 P = dr A (r)∂ A˙ (r) . (3) ⊥ 4πc2 ⊥,i r ⊥,i i=1 1 (b) Show that the total energy of particles and electromagnetic ﬁelds can be written as 1 1 q q H = p q A(r )/c 2 + α β + H (4) 2m | α − α α | 2 r r ⊥ α α α β α=β | − | with

2 1 2 1 H = dr ∂rA(r) + A˙ (r) . (5) ⊥ 8π | | c2 (c) Show that the total angular momentum of particles and electromagnetic ﬁelds can be written as J = r p + J + J , (6) × α ⊥,i ⊥,o α with 1 J = drE A, (7) ⊥,i 4πc ⊥ × 1 3 J = dr E (r ∂r)A . (8) ⊥,o 4πc ⊥,j × ⊥,j j=1

Exercise 5.4: Linear polarization vectors

The linear polarization vectors ǫk,1 and ǫk,2 are unit vectors, such that the three vectors ǫk,1, ǫk,2, and ek = k/k form a right-handed orthonormal set. (a) Show that 2 (k ǫk ) (k ǫk ′ )= k δ ′ . × ,λ × ,λ λλ (b) Show that ǫk (k ǫk ′ )= kδ ′ . ,λ × × ,λ λ,λ (c) Show that kikj (ǫk ) (ǫk ) = (ǫk ) (ǫk ) = δ . ,λ i ,λ j ,λ i ,λ j ij − k2 λ λ

2 Exercise 5.5: Quantized electric and magnetic ﬁelds in waveguide

Consider the electromagnetic ﬁeld in a rectangular waveguide with cross section of size Wy Wz with Wy > Wz. The walls of the waveguide are perfectly conducting. In the direction× along the waveguide, we apply periodic boundary conditions with period L. The Maxwell equations allow solutions for the ﬁelds inside the waveguide only at discrete values of the transverse wavenumbers ky and kz. The lowest-frequency modes appear at ky = π/Wy and kz = 0 and have vector potential of the form Φk(r)cos(ωkt + θ), where θ is an arbitrary phase shift, the longitudinal wavenumber k takes the values 2πn/L with n =0, 1, 2,..., the “mode function” ± ± 1 if k =0, 4 πy 2 Φk(r) = sin ez  cos( kx) if k > 0, WyL Wy ×  sin(kx) if k < 0,

and  2 2 ωk = c k +(π/Wy) . In this exercise, no other modes than these lowest-frequency modes need to be considered. In general, the vector potential A(r,t) will be a superposition of the mode function given above,

A(r,t)= Ak(t)Φk(r), (9) k where the coefficients Ak are real numbers. If one defines the “canonical position” Qk and “canonical momentum” Pk as 1 1 Qk = Ak, Pk = A˙ k, √4πc2 √4πc2 the Hamilton function for the lowest-modes fields in the waveguide reads 1 H = (P 2 + ω2Q2). 2 k k k k (a) Verify that this Hamilton function reproduces the correct equation of motion for the canonical positions Qk and the canonical momenta Pk. Hint: use the explicit time dependence of the modes of the vector potential given in the text above.

3 (b) In the method of canonical quantization, the canonical positions Qk and the canonical momenta Pk are replaced by operators Qˆk and Pˆk. What are the fundamental algebraic relations between these operators?

(c) Instead of using operators Qk and Pk, it is more convenient to use “creation operators” and “annihilation operators” deﬁned as

1 ˆ ˆ † 1 ˆ ˆ aˆk = (ωkQk + iPk), aˆk = (ωkQk iPk). √2~ωk √2~ωk − What are the fundamental algebraic relations of these operators? ˆ † (d) Derive an expression for the vector potential A(r) in terms of the operatorsa ˆk anda ˆk. (e) The vector potential A(r) can be written in the form

πy 4πc2 † A r e ˆ ikx ˆ −ikx ( ) = sin z ~ (a˜ke + a˜ke ), (10) Wy ωkWyL k ˆ ˆ† where the operators a˜k and a˜k satisfy the fundamental algebraic relations of annihilation and creation operators, ˆ ˆ ˆ† ˆ† ˆ ˆ† [a˜k′ , a˜k]=0, [a˜k′ , a˜k]=0, [a˜k′ , a˜k]= δk′k. Show that such an identity follows from the quantum theory you constructed in parts (b)–(d) and derive an expression for the ﬁeld energy in this formulation. (f) Bonus question. Instead of ﬁrst expanding the vector potential A(r) in terms of real-valued basis functions Φk, as was done in Eq. (9), and then transforming to a basis of complex expo- nentials on the level of the quantum theory, as was done in part (e), one may directly formulate the theory in terms of a Fourier transform of the vector potential A(r),

2 πy ikx A(r,t)= sin A˜k(t)e . (11) WyL Wy k This expansion is the classical analogue of the expansion (10) of the quantum vector potential Aˆ(r) you constructed in part (e). Express the classical canonical positions ˜ ˜ ˆ ˆ† and momenta Qk and Pk that correspond to the operators a˜k and a˜k of Eq. (10) in terms of the amplitudes A˜k appearing in the Fourier transform (11) of the classical vector potential A(r).

4 Exercise 5.6: Quantum Electrodynamics in a cavity

In this exercise you develop a quantum theory of electromagnetic ﬁelds in a cavity with perfectly conducting walls. (In optical language, one would refer to the cavity walls as “mirrors”.) Starting point is the expression for the combined energy of particles and ﬁelds,

H = Hmat + H + H⊥, (12) with 1 q 2 H = p α A(r ) , (13) mat 2m α − c α α α 1 qαqβ H = , (14) 2 rα rβ α=β | − | 2 1 2 1 H = dr ∂r A(r) + A˙ (r) . (15) ⊥ 8π | × | c2 In order to arrive at a quantum theory for the electromagneti c fields in the cavity, we first have to introduce “canonical positions” and “canonical momenta” for the fields in the cavity. Hereto, the transverse vector potential is decomposed as

A⊥(r)= A⊥,nΦn(r), (16) n where Φn(r) is a real vector-valued function normalized transverse solution of

ω2 ∂2Φ (r)+ n Φ (r)=0, (17) r n c2 n with the boundary condition that Φn(r) is perpendicular to the boundary of the cavity at the cavity boundary. The set of such vector-valued functions Φn form a complete and orthonormal basis for all transverse vector ﬁelds in the cavity,

drΦ ′ (r) Φ (r)= δ ′ . (18) n n n n

5 (a) Argue that the boundary condition that Φn(r) is perpendicular to the cavity boundary is consistent with the boundary conditions for the electric and magnetic ﬁelds at a perfectly conducting cavity wall.

(b) What is the equation of motion for the ﬁeld component A⊥,n? (c) If one deﬁnes the canonical position and momentum as

A⊥,n A˙ ⊥,n Qn = , Pn = , (19) √4πc2 √4πc2 show that the Hamilton function for the transverse ﬁelds reads 1 H = P 2 + ω2Q2 . (20) ⊥ 2 n n n n

(d) Verify that this Hamilton function, together with Hmat and H, reproduces the correct equations of motion for particles and ﬁelds in the cavity.

(e) Now construct a quantum theory of particles and fields in the cavity using the method of canonical quantization. Derive expressions for the electric and magnetic fields in the † cavity in terms of appropriately defined raising and lowering operatorsa ˆn anda ˆn.

Exercise 5.7: Vacuum ﬂuctuations

In the vacuum state, the expectation values of the transverse electric field E⊥ and the magnetic field B are zero. However, the fields have nonzero fluctuations.

2 2 (a) Express the expectation values of E⊥ and B in the vacuum state as an integration over the frequency ω of the ﬁeld modes. Does your integral converge? At what frequency does the theory break down?

(b) Calculate the autocorrelation function of the electric ﬁeld,

G (τ)= 0 Eˆ (r,t + τ)Eˆ (t)(r,t) 0 . (21) ij { }| ⊥,i ⊥,j |{ } Again, express your result as an integral over the frequency ω of the ﬁeld modes.

6 Exercise 5.8: Squeezed states

The uncertainty of the electric and magnetic field components of an electromagnetic wave is often characterized with the help of the quadratures. The quadratures Xk,λ and Yk,λ are observables which represent the real and imaginary parts of the complex field amplitudes ak,λ. In the quantum theory, the corresponding operators are defined as

1 † 1 † Xˆk,λ = (âk,λ +â ), Yˆk,λ = (âk,λ aˆ ). (22) √2 k,λ i√2 − k,λ Unlike the creation and annihilation operators, the quadratures are hermitian operators, so that we can speak about their expectation values and their fluctuations. In this exercise we will consider a single mode only and drop the indices k and λ. In the vacuum state, the expectation values X and Y both vanish, whereas the uncertainties ∆X and ∆Y are given by 1 ∆X =∆Y = . (23) √2 (a) Verify Eq. (23). The so-called “squeezed states” are states of the radiation field where the uncertainties ∆X or ∆Y (but not both!) of the quadrature components is smaller than in the vacuum state. Squeezed states can be generated with light fields using methods of nonlinear optics. Because of their reduced uncertainty in comparison with the vacuum state, they find application (among others) in high-precision measurement schemes. Theoretically, squeezed states can be generated from the vacuum state 0 by application of the “squeeze operator” Sˆ(ξ), | 1 ∗ 2 †2 Sˆ(ξ)= e 2 (ξ aˆ −ξaˆ ), (24) where ξ is a complex number. (b) Show that Sˆ(ξ) is a unitary operator. Also show that Sˆ(ξ)† = Sˆ( ξ). − The complex number ξ is written as ξ = ρeiθ. We introduce “rotated” raising and lowering operators as † a˜ˆ =âe−iθ/2, a˜ˆ =â†eiθ/2. (25) Similarly, rotated quadrature operators are defined as

1 † 1 a˜ˆ = (Xˆ˜ + iY˜ˆ ), a˜ˆ = (Xˆ˜ iY˜ˆ ). (26) √2 √2 −

7 † (c) Show that the rotated raising and lowering operators a˜ˆ and a˜ˆ satisfy the same commutation relations as the original operatorsa ˆ† anda ˆ.

(d) Show that the rotated quadrature operators are obtained from the original operators by a rotation in the X-Y plane,

Xˆ˜ = Xˆ cos(θ/2) + Yˆ sin(θ/2), Y˜ˆ = Yˆ cos(θ/2) Xˆ sin(θ/2). (27) − (e) Show that Sˆ(ξ)†Xˆ˜Sˆ(ξ)= Xeˆ˜ −ρ, Sˆ(ξ)†Y˜ˆ Sˆ(ξ)= Ye˜ˆ ρ. (28)

(f) Calculate the expectation values of X˜ and Y˜ and calculate their ﬂuctuations ∆X˜ and ∆Y˜ in the “squeezed vacuum state” Sˆ(ξ) 0 . How do the uncertainties in X˜ and Y˜ compare with the vacuum state? Does this| state violate Heisenberg’s uncertainty principle? Can you explain why these states are called “squeezed”?

(g) Describe the time dependence of the squeezed vacuum state Sˆ(ξ) 0 . |

Exercise 5.9: Squeezed number states

This exercise is a continuation of Ex. 5.8. In that exercise you have constructed a “squeezed vacuum state” in which the ﬂuctuations of the quadrature components Xk,λ or Yk,λ are less than in the vacuum state. The squeezed vacuum state was obtained by acting a “squeeze operator” Sˆ(ξ) on the vacuum state 0 . In this exercise (as in Ex. 5.8) we will consider a single mode only and drop the indices| k and λ. In this exercise you construct a set of “squeezed number states” that bear the same relation to the squeezed vacuum as the standard number states do to the standard vacuum.

(a) Show that the operators ˆb = Sˆ(ξ)ˆaSˆ(ξ)† and ˆb† = Sˆ(ξ)ˆa†Sˆ(ξ)† satisfy the commutation relations characteristic of lowering and raising operators.

(b) Show that ˆb annihilates the “squeezed vacuum” you constructed in Ex. 5.8.

(c) Show that the “squeezed number states”, which are obtained by repeated application of the “squeezed raising operator” ˆb† on the squeezed vacuum state are the same states as the ones that one obtains by acting the squeeze operator S(ξ) on a regular number state n . | 8 (d) What is the expectation value of the number of (regular) photons in a squeezed number state?

Exercise 5.10: Squeezed coherent states

This exercise is a continuation of Exs. 5.8 and 5.9. In Ex. 5.8 you have constructed a “squeezed vacuum state” in which the ﬂuctuations of the quadrature components Xk,λ or Yk,λ are less than in the vacuum state. In Ex. 5.9 you constructed “squeezed number states”. In both cases, the squeezed states were obtained from the regular vacuum by acting a “squeeze operator” Sˆ(ξ). Here (as in Exs. 5.8 and 5.9) we will consider a single mode only and drop the indices k and λ. In this exercise you construct a set of “squeezed coherent states” that bear the same relation to the squeezed vacuum as the standard coherent states do to the standard vacuum.

(a) A squeezed coherent state can be obtained by acting the squeeze operator Sˆ(ξ) on a regular coherent state z . Find an expression for the state Sˆ(ξ) z in terms of the squeezed number states.| |

(b) What is the expectation value of the photon number in the squeezed coherent state Sˆ(ξ) z ? | (c) Describe the time dependence of the squeezed coherent state Sˆ(ξ) z . | (d) Consider a coherent state z for which z = x is real at t = 0. In this state, the quadratures X and Y ﬂuctuate| equally around X = x and Y = 0. What are the ﬂuctuations of X and Y for the squeezed version of this state? What is its time dependence? Depending on the choice of the phase of the squeeze parameter ξ one refers to the squeeze as “amplitude squeezing” or “phase squeezing”. Can you explain this??

Exercise 5.11: Electric ﬁeld for a squeezed state

9 (a) Evaluate the expectation value E⊥(r,t) of the electric ﬁeld in the squeezed coherent state Sˆ(ξ) z , where the state z is a coherent state for a single ﬁeld mode with wavenumber| k and linear polarization.|

(b) Evaluate the expectation value E⊥(r,t) of the electric field in the squeezed coherent state Sˆ(ξ) z , where the state z is a coherent state for a single field mode with wavenumber| k and circular polarization.| (c) Repeat questions (a) and (b) for the contribution of the mode to the electric field variance ∆E (r,t)2 . ⊥

Exercise 5.12: Translation operator

The operator Tˆ(s) displaces a quantum state by a displacement vector s. (a) For an observable O(r) that depends on position, show that Tˆ( s)Oˆ(r)Tˆ(s)= Oˆ(r s). − − (b) Show that Tˆ(s) = e−iPˆ fields/~ translates the photon field by a displacement s, where ˆ ~ † Pfield = k,λ kaˆk,λaˆk,λ is the operator for the total momentum of the fields. Hint: Show that the equality Tˆ( s)Oˆ(r)Tˆ(s) = Oˆ(r s) is satisfied for Tˆ(s) = −iPˆ fields/~ − − e and Oˆ(r)= Aˆ ⊥(r) or Oˆ(r)= Eˆ ⊥(r).

Exercise 5.13: Repulsive Casimir force

Zero point fluctuations of the electromagnetic field give an attractive force between two parallel perfectly conducting plates. In this exercise, you consider the force from zero point fluctuations of the electromagnetic field between two parallel plates, one of which is a perfect conductor (dielectric constant ǫ ) and one of which has perfect magnetic permeability (magnetic permeability ).→ ∞ We choose coordinates,→ such ∞ that the z axis is perpendicular to the two plates. The distance between the two plates is a. For the x and y directions you may use periodic boundary conditions, with period L a. ≫ 10 (a) The boundary condition at the perfectly conducting is Ex = Ey = 0. The boundary condition at the perfectly permeable plate is Bx = By = 0. Show that the allowed modes of the electromagnetic field between the two plates have wavenumber 1 π k = n + , z 2 a where n =0, 1, 2,....

in (b) Derive an expression for the zero point energy E0 (a) from the ﬁelds between the two plates.

out (c) Calculate the zero point energy E0 (a) from the ﬁeld outside the plates. in out (d) Express the a-dependent part of the total zero point energy E0(a)= E0 (a)+ E0 (a) as an integral and/or a sum over the frequency ω of the radiation modes. (e) In part (d) you should have found an expression that is formally divergent. The divergence can be removed by inserting a factor e−ηω in the integrand/summand, where η is a positive inﬁnitesimal. Find an expression for the total zero point energy as a power series in η. (f) Now calculate the force between the plates and take the limit η 0. What is the sign of the force? →

Exercise 5.14: Dipole approximation

In the dipole approximation one replaces the matter-radiation interaction by its leading contribution Hˆ1 and neglects the position dependence of the vector potential A(r). In this approximation, the radiation-matter interaction takes the form q Hˆ α pˆ Aˆ (Rˆ ), int ≈− m α ⊥ α α where α labels the particles, qα and mα are the charge and the mass of each particle, and R is the center-of-mass coordinate. In the lecture this expression for Hînt was used to calculate the Golden-Rule spontaneous transition rate from the initial state i to the final state f under emission of a photon in the solid angle element dΩ. The result| is | αc dΓsp. em. = k3 f D ǫ∗ i 2dΩ, (29) i→f 2πe2 | | k,λ| | 11 ˆ where D = α qαˆrα is the electric dipole moment. As an alternative, one may first perform a gauge transformation, after which the interaction Hamiltonian takes the form (again in the dipole approximation)

Hˆ Dˆ E (R). (30) int ≈− ⊥ Use Eq. (30) to calculate the Golden Rule transition rate and compare your answer to Eq. (29). Which calculation is easier?

Exercise 5.15: Magnetic dipole radiation

Interactions between matter (particles) and the radiation ﬁelds are described by the Hamil- tonian Hˆint = Hˆ1 + Hˆ2 + Hˆ3, with q Hˆ = α pˆ Aˆ (ˆr ), 1 − cm α ⊥ α α α Hˆ = ˆ Bˆ (ˆr ), 2 − α α α α q2 Hˆ = α Aˆ (ˆr ) 2. 3 2m c2 | ⊥ α | α α

The transverse vector potential Aˆ ⊥(r) and the magnetic ﬁeld Bˆ (r) are expressed in terms of photon creation and annihilation operators as

2π~c2 Aˆ r ǫ ikr † ǫ∗ −ikr ⊥( ) = 3 aˆk,λ k,λe +ˆak,λ k,λe , ωk L k ,λ ,λ 2π~c2 Bˆ r k ǫ ikr † k ǫ∗ −ikr ( ) = i 3 aˆk,λ( k,λ)e aˆk,λ( k,λ)e . ωk L × − × k ,λ ,λ In the dipole approximation, one replaces the exponent e±ikr by unity. The transition amplitudes following from the perturbation Hˆ1 are then found to be proportional to a matrix element of the electric dipole moment between the initial matter state i and ﬁnal matter state f . If the electric dipole matrix element vanishes, one has to consider| the perturbation | Hˆ2 or higher orders in a small-k expansion of Hˆ1 and Hˆ2. In this exercise you analyze the

12 next-to-leading order term in the small-k expansion of Hˆ1 and the leading-order radiation from Hˆ2.

(a) Find an expression for the matrix element f, 1kλ Hˆ2 i, 0 between a ﬁnal state consist- ing of the matter state f and one photon in mode| | (k,λ ) and the initial matter state i without photons. | |

The interaction Hˆ2 couples to the intrinsic magnetic moment of the particles only. The intrinsic magnetic moment is the moment associated with spin angular momentum. Hence, the matrix element you calculated in (a) does not involve the total magnetic moment of the particles. The orbital component has to come from H1, which describes the interaction of the charge degrees of freedom with the radiation ﬁelds.

±ikr (b) Expand the exponents e in the expressions for Aˆ ⊥ and Bˆ to ﬁrst order in k and calculate the corresponding contribution to the matrix element f, 1k Hˆ i, 0 . Argue λ| 1| that it involves both a contribution from the particles’ orbital magnetic moment and a contribution from the electric quadrupole moment.

(c) Combine the contribution from the orbital magnetic moment with your answer in (a) to ﬁnd an expression for the magnetic dipole transition rate. How does this rate scale with the frequency ω of the emitted photon?

(d) What is the frequency dependence of the electric quadrupole transition rate? What are the corresponding selection rules? Can one have magnetic dipole radiation and electric quadrupole radiation at the same time?

Exercise 5.16: The A 2 perturbation | | The interaction Hamiltonian between matter and radiation consists of three contributions, Hˆ1, Hˆ2, and Hˆ3, see Ex. 5.15. When calculating the mass renormalization and the Lamb shift, we neglected the contribution from Hˆ3. Why is this justiﬁed?

Exercise 5.17: Metastable Hydrogen 2s

The Hydrogen 2s state can not decay to the Hydrogen 1s ground state via single photon decay, because such decay would violate the selection rule m + λ = m , where λ = 1 is f i ± 13 the helicity of the photon and mf and mi are angular momentum quantum numbers of the initial and ﬁnal states. In this exercise you investigate alternative decay mechanisms.

(a) The 2s state can decay to the 2p state. The latter has a slightly lower energy because of the Lamb shift, and the transition is allowed by electric dipole selection rules. Find an estimate for the lifetime of the 2s state because of this decay mode. Is your lifetime realistic? Remark: You can answer this exercise without knowledge of the microscopic origin of the Lamb shift. (b) An alternative decay channel is via two-photon decay: The 2s state decays to the 1s state by emission of two photons of opposite helicity. Argue that such a process does not violate selection rules. (c) In general, the two-photon decay rate between an initial matter state i and a final state f can be characterized by a Golden Rule decay rate | | dΓ =2π T 2dρ dρ dω , fi | fi| 1 2 1 where ω and ω are the frequencies of the emitted photons, ~ω + ~ω = E E , 1 2 1 2 i − f k2 dρ = 1 dΩ . 1 (2π)3~c 1

Find an expression for the matrix element Tﬁ. (d) Now estimate the resulting lifetime of the Hydrogen 2s state. In your estimate, you may use the dipole approximation for all matrix elements and you may restrict any summation over intermediate states to the Hydrogen 2p state. (The latter approximation is not very accurate, though!)

The analysis of two photon decay goes back to Maria Göppert-Mayer, who published her result in 1931. The first precise calculation of the Hydrogen 2s lifetime took until 1959, see J. Shapiro and G. Breit, Phys. Rev. 113, 179 (1959). The main reason why it took so long was because of the difficult summation over intermediate states. A direct measurement of the 2s lifetime has been possible in ultracold magnetically trapped Hydrogen, see C. L. Cesar et al., Phys. Rev. Lett. 77, 225 (1996).

Exercise 5.18: Radiative transitions in Hydrogen

14 In the dipole approximation, the interaction between an H atom and the radiation field is described by the interaction Hamiltonian Hˆ = Eˆ Dˆ , (31) int − where Dˆ = eˆr is the dipole operator for the electron (charge e) and Eˆ the electric field. − − (a) Show that the spontaneous decay rate of an excited state e = nl with energy ε = ε | | e nl to the Hydrogen ground state g = 1s , with energy εg = ε1s, to leading order in Hînt, is given by the expression | | 4π2 Γsp = ω g Dˆ e ǫ∗ 2δ(ε ε ~ω ). e→g L3 k| | | kλ| e − g − k k λ sp (b) Calculate the spontaneous decay rate Γ2p→1s of the Hydrogen 2p state. (c) At a finite temperature T , the number of photons present in a given mode is given by the Planck law (or, equivalently, by the Bose-Einstein distribution function at zero chemical potential ) 1 nkλ = , (32) e~ωk/kBT 1 − where kB is the Boltzmann constant. The spontaneous decay rate Γsp of parts (a) and (b) describes limit T 0, → sp Γ2p→1s =Γ2p→1s(T = 0). sp Give an expression for the finite-temperature decay rate Γ2p→1s(T ) in terms of Γ2p→1s. It is sufficient to write down the final answer to this item; no derivation is needed.

(d) Express the rate Γ1s→2p(T ) at which the inverse transition takes place because of ab- sp sorption of photons in terms of Γ2p→1s. It is sufficient to write down the final answer to this item; no derivation is needed. (e) Use your answers to (c) and (d) to find a relation between the relative probabilities that the H atom is found in the 1s and 2p states at temperature T .

Exercise 5.19: Radiative decay of trapped atoms

15 Consider a point particle of charge e and mass m in an anisotropic three-dimensional harmonic oscillator potential. The Hamiltonian is

pˆ2 1 Hˆ = + m(ω2x2 + ω′2y2 + ω′2z2). 2m 2 The harmonic-oscillator states for the particle have energy

′ E(nx,ny,nz)= ~ω(nx +1/2) + ~ω (ny + nz + 1),

′ where nx,y,z =0, 1, 2,.... The trap frequency in the y and z directions, ω , is much larger than ω, the trap frequency in the x direction, so we may restrict ourselves to the case ny = nz =0 for a description of the lowest-lying excited states. In this exercise, we will use the symbol n to denote the eigenstate with nx = n, ny = nz = 0. |The particle is coupled to a radiation ﬁeld. In the dipole approximation, the interaction Hamiltonian reads Hˆ = eˆr Eˆ(0), int − where electric ﬁeld Eˆ(r) is given in terms of photon creation and annihilation operators as

~ 2π ωk ikr † −ikr Eˆ(r)= i ǫk aˆk e aˆ e . L3 ,λ ,λ − k,λ k λ=1,2

(a) The vector ǫk,λ is a real vector. What is its interpretation, and what are the conditions on its direction?

sp (b) Calculate the rate dΓ1→0/dΩ that the excited state 1 spontaneously decays to the ground state 0 , while a photon is emitted within the| solid angle element dΩ. | (c) For the spontaneous decay you considered in part (b), what is the polarization of a photon that is emitted in the y direction?

(d) How does the decay rate you calculated in part (b) change if there is a nonzero average occupation n(ω) of photon modes with energy ω? You may express your answer in sp terms of Γ1→0. (e) In general, the emission or absorption of photons can cause transitions between diﬀerent harmonic oscillator states n and n′ . What are the selection rules for such transitions? | |

16 Exercise 5.20: Thomson scattering (1)

In the dipole approximation, the scattering cross section for scattering of a point particle is

dσ 2 ∗ 2 = r ǫ ǫki i . dΩ 0 | kf ,λf ,λ | Thomson This result was derived in the dipole approximation. Calculate the differential scattering cross section without making the dipole approximation. Is the difference with the above expression significant? You may specialize to the case that the particle is at rest initially.

Exercise 5.21: Thomson scattering (2)

A photon with wavevector ki scatters off a free electron, initially at rest. After scattering, the photon has a different wavevector kf . The Hamiltonian describing the interaction between the electromagnetic field and the electron reads e e2 Hˆ = pˆ Aˆ (ˆr)+ Aˆ (ˆr) 2, int −mc 2mc2 | | where A(r) is the vector potential of the quantized electromagnetic field.

(a) Give expressions for the initial and ﬁnal states of the photon and the electron. For the electron, specify the wavefunction of the initial and ﬁnal states. Use a normalization that is consistent with the normalization of the photon states.

(b) Explain which of the two terms in the expression for Hînt contributes to the scattering amplitude to first order in Hînt (i.e., in the first-order Born approximation). (c) Calculate the T matrix element for this scattering process in the first-order Born approximation.

(d) Give an expression for the scattering cross section dσ/dΩ in terms of the T matrix. (dΩ refers to the solid angle for the direction of the scattered photon.)

(e) Calculate the scattering cross section dσ/dΩ for this scattering process, according to the ﬁrst-order Born approximation.

17 (f) Is the ﬁrst-order Born approximation suﬃcient for the calculation of the T matrix? If yes, explain. If not, calculate the relevant second-order term.

Exercise 5.22: Rayleigh scattering

If the photon energy ~ω is much smaller than atomic transition energies, the photon scattering cross section is given by the Rayleigh formula, dσ = (mr )2 ω4 dΩ 0 Rayleigh 2 ∗ ∗ i ǫ ˆr n n ǫki,λi ˆr i + i ǫki,λi ˆr n n ǫ ˆr i | kf ,λf | | | | | | kf ,λf | . × εi εn n=i − (a) Derive the Rayleigh formula from the Kramers-Heisenberg formula for the photon scattering cross section.

(b) Show that the tensor 2 i rˆi n n rˆj i ij = e | | | | D εn εi n=i − describes the polarizability of the atom, i.e., its electric dipole moment in response to a static electric ﬁeld.

Exercise 5.23: Photo-electric eﬀect

In the photo-electric effect, a photon is absorbed by an atom and the energy of the photon is used to release an electron initially bound to the atom. In this exercise, you consider the photo-electric effect for a Hydrogen atom in its ground state. The photo-electric effect may be considered as a scattering problem, where the initial state consists of a photon with wavevector k and polarization λ and an electron in the Hydrogen ground state and the final state consists of a free electron in a state with momentum p. (a) Give explicit expressions for the initial and final states for this scattering problem.

18 The interaction between the photon and the electron bound to the Hydrogen atom is given by the Hamiltonian e Hˆ = pˆ Aˆ (ˆr). int −mc

(b) Give an expression for the T matrix Tp,kλ for this situation. You may use the Born approximation (i.e., you may use first-order perturbation theory in the interaction between the photon and the electron). If you prefer, you may also give an expression for the scattering matrix Sp,kλ instead of the T matrix Tp,kλ. (c) What is the relation between the T matrix and the differential cross section dσ/dΩ in this case? (d) Calculate the differential scattering cross section dσ/dΩ for the photo-electric effect.

Exercise 5.24: Two-level system in a classical ﬁeld

Consider a quantum system with two discrete states: an excited state e and a ground state | g . The energy difference between the ground state and the excited state is ~ω0. Without loss| of generality, you may set the energy of the ground state at ~ω /2, and the energy of − 0 the excited state at ~ω0/2, so that the Hamiltonian of the two-level system reads ~ω Hˆ = 0 ( e e g g ). (33) 0 2 | |−| | The system is placed in a classical electric radiation field of frequency ω. In the dipole approximation, the Hamiltonian H1 describing the interaction between the radiation field and the system is Hˆ = Dˆ E(t), 1 − where D is the dipole moment of the two-level system and E(t) = E0 cos(ωt) the electric field. The matrix elements of H1 between the states g and e are then proportional to matrix elements of the dipole operator, | | e Hˆ g = g Hˆ e ∗ = e Dˆ g E cos(ωt). | 1| | 1| − | | 0 (a) Show that Hˆ1 can be written as Hˆ = ~ cos(ωt)( Ω e g Ω∗ g e ), 1 | |− | | and find an expression for the coupling constant Ω. The coupling constant Ω is called “Rabi frequency”. Verify that Ω has the dimension of frequency.

Give explicit equations for the time derivativesa ˙ e anda ˙ g in terms of the amplitudes ae and ag.

(c) Argue that the time evolution of ae and ag is slow (on the scale ω0) if the Rabi frequency Ω and the detuning δ = ω ω are small in comparision to ω . − 0 0 (d) If the conditions mentioned under (c) are fulfilled, terms in the evolution equations for ae and ag that oscillate fast on the scale ω0 average to zero and may be left out. This approximation is known as the “rotating wave approximation”. Solve the remaining set of equations for the time-evolution of ae and ag and interpret your answer. If the system starts in the ground state g , what is the (time-averaged) probability to find it in the excited state e ? | | (f) Simplify your expression for Ψ(t) in the limit that the detuning δ is much larger than the Rabi frequency Ω. What| is the effective energy of the ground state in this case? The shift of the ground state energy is known as the “light shift” or “AC Stark effect”. It is an often used tool in quantum optics and atomic physics to manipulate, confine, or guide atoms.

Exercise 5.25: Diﬀraction of an atomic beam oﬀ a standing wave

A beam of two-state atoms in their ground state passes at right angles to a standing wave of light at frequency ω, see the ﬁgure below. The standing wave is described by a mode function Φ(r) as in Ex. 5.6. We choose our coordinates such that the atoms move in the positive z direction and the standing beam is directed in the x direction. The intensity of the standing light wave is maximal near z = 0. Hence, the mode function has the form

Φ(r) ǫ cos(kx)f(z), k = ω/c, ∝ where the polarization vector ǫ is in the xy plane and f(z) is a slowly varying function of z, peaked near z = 0.

20 standing light wave

beam of atoms

x z

(a) The detuning δ = ω ω0, where ~ω0 is the energy diﬀerence between the atom’s excited state and the ground− state, is supposed to be large. Find an eﬀective Hamiltonian for the atom’s center-of-mass motion.

(b) Upon passing through the standing light wave, the atoms can scatter. Show that the atom’s momentum in the x direction (i.e., perpendicular to their initial propagation direction) after passing through the standing light wave can only change by integer multiples of 2~k. Interpret this result in terms of absorption and reemission of photons.

Exercise 5.26: Two-slit experiment for atoms

Consider an atom beam incident on two slits in a screen. The slits are a distance d apart. The atoms are detected at a distance L d behind the screen, see the left panel of the ﬁgure below. Interference between pathes through≫ the two slits gives an interference pattern for the detected intensity. The incident atom beam consists of atoms of momentum ~k moving perpendicular to the screen.

(a) The probability p(x) to detect an atom at a distance x from a position midway between the two slits is proportional to ψ(x) 2, where ψ(x) is the wavefunction of the atom at | | a distance L from the slits. Argue that kxd p(x) cos2 . ∝ 2L

21 d x

(b) An optical cavity is placed in front of one of the slits, see the right panel of the figure. The cavity contains a field mode at a frequency near an atomic transition, but detuned sufficiently far that no transitions are induced. Argue that the presence of the cavity induces an additional phase change for atoms that travel through the cavity that is proportional to the number of photons n in the field mode,

δφ = nχ.

(You do not need to ﬁnd an explicit expression for the proportionality constant χ.)

(c) In the presence of the cavity, the probability p(x) to detect an atom at position x is changed. Show how the presence of the cavity modiﬁes the x dependence of p(x) with respect to the probability density given in part (a). Your expression may include an x-independent proportionality constant that you do not need to calculate.

(d) When the state of the field in the cavity before the passage of an atom is given by the linear combination Φ(0) = c(0) n , | n | n where n is a state with n photons in the field mode, the atom and the field are | “entangled” after passage of the atom through the cavity: Its “state” at a distance L from the screen can be written as

Ψ(0) = c(0) dxψ (x) n,x , | n n | n

where ψn(x) is the atomic wavefunction for the case that there are n photons in the cavity. Measuring the atom at position x1 collapses the atom part of the state to x1 . What is the state Φ(1) of the radiation ﬁeld in the cavity after an atom that passed| | 22 through the cavity has been detected at a position x1? You may express your answer in terms of the wavefunctions ψn(x). Hint: Recall that a measurement leads to a collaps of the wavefunction.

(e) If initially the radiation field in the cavity was in a state with a well-defined number of photons, will the cavity still be in a state with a well-defined number of photons after one such a measurement? Or will the photon number become uncertain?

(f) If initially the radiation ﬁeld in the cavity was in a state with an uncertain number of photons, will a measurement of the position of the atom change that state? And will it change the uncertainty?

A detailed analysis shows that, after repeated passage of atoms through the cavity and a measurement of the position of the atom, only a single value of the photon number n will be singled out: The field in the cavity is prepared in a pure number state n . What number | that is is known from the sequence of atom positions x1, x2, x3, .... Such a measurement is called a “quantum non-demolition measurement”, because such a measurement preserves the state of the radiation field after measurement. (It does not measure the number of photons by absorbing them, as is the standard method to detect photons.) Such a manipulation of the radiation field has been performed experimentally in S. Haroche, M. Brune, and J. M. Raimond, Appl. Phys. B 54, 355 (1992) and M. Brune, S. Haroche, J. M. Raimond, L. Davidovich, and N Zagury, Phys. Rev. A 45, 5193 (1992).

Exercise 5.27: Dicke Superradiance

Consider one atom in a (large) cavity of linear dimension L. The atom is modeled as a two level system with Hamiltonian ~ω Hˆ = 0 ( e e g g ) (35) a 2 | |−| | where e and g refer to the excited and ground states of the atom, respectively. The modes | | of the radiation field in the cavity are labeled with the number n, and the Hamiltonians Hˆr and Hâr describing the radiation field and the coupling between the radiation field and the atom can be taken to be λ Hˆ = ~ω aˆ† aˆ , Hˆ = ( e aˆ g + g aˆ† e ), (36) r n n n ar L3/2 | n | | n | n n

23 respectively, where for simplicity the coupling parameter λ (which is proportional to the dipole matrix element between the states e and g ) is taken to be independent of n. Initially, no photons are present in the cavity| and the| atom is prepared in the excited state e . | (a) Calculate the (initial) decay rate Γ of the excited state. Express your answer in terms of the coupling parameter λ and the density of states ν(ω) for the radiation modes, 1 ν(ω)= δ(ω ω ). L3 − n n

(b) Use your answer to (a) to ﬁnd the probability Pe(t) that the atom is in the excited state as a function of time.

Now consider two atoms in the same cavity. The atoms are modeled as two level systems with Hamiltonian

2 ~ω Hˆ = Hˆ , Hˆ = 0 ( e,i e,i g,i g,i ), i =1, 2, (37) a a,i a,i 2 | |−| | i=1 where e,i and g,i , i = 1, 2, refer to the excited and ground states of the two atoms, | | respectively. The separation between the two atoms is assumed to be much less than λ0 = 2πc/ω0. The atoms are coupled to the electromagnetic ﬁeld via the Hamiltonian

2 λ Hˆ = Hˆ , Hˆ = ( e,i aˆ g,i + g,i aˆ† e,i ) (38) ar ar,i ar,i L3/2 | n | | n | i=1 n (c) Explain why the same coupling constant λ can be used for both atoms i =1, 2. Would that be allowed if their separation is larger than λ0? (d) If initially only atom no. 1 is in the excited state and there are no photons present in the cavity, what are the probabilities Pe,1(t) and Pe,2(t) to ﬁnd atom 1 and atom 2 in the excited state (irrespective of the state of the other atom), respectively, as a function of time?

(e) Answer the same question for the case that all atoms are in the excited state initially.

(f) Generalize your answers to (d) and (e) for the case that there are N atoms, all within a distance λ0 from each other.

24 The effect you find was first discovered by R. H. Dicke, see Phys. Rev. 93, 99 (1954).

Exercise 5.28: Excitations on the surface of liquid Helium

The dispersion relation for waves propagating on the surface of a liquid is

3/2 A ωq = q , with q = q , ρ0 | |

where ωq and q =(qx,qy) are, respectively, the frequency and the wave vector of the surface wave and A and ρ0 are constants that represent the surface tension and the mass density of the liquid. A quantum-mechanical theory of surface waves is formulated in terms of bosonic † creation and annihilation operatorsa ˆq anda ˆq. These operators describe the creation and annihilation of a “ripplon”, a “quantum of surface oscillation”. We apply periodic boundary conditions with periods Lx and Ly in the directions parallel to the surface. The amplitude h(r) of the liquid surface oscillation at point r = (x,y) can then be expressed in terms of † the operators aq and aq, and reads

qmax ~q hˆ(r)= aˆ† e−iqr +â eiqr . 2ρ ω L L q q q 0 q x y The high-momentum cut-off qmax accounts for the fact that for larger values of q, hence smaller wave lengths, the fluid-dynamical description of the surface ceases to hold. The energy of the ripplon excitations is

qmax ˆ ~ † HS = ωqaˆqaˆq. q (a) At zero temperature, no ripplon excitations are present. Evaluate the root mean square amplitude ∆h =(h2)1/2 of the surface at zero temperature. Electrons can be trapped electrostatically just above the surface of liquid helium. A trapped electron can move freely along the surface, while in the direction normal to the interface its motion is restricted to bound states. A simple model for the Hamiltonian of an electron at coordinate x =(r,z) is pˆ2 mω2 2 Hˆ = + Vˆ (ˆr, zˆ), with Vˆ (r,z)= z hˆ(r) hˆ2(r) . 2m 2 − − where p is the electron momentum, m its mass, and z the coordinate normal to the surface.

25 (b) Show that the Hamiltonian H may be rewritten as pˆ2 1 Hˆ = + mω2zˆ2 + Vˆ (ˆr, zˆ) 2m 2 er

where the “electron-ripplon interaction” Vˆer(r,z) reads

~q Vˆ (r,z)= mω2z aˆ† e−iqr +ˆa eiqr . (39) er 2ρ ω L L q q − q 0 q x y

1 ikr ψn,k(r,z)= r,z n, k = ψn(z)e , | LxLy

where ψn(z) is an eigenfunction of the one-dimensional Harmonic oscillator and k = (kx, ky) is a two-dimensional wave vector parallel to the surface. What are the corresponding energies En,k? Electrons trapped near the surface can scatter oﬀ surface excitations. According to the electron-ripplon interaction (39), such scattering involves the creation or annihilation of a ripplon excitation. At zero temperature, no ripplons are present in the initial state, so that ripplon creation is the only possibility. (d) Consider the elementary scattering process

n, k n′, k′ q (40) | →| | in which a ripplon with wavevector q is created. Give the selection rules and/or conservation laws for the quantum numbers n and n′ as well as for the momenta ~k, ~k′, and ~q for this scattering process.

(e) Give the zero-temperature Golden-Rule decay rate Γ1→0(0) for an electron state n, k with n = 1 and k = 0, due to spontaneous emission of a ripplon. You may assume| that the energy of the emitted ripplon ~ω ~2q2/2m. q ≫

(f) Express the ﬁnite-temperature decay rate Γ1→0(T ) for the same decay process in terms of the zero-temperature rate you calculated in (e).