PHYS 130C

Part 3: Quantum Optics

Quantization of Light

Classical Electromagnetic Wave

Lagrangian Description

Electromagnetic field is the physical observable:

E - electric field,

B - magnetic field,

Note: the speed of light c=1 is set to unity.

Gauge field: (Φ,A) as generalized coordinates (state variables)

Φ - scalar potential,

A - vector potential.

Maxwell Equations

The first two equations follows from Eq. (), by definition.

Verify that Eq. () implies the first two equations in Eq. ().

∇×E=∇×(-∇Φ-(∂_t)A)
=-∇×(∇Φ)-(∂_t)(∇×A)
=-(∂_t)(∇×A)
=-(∂_t)B,

∇·B=∇·(∇×A)=0.

The last two equations follows from the variational principle δℒ=0,

((δℒ)/(δ,Φ))=0=>∇·E=0,
((δℒ)/(δ,A))=0=>∇×B-(∂_t)E=0.

Drive Eq. ().

ℒ=(1/2)((((-∇Φ-(∂_t)A))^2)-(((∇×A))^2)).

((δℒ)/(δ,Φ))=(-∇Φ-(∂_t)A)·((δ(-∇Φ))/(δ,Φ))
=-E·((∇δ,Φ)/(δ,Φ))
=-(Subscript[E, i])(∂_i) 𝟙
=(∂_i)(Subscript[E, i])
=∇·E=0,

((δℒ)/(δ,A))=(-∇Φ-(∂_t)A)·((δ(-(∂_t)A))/(δ,A))-(∇×A)·((δ(∇×A))/(δ,A))
=-E·(((∂_t)δ,A)/(δ,A))-B·((∇×δ,A)/(δ,A))
=-(Subscript[E, j])(∂_t)((δ,(Subscript[A, j]))/(δ,(Subscript[A, i])))-(Subscript[ϵ, k,l,m])(Subscript[B, k])(∂_l)((δ,(Subscript[A, m]))/(δ,(Subscript[A, i])))
=-(Subscript[E, j])(Subscript[δ, i,j])(∂_t)𝟙-(Subscript[ϵ, k,l,m])(Subscript[B, k])(Subscript[δ, m,i])(∂_l)𝟙
=(∂_t)(Subscript[E, j])(Subscript[δ, i,j])+(Subscript[ϵ, k,l,m])(∂_l)(Subscript[B, k])(Subscript[δ, m,i])
=(∂_t)(Subscript[E, i])+(Subscript[ϵ, k,l,i])(∂_l)(Subscript[B, k])
=(∂_t)(Subscript[E, i])-(Subscript[ϵ, i,l,k])(∂_l)(Subscript[B, k])
=(∂_t)E-∇×B=0.

Gauge Structure and Gauge Fixing

Show that the gauge transformation Eq. () leaves Eq. () invariant.

E=-∇Φ-(∂_t)A
→-∇(Φ-(∂_t)θ)-(∂_t)(A+∇θ)
=-∇Φ+∇(∂_t)θ-(∂_t)A-(∂_t)∇θ
=-∇Φ-(∂_t)A
=E,

B=∇×A
→∇×(A+∇θ)
=∇×A+∇×(∇θ)
=∇×A
=B,

Gauge structure is a redundancy in the gauge theory: there are multiple state variables (gauge field Φ,A) encoding the same physical observables (electromagnetic field E,B).

Gauge fixing is a procedure to eliminate the gauge redundancy, by using gauge transformation to (partially) fix the gauge field configuration.

Freedom to use: θ field (through out the spacetime).

If Φ≠0, use Φ->Φ-(∂_t)θ to fix Φ=0, by setting

θ=(∫_0^t)dt Φ+(Subscript[θ, t=0]),

where (Subscript[θ, t=0]) is still free to tune through out the space.

With Φ=0 fixed, E=-(∂_t)A, then the Coulomb law implies

∇·E=0=>-(∂_t)∇·A=0=>∇·A=∇·(Subscript[A, t=0]).

At the t=0 time slice, if ∇·(Subscript[A, t=0])≠0, use (Subscript[A, t=0])->(Subscript[A, t=0])+∇(Subscript[θ, t=0]) to fix ∇·(Subscript[A, t=0])=0, by solving for (Subscript[θ, t=0]) from

∇·(Subscript[A, t=0])+(∇^2)(Subscript[θ, t=0])=0.

Then ∇·A=0 is also fixed.

[Almost all freedom of θ has been used, only the global shift of θ is still free, which corresponds to a global U(1) symmetry associated with electric charge conservation.]

Hamiltonian Description

Derive the Hamiltonian density Eq. () from the Lagrangian density Eq. ().

ℋ=((∂ℒ)/(∂((∂_t)A)))·((∂_t)A)-ℒ
=-(-∇Φ-(∂_t)A)·((∂_t)A)-(1/2)((E^2)-(B^2))
=E·(E+∇Φ)-(1/2)(E^2)+(1/2)(B^2)
=(1/2)(E^2)+(1/2)(B^2)+E·∇Φ
=(1/2)(E^2)+(1/2)(B^2)-Φ ∇·E
=(1/2)(E^2)+(1/2)(B^2)

ℋ=(1/2)((E^2)+(((∇×A))^2)).

Hamiltonian dynamics

(∂_t)A=-((∂ℋ)/(∂E))=>(∂_t)A=-E,
(∂_t)E=((∂ℋ)/(∂A))=>(∂_t)E=-∇×(∇×A)=(∇^2)A-∇(∇·A).

Use the Coulomb gauge ∇·A=0, and combine the equations of motion, the vector potential satisfy a wave equation,

Electromagnetic Wave

The angular frequency (Subscript[ω, k]) must satisfy the dispersion relation

where the speed of light has been set to c=1, compared to the general form of (Subscript[ω, k])=c|k|.

Verify that Eq. () is a general solution of Eq. (), given Eq. ().

((∂_t^2)-(∇^2))A=(∑_k)(Subscript[A, k])((∂_t^2)-(∇^2))(e^(-i (Subscript[ω, k])t+i k·r))
=(∑_k)(Subscript[A, k])((((-i (Subscript[ω, k])))^2)-(((i k))^2))(e^(-i (Subscript[ω, k])t+i k·r))
=(∑_k)(Subscript[A, k])(-(Subscript[ω, k]^2)+(k^2))(e^(-i (Subscript[ω, k])t+i k·r))
=0,

(Subscript[A, k]) is the wave amplitudes (i.e. the Fourier components of A, as a complex vector) at each wave vector k. The gauge constraint ∇·A=0 further requires

meaning that the electromagnetic wave is transverse. For any k, there are only two transverse directions, hence, two independent polarization directions, labeled by unit vectors (Subscript[e, k,α]) (α=1,2), such that

(Subscript[A, k])=(Subscript[A, k,1])(Subscript[e, k,1])+(Subscript[A, k,2])(Subscript[e, k,2])=(∑_(α=1,2))(Subscript[A, k,α])(Subscript[e, k,α]),

where (Subscript[A, k,α]) is the wave amplitude of polarization α with wave vector k.

Block[{arr},
arr[pts_]:=Arrow[Tube[pts,0.018]];Graphics3D[{{Opacity[0.5],Glow[White],EdgeForm[Directive[Gray,AbsoluteThickness[1]]],Polygon[Prepend[#,0]&/@CirclePoints[{1.2Sqrt[2],π/4},4]]},{Arrowheads[0.05],{Red,arr[{{-1,0,0},{3,0,0}}],Text["\!\( k\)",{3,0,0},{0,-1}]},{Black,arr[{{0,-1,0},{0,1,0}}],arr[{{0,0,-1},{0,0,1}}],Text["\!\(e\_\(k,1\)\)",{0,1,0},{-1.4,0}],Text["\!\(e\_\(k,2\)\)",{0,0,1},{-1.4,0}]}}},Boxed->False,ViewPoint->{1.3, -2.4, 2.},Lighting->"Neutral",ImageSize->160]]

The corresponding solution of electromagnetic field follows from E=-(∂_t)A and B=∇×A,

E(r,t)=i (∑_k)(Subscript[ω, k])(Subscript[A, k])(e^(-i (Subscript[ω, k])t+i k·r)),
B(r,t)=i (∑_k)k×(Subscript[A, k])(e^(-i (Subscript[ω, k])t+i k·r)).

Here is an illustration of linearly polarized electromagnetic wave.

Manipulate[Block[{arr,A,e},
arr[pts_]:=Arrow[Tube[pts,0.03]];
A[x_]:=Cos[π x/2-2π t];
e=<|1->{0,1,0},2->{0,0,1}|>[α];Graphics3D[{{Glow@Black,Arrowheads[0.05],{Red,arr[{{-1,0,0},{6,0,0}}],Text["\!\( k\)",{6,0,0},{0,-1}]},
Arrowheads[0.035],Opacity[0.5],Table[Translate[{{Blue,arr[{{0,0,0},# e}]},{Green,arr[{{0,0,0},# Cross[{1,0,0},e]}]}}&@A[x],{x,0,0}],{x,-1,5,0.5}]},
{{Blue,Line[Table[{x,0,0}+A[x] e,{x,-1,5,0.1}]],Text["\!\( E\)",A[0] e,{0,-Sign[A[0]]}]},{Green,Line[Table[{x,0,0}+A[x] Cross[{1,0,0},e],{x,-1,5,0.1}]],
Text["\!\( B\)",A[0] Cross[{1,0,0},e],{0,(-1)^α Sign[A[0]]}]}}},Boxed->False,PlotRange->{{-1.2,6.2},{-1.2,1.2},{-1.2,1.2}},ViewPoint->{1.3, -2.4, 2.},Lighting->"Neutral",ImageSize->200]],{{α,2},{1,2}},{{t,1/8},0,1,ImageSize->Small},Paneled->False]

Quantization of Electromagnetic Field

Canonical Quantization (Real Space)

Identify the classical phase space: a classical system described by generalized coordinates (Subscript[q, i]) and their conjugate momenta (Subscript[p, i]):=∂L/∂(Subscript[(OverDot[q]), i]).

Promote classical variables to quantum operators:

(Subscript[q, i])→(Subscript[(OverHat[q]), i]), (Subscript[p, i])->(Subscript[(OverHat[p]), i]),
H->(OverHat[H])=H((Subscript[(OverHat[q]), i]),(Subscript[(OverHat[p]), i])).

Impose canonical commutation relations between conjugate pairs of coordinates and momenta (setting ℏ=1):

[(Subscript[(OverHat[q]), i]),(Subscript[(OverHat[q]), j])]=[(Subscript[(OverHat[p]), i]),(Subscript[(OverHat[p]), j])]=0,
[(Subscript[(OverHat[q]), i]),(Subscript[(OverHat[p]), j])]=i (Subscript[δ, i,j])𝟙.

Field Operators: define the following (vectorial) operators

A(r)=((Subscript[A, x])(r),(Subscript[A, y])(r),(Subscript[A, z])(r)),
E(r)=((Subscript[E, x])(r),(Subscript[E, y])(r),(Subscript[E, z])(r)),

at each point r in the space.

Each component is a Hermitian operator (corresponding to a real variable in the classical limit)

(Subscript[A, i]^†)(r)=(Subscript[A, i])(r),
(Subscript[E, i]^†)(r)=(Subscript[E, i])(r).

In general, A and E are non-commuting operators. They only commute (become independent) if they are

at different spacial positions,

or along perpendicular directions.

Canonical Quantization (Momentum Space)

Forward transformation:

Backward transformation:

Note: (∑_k):=(((2π))^(-3))∫(d^3)k to properly normalize.

(Subscript[A, k]) and (Subscript[E, k]) are no longer Hermitian operators by themselves. Instead, their Hermitian conjugates are

(Subscript[A, k]^†)=(Subscript[A, -k]),
(Subscript[E, k]^†)=(Subscript[E, -k]).

Commutation relations (with ℏ=1):

[(Subscript[A, i,k]),(Subscript[A, j,k']^†)]=[-(Subscript[E, i,k]),-(Subscript[E, j,k']^†)]=0,
[(Subscript[A, i,k]),-(Subscript[E, j,k']^†)]=i (Subscript[δ, i,j])(Subscript[δ, k,k'])𝟙.

Verify Eq. (), given Eq. ().

[(Subscript[A, i,k]),-(Subscript[E, j,k']^†)]=[(Subscript[A, i,k]),-(Subscript[E, j,-k'])]
=∫(d^3)r∫(d^3)r'[(Subscript[A, i])(r),-(Subscript[E, j])(r')](e^(-i k·r))(e^(+i k'·r'))
=∫(d^3)r∫(d^3)r'(i (Subscript[δ, i,j])δ(r-r')𝟙)(e^(-i k·r+i k'·r'))
=∫(d^3)r (i (Subscript[δ, i,j])𝟙)(e^(-i k·r+i k'·r))
=i (Subscript[δ, i,j])𝟙∫(d^3)r (e^(-i (k-k')·r))
=i (Subscript[δ, i,j])(Subscript[δ, k,k'])𝟙.

(Subscript[A, k,α]) and (Subscript[E, k,α]) are not Hermitian operators. As inherited from Eq. (), their Hermitian conjugates are

(Subscript[A, k,α]^†)=(Subscript[A, -k,α]),
(Subscript[E, k,α]^†)=(Subscript[E, -k,α]).

Commutation relations (with ℏ=1):

[(Subscript[A, k,α]),(Subscript[A, k',α']^†)]=[-(Subscript[E, k,α]),-(Subscript[E, k',α']^†)]=0,
[(Subscript[A, k,α]),-(Subscript[E, k',α']^†)]=i (Subscript[δ, α,α'])(Subscript[δ, k,k'])𝟙.

Hamiltonian Operator

Derive Eq. ().

H=(1/2)∫(d^3)r(((E(r))^2)+(∇×A)(((r))^2))
=(1/2)∫(d^3)r(∑_(k,k'))((Subscript[E, k'])·(Subscript[E, k])+(i k'×(Subscript[A, k']))·(i k×(Subscript[A, k])))(e^(i (k+k')·r))
=(1/2)(∑_(k,k'))((Subscript[E, k'])·(Subscript[E, k])+(i k'×(Subscript[A, k']))·(i k×(Subscript[A, k])))(Subscript[δ, k,-k'])
=(1/2)(∑_k)((Subscript[E, -k])·(Subscript[E, k])+(-i k×(Subscript[A, -k]))·(i k×(Subscript[A, k])))
=(1/2)(∑_k)((Subscript[E, k]^†)·(Subscript[E, k])+(k×(Subscript[A, k]^†))·(k×(Subscript[A, k])))
=(1/2)(∑_k)(∑_(α,α'))((Subscript[e, k,α'])·(Subscript[e, k,α])(Subscript[E, k,α']^†)(Subscript[E, k,α])+(k×(Subscript[e, k,α']))·(k×(Subscript[e, k,α]))(Subscript[A, k,α']^†)(Subscript[A, k,α])).

(Subscript[e, k,α'])·(Subscript[e, k,α])=(Subscript[δ, α,α']),
(k×(Subscript[e, k,α']))·(k×(Subscript[e, k,α]))=(k·k)((Subscript[e, k,α'])·(Subscript[e, k,α]))-(k·(Subscript[e, k,α']))(k·(Subscript[e, k,α]))=(k^2)(Subscript[δ, α,α']).

H=(1/2)(∑_k)(∑_(α,α'))((Subscript[δ, α,α'])(Subscript[E, k,α']^†)(Subscript[E, k,α])+(k^2)(Subscript[δ, α,α'])(Subscript[A, k,α']^†)(Subscript[A, k,α]))
=(1/2)(∑_k)(∑_α)((Subscript[E, k,α]^†)(Subscript[E, k,α])+(k^2)(Subscript[A, k,α]^†)(Subscript[A, k,α])).

H=(1/2)(∑_(k,α))((Subscript[E, k,α]^†)(Subscript[E, k,α])+(Subscript[ω, k]^2)(Subscript[A, k,α]^†)(Subscript[A, k,α])).

H accounts for contributions from all possible wave modes, labeled by the wave vector k and the polarization index α.

(Subscript[A, k,α]) and (Subscript[E, k,α]) are quantum operators satisfying the commutation relation in Eq. ().

(Subscript[E, k,α]^†)(Subscript[E, k,α]) originated from the (E^2) term, representing the kinetic energy of the electromagnetic field.

(Subscript[ω, k]^2)(Subscript[A, k,α]^†)(Subscript[A, k,α]) originated from the (B^2) term, representing the potential energy of the electromagnetic field.

Photons

(Subscript[A, k,α])=(1/(Sqrt[2]))(Subscript[ω, k]^(-1/2))((Subscript[a, k,α])+(Subscript[a, -k,α]^†)),
(Subscript[E, k,α])=(i/(Sqrt[2]))(Subscript[ω, k]^(1/2))((Subscript[a, k,α])-(Subscript[a, -k,α]^†)).

Derive Eq. () by inverting Eq. ().

(Subscript[a, k,α])=(1/(Sqrt[2]))((Subscript[ω, k]^(1/2))(Subscript[A, k,α])-i (Subscript[ω, k]^(-1/2))(Subscript[E, k,α])),
(Subscript[a, -k,α]^†)=(1/(Sqrt[2]))((Subscript[ω, k]^(1/2))(Subscript[A, k,α])+i (Subscript[ω, k]^(-1/2))(Subscript[E, k,α])),

They satisfy the following commutation relations

Derive Eq. () from Eq. (), given the definition Eq. ().

[(Subscript[a, k,α]),(Subscript[a, k',α'])]=(1/2)[(Subscript[ω, k]^(1/2))(Subscript[A, k,α])-i (Subscript[ω, k]^(-1/2))(Subscript[E, k,α]),(Subscript[ω, k']^(1/2))(Subscript[A, k',α'])-i (Subscript[ω, k']^(-1/2))(Subscript[E, k',α'])]
=(1/2)((Subscript[ω, k]^(1/2))(Subscript[ω, k']^(1/2))[(Subscript[A, k,α]),(Subscript[A, k',α'])]+i (Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))[(Subscript[A, k,α]),-(Subscript[E, k',α'])]+i (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(1/2))[-(Subscript[E, k,α]),(Subscript[A, k',α'])]- (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(-1/2))[-(Subscript[E, k,α]),-(Subscript[E, k',α'])])
=(1/2)((Subscript[ω, k]^(1/2))(Subscript[ω, k']^(1/2))[(Subscript[A, k,α]),(Subscript[A, -k',α']^†)]+i (Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))[(Subscript[A, k,α]),-(Subscript[E, -k',α']^†)]-i (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(1/2))[(Subscript[A, k',α']),-(Subscript[E, -k,α]^†)]- (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(-1/2))[-(Subscript[E, k,α]),-(Subscript[E, -k',α']^†)])
=(1/2)(0-(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))(Subscript[δ, α,α'])(Subscript[δ, k,,-k'])𝟙+(Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(1/2))(Subscript[δ, α,α'])(Subscript[δ, k,,-k'])𝟙-0)
=(1/2)(-1+1)(Subscript[δ, α,α'])(Subscript[δ, k,,-k'])𝟙
=0.

[(Subscript[a, k,α]^†),(Subscript[a, k',α']^†)]=(Subscript[a, k,α]^†)(Subscript[a, k',α']^†)-(Subscript[a, k',α']^†)(Subscript[a, k,α]^†)
=(SuperDagger[((Subscript[a, k',α'])(Subscript[a, k,α])-(Subscript[a, k,α])(Subscript[a, k',α']))])
=(SuperDagger[[(Subscript[a, k',α']),(Subscript[a, k,α])]])
=0.

[(Subscript[a, k,α]),(Subscript[a, k',α']^†)]=(1/2)[(Subscript[ω, k]^(1/2))(Subscript[A, k,α])-i (Subscript[ω, k]^(-1/2))(Subscript[E, k,α]),(Subscript[ω, k']^(1/2))(Subscript[A, k',α']^†)+i (Subscript[ω, k']^(-1/2))(Subscript[E, k',α']^†)]
=(1/2)((Subscript[ω, k]^(1/2))(Subscript[ω, k']^(1/2))[(Subscript[A, k,α]),(Subscript[A, k',α']^†)]-i (Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))[(Subscript[A, k,α]),-(Subscript[E, k',α']^†)]+i (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(1/2))[-(Subscript[E, k,α]),(Subscript[A, k',α']^†)]+ (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(-1/2))[-(Subscript[E, k,α]),-(Subscript[E, k',α']^†)])
=(1/2)((Subscript[ω, k]^(1/2))(Subscript[ω, k']^(1/2))[(Subscript[A, k,α]),(Subscript[A, k',α']^†)]-i (Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))[(Subscript[A, k,α]),-(Subscript[E, k',α']^†)]-i (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(1/2))[(Subscript[A, -k',α']),-(Subscript[E, -k,α]^†)]+ (Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(-1/2))[-(Subscript[E, k,α]),-(Subscript[E, k',α']^†)])
=(1/2)(0+(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))(Subscript[δ, α,α'])(Subscript[δ, k,k'])𝟙+(Subscript[ω, k]^(-1/2))(Subscript[ω, k']^(1/2))(Subscript[δ, α,α'])(Subscript[δ, k,k'])𝟙+0)
=(1/2)(1+1)(Subscript[δ, α,α'])(Subscript[δ, k,k'])𝟙
=(Subscript[δ, α,α'])(Subscript[δ, k,k'])𝟙.

Photon number operator:

By quantum bootstrap, Eq. () requires that the eigenvalues of (Subscript[n, k,α]) are quantized to natural numbers

(Subscript[n, k,α])=0,1,2,…∈ℕ.

Photons are Bosons: their creation (Subscript[a, k,α]^†) and annihilation (Subscript[a, k,α]) operators satisfy the bosonic commutation relations, and each photon mode (labeled by k and α) can be occupied by an arbitrary number (Subscript[n, k,α]) of photons.

Photon Energy

H=∫(d^3)r(1/2)((E^2)+(B^2)).

Every photon mode is (mathematically) equivalent to a simple harmonic oscillator with quantized energy levels.

Adding/removing each photon ((Subscript[n, k,α])->(Subscript[n, k,α])±1) will cause the total energy H to increase/decrease by (Subscript[ω, k])=|k| (or ℏ (Subscript[ω, k])=ℏ c|k|, if the units are restored).

=> Energy quantization: each photon of wave vector k and polarization α carries ℏ (Subscript[ω, k]) unit of energy.

Even if (Subscript[n, k,α])=0 (in the photon vacuum state), there is still (1/2)ℏ (Subscript[ω, k]) energy associated with each photon mode, known as the vacuum energy.

Field Operators

Vector potential

Electromagnetic field

Verify Eq. () and Eq. ().

A(r)=(∑_k)(Subscript[A, k])(e^(i k·r))
=(∑_(k,α))(Subscript[A, k,α])(Subscript[e, k,α])(e^(i k·r))
=(1/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(-1/2))(Subscript[e, k,α])((Subscript[a, k,α])+(Subscript[a, -k,α]^†))(e^(i k·r))
=(1/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(-1/2))(Subscript[e, k,α])((Subscript[a, k,α])(e^(i k·r))+(Subscript[a, k,α]^†)(e^(-i k·r))),

E(r)=(∑_k)(Subscript[E, k])(e^(i k·r))
=(∑_(k,α))(Subscript[E, k,α])(Subscript[e, k,α])(e^(i k·r))
=(i/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(1/2))(Subscript[e, k,α])((Subscript[a, k,α])-(Subscript[a, -k,α]^†))(e^(i k·r))
=(i/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(1/2))(Subscript[e, k,α])((Subscript[a, k,α])(e^(i k·r))-(Subscript[a, k,α]^†)(e^(-i k·r))).

B(r)=∇×((1/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(-1/2))(Subscript[e, k,α])((Subscript[a, k,α])(e^(i k·r))+(Subscript[a, k,α]^†)(e^(-i k·r))))
=(i/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(-1/2))k×(Subscript[e, k,α])((Subscript[a, k,α])(e^(i k·r))-(Subscript[a, k,α]^†)(e^(-i k·r))).

Photon Momentum

K=∫(d^3)r E×B.

Derive Eq. ().

K=∫(d^3)r E×B
=-(1/2)∫(d^3)r (∑_(k,α))(∑_(k',α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))((Subscript[e, k,α])×(k'×(Subscript[e, k',α'])))((Subscript[a, k,α])(e^(i k·r))-(Subscript[a, k,α]^†)(e^(-i k·r)))((Subscript[a, k',α'])(e^(i k'·r))-(Subscript[a, k',α']^†)(e^(-i k'·r))).

(Subscript[(a×(b×c)), i])=(Subscript[ϵ, i,j,k])(Subscript[ϵ, k,l,m])(Subscript[a, j])(Subscript[b, l])(Subscript[c, m])
=((Subscript[δ, i,l])(Subscript[δ, j,m])-(Subscript[δ, i,m])(Subscript[δ, j,l]))(Subscript[a, j])(Subscript[b, l])(Subscript[c, m])
=(Subscript[a, j])(Subscript[b, i])(Subscript[c, j])-(Subscript[a, j])(Subscript[b, j])(Subscript[c, i])
=(Subscript[(b (a·c)-c (a·b)), i]),

(Subscript[e, k,α])×(k'×(Subscript[e, k',α']))=k'((Subscript[e, k,α])·(Subscript[e, k',α']))-(Subscript[e, k',α'])((Subscript[e, k,α])·k').

K=-(1/2)(∑_(k,α))(∑_(k',α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))(k'((Subscript[e, k,α])·(Subscript[e, k',α']))-(Subscript[e, k',α'])((Subscript[e, k,α])·k'))∫(d^3)r((Subscript[a, k,α])(Subscript[a, k',α'])(e^(i (k+k')·r))-(Subscript[a, k,α])(Subscript[a, k',α']^†)(e^(i (k-k')·r))-(Subscript[a, k,α]^†)(Subscript[a, k',α'])(e^(-i (k-k')·r))+(Subscript[a, k,α]^†)(Subscript[a, k',α']^†)(e^(-i (k+k')·r)))
=(1/2)(∑_(k,α))(∑_(k',α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))(k'((Subscript[e, k,α])·(Subscript[e, k',α']))-(Subscript[e, k',α'])((Subscript[e, k,α])·k'))(((Subscript[a, k,α])(Subscript[a, k',α']^†)+(Subscript[a, k,α]^†)(Subscript[a, k',α']))(Subscript[δ, k,k'])-((Subscript[a, k,α])(Subscript[a, k',α'])+(Subscript[a, k,α]^†)(Subscript[a, k',α']^†))(Subscript[δ, k,-k']))
=(1/2)(∑_k)(∑_(α,α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k]^(-1/2))(k ((Subscript[e, k,α])·(Subscript[e, k,α']))-(Subscript[e, k,α'])((Subscript[e, k,α])·k))(((Subscript[a, k,α])(Subscript[a, k,α']^†)+(Subscript[a, k,α]^†)(Subscript[a, k,α']))-((Subscript[a, k,α])(Subscript[a, -k,α'])+(Subscript[a, k,α]^†)(Subscript[a, -k,α']^†))).

K=(1/2)(∑_k)(∑_(α,α'))k (Subscript[δ, α,α'])(((Subscript[a, k,α])(Subscript[a, k,α']^†)+(Subscript[a, k,α]^†)(Subscript[a, k,α']))-((Subscript[a, k,α])(Subscript[a, -k,α'])+(Subscript[a, k,α]^†)(Subscript[a, -k,α']^†)))
=(1/2)(∑_(k,α))k (((Subscript[a, k,α])(Subscript[a, k,α]^†)+(Subscript[a, k,α]^†)(Subscript[a, k,α]))-((Subscript[a, k,α])(Subscript[a, -k,α])+(Subscript[a, k,α]^†)(Subscript[a, -k,α]^†))).

(∑_(k,α))k ((Subscript[a, k,α])(Subscript[a, -k,α])+(Subscript[a, k,α]^†)(Subscript[a, -k,α]^†))
=(∑_(k,α))(-k)((Subscript[a, -k,α])(Subscript[a, k,α])+(Subscript[a, -k,α]^†)(Subscript[a, k,α]^†))
=-(∑_(k,α))k ((Subscript[a, k,α])(Subscript[a, -k,α])+(Subscript[a, k,α]^†)(Subscript[a, -k,α]^†)).

K=(1/2)(∑_(k,α))k ((Subscript[a, k,α])(Subscript[a, k,α]^†)+(Subscript[a, k,α]^†)(Subscript[a, k,α]))
=(∑_(k,α))k ((Subscript[n, k,α])+(1/2))
=(∑_(k,α))k (Subscript[n, k,α]).

Adding/removing each photon ((Subscript[n, k,α])->(Subscript[n, k,α])±1) will cause the total  momentum K to increase/decrease by k (or ℏ k, if the units are restored).

=> Each photon of wave vector k and polarization α carries ℏ k amount of momentum.

Photon Spin

S=∫(d^3)r E×A.

Derive Eq. ().

S=∫(d^3)r E×A
=(i/2)∫(d^3)r (∑_(k,α))(∑_(k',α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))((Subscript[e, k,α])×(Subscript[e, k',α']))((Subscript[a, k,α])(e^(i k·r))-(Subscript[a, k,α]^†)(e^(-i k·r)))((Subscript[a, k',α'])(e^(i k'·r))+(Subscript[a, k',α']^†)(e^(-i k'·r)))
=(i/2)(∑_(k,α))(∑_(k',α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))((Subscript[e, k,α])×(Subscript[e, k',α']))∫(d^3)r ((Subscript[a, k,α])(Subscript[a, k',α'])(e^(i (k+k')·r))+(Subscript[a, k,α])(Subscript[a, k',α']^†)(e^(i (k-k')·r))-(Subscript[a, k,α]^†)(Subscript[a, k',α'])(e^(-i (k-k')·r))-(Subscript[a, k,α]^†)(Subscript[a, k',α']^†)(e^(-i (k+k')·r)))
=(i/2)(∑_(k,α))(∑_(k',α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k']^(-1/2))((Subscript[e, k,α])×(Subscript[e, k',α']))(((Subscript[a, k,α])(Subscript[a, k',α']^†)-(Subscript[a, k,α]^†)(Subscript[a, k',α']))(Subscript[δ, k,k'])+((Subscript[a, k,α])(Subscript[a, k',α'])-(Subscript[a, k,α]^†)(Subscript[a, k',α']^†))(Subscript[δ, k,-k']))
=(i/2)(∑_k)(∑_(α,α'))(Subscript[ω, k]^(1/2))(Subscript[ω, k]^(-1/2))((Subscript[e, k,α])×(Subscript[e, k,α']))(((Subscript[a, k,α])(Subscript[a, k,α']^†)-(Subscript[a, k,α]^†)(Subscript[a, k,α']))-((Subscript[a, k,α])(Subscript[a, -k,α'])-(Subscript[a, k,α]^†)(Subscript[a, -k,α']^†)))

(Subscript[e, k,1])×(Subscript[e, k,2])=(k/(|k|)),

S=i(∑_k)(k/(|k|))((Subscript[a, k,1])(Subscript[a, k,2]^†)-(Subscript[a, k,1]^†)(Subscript[a, k,2])).

Define the circular polarization basis

(Subscript[a, k,±])=(1/(Sqrt[2]))((Subscript[a, k,1])+i (Subscript[a, k,2])),
(Subscript[a, k,±]^†)=(1/(Sqrt[2]))((Subscript[a, k,1]^†)-i (Subscript[a, k,2]^†)),

where ± labels the left/right circular polarized light.

Each left/right circular polarized photon carries spin-1 angular momentum along/against the wave vector direction k/|k| (the light propagation direction).

Quantum Vacuum Fluctuations

Uncertainty Relation

var (Subscript[E, k]):=⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩-⟨(Subscript[E, k]^†)⟩·⟨(Subscript[E, k])⟩,
var (Subscript[B, k]):=⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩-⟨(Subscript[B, k]^†)⟩·⟨(Subscript[B, k])⟩.

Prove the uncertainty relation Eq. ().

(Ket[ϕ])=((Subscript[E, k'])+i x (Subscript[A, k]))(Ket[ψ]),

(Bra[ϕ])·(Ket[ϕ])=(Bra[ψ])((Subscript[E, k']^†)-i x (Subscript[A, k]^†))·((Subscript[E, k'])+i x (Subscript[A, k]))(Ket[ψ])
=(Bra[ψ])((Subscript[E, k']^†)·(Subscript[E, k'])+i x (Subscript[E, k']^†)·(Subscript[A, k])-i x (Subscript[A, k]^†)·(Subscript[E, k'])+ (x^2) (Subscript[A, k]^†)·(Subscript[A, k]))(Ket[ψ]).

⟨(Subscript[E, k']^†)·(Subscript[E, k'])⟩+x ⟨i((Subscript[E, k']^†)·(Subscript[A, k])-(Subscript[A, k]^†)·(Subscript[E, k']))⟩+(x^2)⟨(Subscript[A, k]^†)·(Subscript[A, k])⟩≥0.

⟨(Subscript[E, -k']^†)·(Subscript[E, -k'])⟩+x ⟨i((Subscript[E, -k']^†)·(Subscript[A, -k])-(Subscript[A, -k]^†)·(Subscript[E, -k']))⟩+(x^2)⟨(Subscript[A, -k]^†)·(Subscript[A, -k])⟩≥0.

c+b x +a (x^2)≥0

c=⟨(Subscript[E, k']^†)·(Subscript[E, k'])+(Subscript[E, -k']^†)·(Subscript[E, -k'])⟩,
b=⟨i((Subscript[E, k']^†)·(Subscript[A, k])-(Subscript[A, k]^†)·(Subscript[E, k'])+(Subscript[E, -k']^†)·(Subscript[A, -k])-(Subscript[A, -k]^†)·(Subscript[E, -k']))⟩,
a=⟨(Subscript[A, k]^†)·(Subscript[A, k])+(Subscript[A, -k]^†)·(Subscript[A, -k])⟩.

b=⟨i((Subscript[E, k']^†)·(Subscript[A, k])-(Subscript[A, k]^†)·(Subscript[E, k'])+(Subscript[E, k'])·(Subscript[A, k]^†)-(Subscript[A, k])·(Subscript[E, k']^†))⟩
=⟨i(Tr[(Subscript[A, k]),-(Subscript[E, k']^†)]+Tr[(Subscript[A, k]^†),-(Subscript[E, k'])])⟩
=⟨i(Tr(i (Subscript[δ, k,k'])𝟙)+Tr(i (Subscript[δ, k,k'])𝟙))⟩
=-2(Subscript[δ, k,k'])⟨Tr 𝟙⟩
=-4(Subscript[δ, k,k']),

(Subscript[B, k]^†)(Subscript[B, k])=(-i k×(Subscript[A, k]^†))·(i k×(Subscript[A, k]))
=(k×(Subscript[A, k]^†))·(k×(Subscript[A, k]))
=(k^2)(Subscript[A, k]^†)·(Subscript[A, k])-(k·(Subscript[A, k]))(k·(Subscript[A, k]^†))
=(Subscript[ω, k]^2)(Subscript[A, k]^†)·(Subscript[A, k]),

a=(1/(Subscript[ω, k]^2))⟨(Subscript[B, k]^†)·(Subscript[B, k])+(Subscript[B, -k]^†)·(Subscript[B, -k])⟩.

(b^2)-4 a c<=0
=>16(Subscript[δ, k,k'])-(4/(Subscript[ω, k]^2))⟨(Subscript[B, k]^†)·(Subscript[B, k])+(Subscript[B, -k]^†)·(Subscript[B, -k])⟩⟨(Subscript[E, k']^†)·(Subscript[E, k'])+(Subscript[E, -k']^†)·(Subscript[E, -k'])⟩≤0.

((⟨(Subscript[E, k']^†)·(Subscript[E, k'])⟩+⟨(Subscript[E, -k']^†)·(Subscript[E, -k'])⟩)/2)((⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩+⟨(Subscript[B, -k]^†)·(Subscript[B, -k])⟩)/2)≥(Subscript[ω, k]^2)(Subscript[δ, k,k']).

(Subscript[E, k])→(Subscript[E, k])-⟨(Subscript[E, k])⟩𝟙,
(Subscript[B, k])→(Subscript[B, k])-⟨(Subscript[B, k])⟩𝟙,

⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩→
⟨((Subscript[E, k]^†)-⟨(Subscript[E, k]^†)⟩𝟙)·((Subscript[E, k])-⟨(Subscript[E, k])⟩𝟙)⟩
=⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩-⟨(Subscript[E, k]^†)⟩·⟨(Subscript[E, k])⟩
=var (Subscript[E, k]),
⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩→
⟨((Subscript[B, k]^†)-⟨(Subscript[B, k]^†)⟩𝟙)·((Subscript[B, k])-⟨(Subscript[B, k])⟩𝟙)⟩
=⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩-⟨(Subscript[B, k]^†)⟩·⟨(Subscript[B, k])⟩
=var (Subscript[B, k]).

((var (Subscript[E, k'])+var (Subscript[E, -k']))/2)((var (Subscript[B, k'])+var (Subscript[B, -k']))/2)≥(Subscript[ω, k]^2)(Subscript[δ, k,k']).

Noise trade-off: It highlights the inherent quantum noise present in electromagnetic fields — reducing the noise in one field (E or B) inevitably leads to an increase in the noise of the other.

Frequency dependence: The bound (Subscript[ω, k]^2) grows with the mode frequency — higher frequency field exhibits stronger quantum noise.

(Ket[(Subscript[n, k,1]),(Subscript[n, k,2]);(Subscript[n, -k,1]),(Subscript[n, -k,2])]).

The uncertainties of electric and magnetic fields are given by

var (Subscript[E, k])=
var (Subscript[B, k])=(∑_α)((Subscript[ω, k])/2)((Subscript[n, k,α])+(Subscript[n, -k,α])+1),

therefore the uncertainty relation holds for all Fock states (∀(Subscript[n, k,α])∈ℕ)

var (Subscript[E, k])var (Subscript[B, k])=(((Subscript[ω, k]^2)((∑_α)(1/2)((Subscript[n, k,α])+(Subscript[n, -k,α])+1)))^2)≥(Subscript[ω, k]^2).

Calculate the variances in Eq. () on the state Eq. ().

(Subscript[A, k,α])=(1/(Sqrt[2]))(Subscript[ω, k]^(-1/2))((Subscript[a, k,α])+(Subscript[a, -k,α]^†)),
(Subscript[E, k,α])=(i/(Sqrt[2]))(Subscript[ω, k]^(1/2))((Subscript[a, k,α])-(Subscript[a, -k,α]^†)),

(Subscript[A, k,α])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])
=(1/(Sqrt[2]))(Subscript[ω, k]^(-1/2))((Subscript[a, k,α])+(Subscript[a, -k,α]^†))(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])
=(1/(Sqrt[2]))(Subscript[ω, k]^(-1/2))((Sqrt[(Subscript[n, k,α])])(Ket[(Subscript[n, k,α])-1;(Subscript[n, -k,α])])+(Sqrt[(Subscript[n, -k,α])+1])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])+1])),
(Subscript[E, k,α])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])
=(i/(Sqrt[2]))(Subscript[ω, k]^(1/2))((Subscript[a, k,α])-(Subscript[a, -k,α]^†))(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])
=(i/(Sqrt[2]))(Subscript[ω, k]^(1/2))((Sqrt[(Subscript[n, k,α])])(Ket[(Subscript[n, k,α])-1;(Subscript[n, -k,α])])-(Sqrt[(Subscript[n, -k,α])+1])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])+1])).

Since (Ket[(Subscript[n, k,α])-1;(Subscript[n, -k,α])]) and (Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])+1]) are orthogonal to (Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])]),

(Bra[(Subscript[n, k,α]);(Subscript[n, -k,α])])(Subscript[A, k,α])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])=(Bra[(Subscript[n, k,α]);(Subscript[n, -k,α])])(Subscript[E, k,α])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])=0,

or simply written as

⟨(Subscript[A, k,α])⟩=⟨(Subscript[E, k,α])⟩=0.

By evaluating the square norm of these states,

(Bra[(Subscript[n, k,α]);(Subscript[n, -k,α])])(Subscript[A, k,α]^†)(Subscript[A, k,α])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])
=(1/(2(Subscript[ω, k])))((Sqrt[(Subscript[n, k,α])])(Bra[(Subscript[n, k,α])-1;(Subscript[n, -k,α])])+(Sqrt[(Subscript[n, -k,α])+1])(Bra[(Subscript[n, k,α]);(Subscript[n, -k,α])+1]))((Sqrt[(Subscript[n, k,α])])(Ket[(Subscript[n, k,α])-1;(Subscript[n, -k,α])])+(Sqrt[(Subscript[n, -k,α])+1])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])+1]))
=(1/(2(Subscript[ω, k])))((Subscript[n, k,α])+(Subscript[n, -k,α])+1),
(Bra[(Subscript[n, k,α]);(Subscript[n, -k,α])])(Subscript[E, k,α]^†)(Subscript[E, k,α])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])])
=((Subscript[ω, k])/2)((Sqrt[(Subscript[n, k,α])])(Bra[(Subscript[n, k,α])-1;(Subscript[n, -k,α])])-(Sqrt[(Subscript[n, -k,α])+1])(Bra[(Subscript[n, k,α]);(Subscript[n, -k,α])+1]))((Sqrt[(Subscript[n, k,α])])(Ket[(Subscript[n, k,α])-1;(Subscript[n, -k,α])])-(Sqrt[(Subscript[n, -k,α])+1])(Ket[(Subscript[n, k,α]);(Subscript[n, -k,α])+1]))
=((Subscript[ω, k])/2)((Subscript[n, k,α])+(Subscript[n, -k,α])+1),

or simply written as

⟨(Subscript[A, k,α]^†)(Subscript[A, k,α])⟩=(1/(2(Subscript[ω, k])))((Subscript[n, k,α])+(Subscript[n, -k,α])+1),
⟨(Subscript[E, k,α]^†)(Subscript[E, k,α])⟩=((Subscript[ω, k])/2)((Subscript[n, k,α])+(Subscript[n, -k,α])+1).

var (Subscript[E, k])=⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩-⟨(Subscript[E, k]^†)⟩·⟨(Subscript[E, k])⟩
=(∑_(α,α'))(Subscript[e, k,α])·(Subscript[e, k,α'])(⟨(Subscript[E, k,α]^†)(Subscript[E, k,α'])⟩-⟨(Subscript[E, k,α]^†)⟩⟨(Subscript[E, k,α'])⟩)
=(∑_(α,α'))(Subscript[δ, α,α'])(⟨(Subscript[E, k,α]^†)(Subscript[E, k,α'])⟩-(SuperStar[⟨(Subscript[E, k,α])⟩])⟨(Subscript[E, k,α'])⟩)
=(∑_α)(⟨(Subscript[E, k,α]^†)(Subscript[E, k,α])⟩-(SuperStar[⟨(Subscript[E, k,α])⟩])⟨(Subscript[E, k,α])⟩)
=(∑_α)((Subscript[ω, k])/2)((Subscript[n, k,α])+(Subscript[n, -k,α])+1),

var (Subscript[B, k])=⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩-⟨(Subscript[B, k]^†)⟩·⟨(Subscript[B, k])⟩
=(k^2)(⟨(Subscript[A, k]^†)·(Subscript[A, k])⟩-⟨(Subscript[A, k]^†)⟩·⟨(Subscript[A, k])⟩)
=(Subscript[ω, k]^2)(∑_(α,α'))(Subscript[e, k,α])·(Subscript[e, k,α'])(⟨(Subscript[A, k,α]^†)(Subscript[A, k,α'])⟩-⟨(Subscript[A, k,α]^†)⟩⟨(Subscript[A, k,α'])⟩)
=(Subscript[ω, k]^2)(∑_(α,α'))(Subscript[δ, α,α'])(⟨(Subscript[A, k,α]^†)(Subscript[A, k,α'])⟩-(SuperStar[⟨(Subscript[A, k,α])⟩])⟨(Subscript[A, k,α'])⟩)
=(Subscript[ω, k]^2)(∑_α)(⟨(Subscript[A, k,α]^†)(Subscript[A, k,α])⟩-(SuperStar[⟨(Subscript[A, k,α])⟩])⟨(Subscript[A, k,α])⟩)
=(Subscript[ω, k]^2)(∑_α)(1/(2(Subscript[ω, k])))((Subscript[n, k,α])+(Subscript[n, -k,α])+1)
=(∑_α)((Subscript[ω, k])/2)((Subscript[n, k,α])+(Subscript[n, -k,α])+1).

Specifically, the uncertainty relation is saturated when

∀:(Subscript[n, k,α])=0,

i.e. on the photon vacuum state (Ket[vac])=(Ket[0,0;0,0]). We say (Ket[vac]) is a minimal uncertainty state.

The finite amount of vacuum energy is also a consequence of the uncertainty relation. Given that

⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩≥var (Subscript[E, k])var (Subscript[B, k])≥(Subscript[ω, k]^2),

The total energy is therefore bounded

E=⟨H⟩=(1/2)(∑_k)(⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩+⟨(Subscript[B, k]^†)·(Subscript[B, k])⟩)
≥(1/2)(∑_k)(⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩+((Subscript[ω, k]^2)/(⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩)))
(Overscript[=, (Subscript[x, k])=⟨(Subscript[E, k]^†)·(Subscript[E, k])⟩/(Subscript[ω, k])])(∑_k)((Subscript[ω, k])/2)((Subscript[x, k])+(1/(Subscript[x, k])))
≥(∑_k)(Subscript[ω, k])=(Subscript[E, vac]).

Plot[x+1/x,{x,0,3},PlotStyle->Black,FrameLabel->{"\!\( x\)","\!\(x+1/x\)"},GridLines->{{1},{2}},PlotLabel->"\!\(min (x+1/x)=2\)",ImageSize->180]

The lower bound turns out to match the vacuum energy (Subscript[E, vac]), as discussed in Eq. ().

Casimir Effect: Vacuum Energy is Real

in (1+1) spacetime dimension,

and for scalar field (like for sound waves).

(Subscript[ψ, n])(x)=sin(((n π x)/d)).

Manipulate[Plot[0.75Sin[n π x]Cos[2π n t],{x,0,1},AspectRatio->0.4,ImageSize->160,PlotRange->{-1.5,1.2},Frame->False,Ticks->False,Axes->{True,False},AxesStyle->LightGray,PlotStyle->Red,Prolog->{Translate[{Arrowheads[{-0.05,0.05}],AbsoluteThickness[1],Arrow[{#,0}&/@{0,1}],Text[Style["\!\( d\)",Background->White],{0.5,0}]},{0,-1.3}],Text[StringTemplate["\!\(n=`1`\)"][n],{0.5,1.2},{0,1}]},Epilog->{Scale[{Line[{0,#}&/@{-1.5,1.2}],Translate[{AbsoluteThickness[0.8],Line[{#,#}&/@{0,-0.2}]},{0,#}]&/@Range[-1.5,1.2,0.1]},{#,1},{0.5,0}]&/@{1,-1}}],{{n,3},Range[1,5]},{t,0,1,ImageSize->Small},Paneled->False]

n=1,2,3,… is the mode index, labeling different wave modes on which bosons (like phonons) can occupy.

Oscillation frequency (assuming linear dispersion)

(Subscript[ω, n])=(Subscript[k, n])=((n π)/d).

Plot[-π/(12 d),{d,0,1},PlotStyle->Black,FrameLabel->{"\!\( d\)","\!\(E\_vac\)"},ImageSize->200]

A brute-force yet incorrect answer (by Srinivasa Ramanujan):

({{S, =, 1, +, 2, +, 3, +, 4, +, 5, +, 6, +, …}, {4S, =,  ,  , 4,  ,  , +, 8,  ,  , +, 12, +, …}, {-3S, =, 1, -, 2, +, 3, -, 4, +, 5, -, 6, +, …}, { , (Overscript[=, x=1]), 1, -, 2x, +, 3(x^2), -, 4(x^3), +, 5(x^4), -, 6(x^5), +, …}, { , =, (((1+x))^(-2)), , ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  }, { , (Overscript[=, x=1]), 1/4, ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  }})

Therefore, S=-1/12. However, S is not a convergent series to start with, so the above manipulations are illegitimate.

Regularization — a correct understanding (by Terence Tao, see Ref. []): introduce a regularization function η(n/N) that smoothly suppresses large terms in the summation across a cut-off scale N, while approximates the original divergent series as N→∞:

(∑_(n=1)^∞)n η(n/N)(Overscript[→, N->∞])(∑_(n=1)^∞)n.

Examples:

(∑_(n=1)^∞)n (e^(-n/N))=(N^2)-(1/12)+(1/(240(N^2)))+𝒪((1/(N^4))),
(∑_(n=1)^∞)(n/((((n/N))^3)+1))=((2π (N^2))/(3(Sqrt[3])))-(1/12)+𝒪((1/(N^4))),
(∑_(n=1)^∞)(n/((((n/N))^4)+1))=((π (N^2))/4)-(1/12)+𝒪((1/(N^4))),
(∑_(n=1)^∞)(n/((((((n/N))^2)+1))^2))=((N^2)/2)-(1/12)-(1/(60(N^2)))+𝒪((1/(N^4))),
…

Verify Eq. () by Mathematica.

Series[Sum[n/Exp[-n/Ν],{n,1,∞}],{Ν,∞,3}]

Series[Sum[n/((n/Ν)^3+1),{n,1,∞}],{Ν,∞,3}]//PowerExpand//FullSimplify

Series[Sum[n/((n/Ν)^4+1),{n,1,∞}],{Ν,∞,3}]//PowerExpand//FullSimplify

Series[Sum[n/((n/Ν)^2+1)^2,{n,1,∞}],{Ν,∞,3}]//PowerExpand//FullSimplify

The leading term diverge as (N^2) (confirming the divergent nature of the series), but its coefficient depends on the choice of the regularization function η(n/N), making it non-universal.

The sub-leading term is always -1/12, which is universal, independent of the choice of regularization. — This represents an intrinsic and invariant feature hidden beneath the apparent divergence of the series.

In reality, the mode frequency can grow indefinitely towards infinity (e.g. materials impose a natural cutoff at plasma frequency), the frequency summation must be regularized

(Subscript[E, vac])(d)=(∑_(n=1)^∞)(Subscript[ω, n])(e^(-a (Subscript[ω, n])))=(d/(π (a^2)))-(π/(12d))+….

Consider the vacuum energy both inside and outside the plates,

(Subscript[E, vac]^total)(d)=(Subscript[E, vac])(d)+(Subscript[E, vac])(L-d),

assuming L→∞ is the size of the full space outside, the attractive force between the plates is the only thing we can measure

(Subscript[F, Casimir])(d)=-(∂_d)(Subscript[E, vac]^total)(d)
=(-(1/(π (a^2)))-(π/(24(d^2)))+….)+((1/(π (a^2)))+(π/(24(((L-d))^2)))+….)
(Overscript[=, a->0,L->∞])-(π/(24(d^2))),

where the leading non-universal divergence cancels exactly.

Terence Tao. The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation. (2010)

S. K. Lamoreaux. Demonstration of the Casimir Force in the 0.6 to 6 μm Range. Phys. Rev. Lett. 78, 5 (1998).

Umar Mohideen, Anushree Roy. Precision Measurement of the Casimir Force from 0.1 to 0.9 microns. arXiv:physics/9805038 (1998).

Quantum Coherence of Light

Light-Matter Interaction

Dipole Approximation

E(r)=(i/(Sqrt[2]))(∑_(k,α))(Subscript[ω, k]^(1/2))(Subscript[e, k,α])((Subscript[a, k,α])(e^(i k·r))-(Subscript[a, k,α]^†)(e^(-i k·r))).

Mode Parity: Eq. () can be further decomposed into

E(r)=(Subscript[E, even])(r)+(Subscript[E, odd])(r),

the even parity part (i.e. (Subscript[E, even])(r)=(Subscript[E, even])(-r))

(Subscript[E, even])(r)=((Subscript[ω, k]^(1/2))/(Sqrt[2]))cos(k·r)(Subscript[e, k,α]) i ((Subscript[a, k,α])-(Subscript[a, k,α]^†)),

the odd parity part (i.e. (Subscript[E, odd])(r)=-(Subscript[E, odd])(-r))

(Subscript[E, odd])(r)=-((Subscript[ω, k]^(1/2))/(Sqrt[2]))sin(k·r)(Subscript[e, k,α]) ((Subscript[a, k,α])+(Subscript[a, k,α]^†)).

Single-Mode Approximation: focus on a specific mode of fixed wavelength, parity (e.g. odd), and polarization,

ω - photon energy (mode frequency),

e - the polarization vector of the mode,

(SuperDagger[a]),a - the creation and annihilation operators of the mode, such that the electromagnetic field energy is given by

g - the light-matter coupling strength (can have spacial dependence, following the mode profile in the space)

g=((((ω/2)))^(1/2))sin(k·r)

d - dipole operator describing how matter responses to electric field along the polarization e direction:

d=e·d.

A Two-Level Atom

Block[{L=8,ψ},
ψ[n_][x_]:=(2x)^n Exp[-Abs[x]]/Sqrt[If[n==0,1/2,n Gamma[2n]]];
Graphics[{Translate[{Arrowheads[{-0.1,0.1}],AbsoluteThickness[1],Arrow[{0,#}&/@{0,1}],Text["\!\( ω\_0\)",{0,0.5},{-1.2,0}]},{0.9L,0}],Translate[{{AbsoluteThickness[1],Line[{#,0}&/@{-L,L}]},{Gray,Cases[Plot[0.2ψ[#][x],{x,-L,L},PlotStyle->Black,PlotRange->All],_Line,∞]}},{0,#}]&/@{0,1},
Text["\!\((Ket[g])\)",{L,0},{-2,0}],Text["\!\((Ket[e])\)",{L,1},{-2,0}],Translate[{Arrowheads[0.1],AbsoluteThickness[1],Arrow[{0,#}&/@{-0.2,1.4}],Translate[Line[Offset[{#,0}]&/@{0,2}],{0,#}]&/@{0,0.5,1},Text[Style["\!\(-\(ω\_0\/2\)\)","InlineFormula"],{0,0},{1,0}],Text[Style["\!\(0\)","InlineFormula"],{0,0.5},{2,0}],Text[Style["\!\(\(ω\_0\/2\)\)","InlineFormula"],{0,1},{1,0}],Text["Energy",{0,1.4},{-1.2,0}]},{-1.2L,0}]},AspectRatio->1.2,ImageSize->128]
]

(Ket[g]) - ground state of the atom,

(Ket[e]) - exited state of the atom,

(Subscript[ω, 0])=(Subscript[E, e])-(Subscript[E, g]) - atom excitation energy (energy level splitting, transition frequency).

Inversion operator: (σ^z) splits the energy levels between (Ket[g]) and (Ket[e]),

Transition operator: (σ^x) mixes (Ket[g]) and (Ket[e]), leading to different dipole moments (by deforming the electron density):

Block[{L=6,ψ,wplt,dplt},
ψ[n_][x_]:=(2x)^n Exp[-Abs[x]]/Sqrt[If[n==0,1/2,n Gamma[2n]]];
wplt[f_]:=Graphics[{{AbsoluteThickness[1],Line[{#,0}&/@{-L,L}]},{Gray,Cases[Plot[f[x],{x,-L,L},PlotStyle->Black,PlotRange->All],_Line,∞]}},PlotRange->{-1.5,1.5},AspectRatio->0.5];
dplt[f_]:=Graphics[{{AbsoluteThickness[1],Line[{#,0}&/@{-L,L}]},{Cases[Plot[(f[x])^2,{x,-L,L},PlotStyle->Blue,Filling->Axis,PlotStyle->Black,PlotRange->All],_GraphicsComplex,∞]},
Block[{d=NIntegrate[x f[x]^2,{x,-L,L}]},{{Blue,Text[Style["-","SansText"],{#,0},{0,0.6}]&/@({-2,1,2,4}d)},Translate[{Arrowheads[0.08],
Arrow[{#,0}&/@{d,0}],If[d>0,Text["\!\(d<0\)",{0,0},{1.2,0}],Text["\!\(d>0\)",{0,0},{-1.2,0}]]},{0,0.4}]}],
{Red,Disk[{0,0},0.25{2.8,1}]},{White,Text[Style["+","SansText",Bold],{0,0},{-0.1,0.1}]}},PlotRange->{-1.2,1.2},AspectRatio->0.5];
Style[Grid[{{"Eigenvalue","Eigenstate","Wave func.","Density dist."},{"\!\(σ\^x=+1\)","\!\(1\/\@2((Ket[g])+(Ket[e]))\)",wplt[(ψ[0][#]+ψ[1][#])/Sqrt[2]&],dplt[(ψ[0][#]+ψ[1][#])/Sqrt[2]&]},
{"\!\(σ\^x=-1\)","\!\(1\/\@2((Ket[g])-(Ket[e]))\)",wplt[(ψ[0][#]-ψ[1][#])/Sqrt[2]&],dplt[(ψ[0][#]-ψ[1][#])/Sqrt[2]&]}}],"DisplayFormula"]
]

This implies d∝-(σ^x) (any proportionality constant here can be absorbed into the coupling constant g below), and the light-matter compiling Hamiltonian in Eq. () becomes

for the two-level atom.

Jaynes-Cummings Model

H=((Subscript[ω, 0])/2) (σ^z)+ω ((SuperDagger[a])a +(1/2))-g (σ^x) ((SuperDagger[a])+a).

Raising and Lowering Operators: Introduce (σ^±) to raise and lower the electron between levels

such that, by definition Eq. (),

(σ^x)=(SuperPlus[σ])+(SuperMinus[σ]).

Decoupled Operator Dynamics: In the decoupled limit g->0, the operators evolves as

a(t)=a(0)(e^(-i ω t)),(SuperDagger[a])(t)=(SuperDagger[a])(0)(e^(i ω t));
(σ^±)(t)=(σ^±)(0)(e^(±i (Subscript[ω, 0]) t)).

Show that Eq. () satisfies the Heisenberg equation (∂_t)A=i [H,A] that governs the operator dynamics, for H in the g→0 limit.

H=(Subscript[ω, 0]) (σ^z)+ω (SuperDagger[a])a,

[H,a]=ω[(SuperDagger[a])a,a]=ω[(SuperDagger[a]),a]a=-ω a,
[H,(SuperDagger[a])]=ω[(SuperDagger[a])a,(SuperDagger[a])]=ω (SuperDagger[a])[a,(SuperDagger[a])]=ω (SuperDagger[a]),
[H,(σ^±)]=((Subscript[ω, 0])/2)[(σ^z),(σ^±)]=((Subscript[ω, 0])/2)(±2(σ^±))=± (Subscript[ω, 0])(σ^±).

(∂_t)a=-i ω a=i [H,a],
(∂_t)(SuperDagger[a])=i ω (SuperDagger[a])=i [H,(SuperDagger[a])],
(∂_t)(σ^±)=± i (Subscript[ω, 0]) (σ^±)=i [H,(σ^±)].

Rotating Wave Approximation: The light-matter coupling Hamiltonian (Subscript[H, cp]) can be expanded as

(Subscript[H, cp])=-g ((SuperPlus[σ])+(SuperMinus[σ]))((SuperDagger[a])+a)
=-g ((SuperPlus[σ])(SuperDagger[a])+(SuperMinus[σ])(SuperDagger[a])+(SuperPlus[σ])a+(SuperMinus[σ])a).

For small g, the approximate time dependence of these terms are

(SuperPlus[σ])(SuperDagger[a])∼(e^(i (+(Subscript[ω, 0])+ω) t)),
(SuperMinus[σ])(SuperDagger[a])∼(e^(i (-(Subscript[ω, 0])+ω) t)),
(SuperPlus[σ])a∼(e^(i (+(Subscript[ω, 0])-ω) t)),
(SuperMinus[σ])a∼(e^(i (-(Subscript[ω, 0])-ω) t)).

When the level spacing (Subscript[ω, 0]) and the photon frequency ω are comparable (i.e. (Subscript[ω, 0])≈ω), (e^(±i ((Subscript[ω, 0])+ω)t)) ((SuperPlus[σ])(SuperDagger[a]) and (SuperMinus[σ])a) oscillate much more rapidly than (e^(±i((Subscript[ω, 0])-ω)t)) ((SuperMinus[σ])(SuperDagger[a]) and (SuperPlus[σ])a), leading to their effect averaging out to zero over time. Thus (Subscript[H, cp]) reduces to

(Subscript[H, cp])=-g ((SuperMinus[σ])(SuperDagger[a])+(SuperPlus[σ])a),

under the rotating wave approximation.

(SuperMinus[σ])(SuperDagger[a]):=(Ket[g])(Bra[e])⊗(SuperDagger[a]) - atom decays from (Ket[e]) to (Ket[g]) to emit a photon.

(SuperPlus[σ])a:=(Ket[e])(Bra[g])⊗a - atom excites from (Ket[g]) to (Ket[e]) to absorb a photon.

Rabi Oscillation

(Ket[e])⊗(Ket[n]): excited state atom with n photons,

(Ket[g])⊗(Ket[n+1]): ground state atom with n+1 photons.

ℋ=span{(Ket[e])⊗(Ket[n]), (Ket[g])⊗(Ket[n+1])}.

H≏(({{((Subscript[ω, 0])/2)+ω (n+(1/2)), -g(Sqrt[n+1])}, {-g(Sqrt[n+1]), -((Subscript[ω, 0])/2)+ω (n+(3/2))}})),

I=(({{1, 0}, {0, 1}})), Z=(({{1, 0}, {0, -1}})), X=(({{0, 1}, {1, 0}})),

(Subscript[E, 0]):=ω (n+1) is a background energy of the system,

Δ:=(Subscript[ω, 0])-ω is difference between the atomic level resonant frequency (Subscript[ω, 0]) and the photon frequency ω, also called the detuning.

g is the coupling strength, and n is the photon number.

Manipulate[DynamicModule[{H,P},
H=Δ/2 PauliMatrix[3]-g Sqrt[n+1] PauliMatrix[1];
P[s_]:=(PauliMatrix[0]+s PauliMatrix[3])/2;
Row[{ListLinePlot[Thread@Table[{{t,Re[Conjugate[#].P[1].#]},{t,Re[Conjugate[#].P[-1].#]}}&@MatrixExp[I H t,{1.,0}],{t,0,5,0.02}],PlotRange->{0,1},PlotRangePadding->Scaled[.02],FrameLabel->{"\!\( t\)","probability"},BaselinePosition->Scaled[0.6],ImageSize->180],Grid[{Graphics[{#1,Line[{#,0}&/@{0,1}]},AspectRatio->0.1,ImageSize->15],Style[#2,"DisplayFormula"]}&@@@{{Red,"\!\((Ket[e])(Bra[e])\)"},{Blue,"\!\((Ket[g])(Bra[g])\)"}}]}]],{{Δ,0},-5,5,ImageSize->Small},{{g,1},0,1,ImageSize->Small},{{n,4},Range[0,4]},Paneled->False]

The Rabi oscillation frequency is given by

Prove Eq. ().

(Subscript[ω, ±])=(Subscript[E, 0])±(Sqrt[((((Δ/2)))^2)+(g^2)(n+1)]).

(Ket[ψ(0)])=(SubPlus[c])(Ket[(SubPlus[ψ])])+(SubMinus[c])(Ket[(SubMinus[ψ])]),

(Ket[ψ(t)])=(SubPlus[c])(e^(-i (SubPlus[ω])t))(Ket[(SubPlus[ψ])])+(SubMinus[c])(e^(-i (SubMinus[ω])t))(Ket[(SubMinus[ψ])]),

⟨O⟩(t)=(Bra[ψ(t)])O(Ket[ψ(t)])
=(((|(SubPlus[c])|)^2)(Bra[(SubPlus[ψ])])O(Ket[(SubPlus[ψ])])+((|(SubMinus[c])|)^2)(Bra[(SubMinus[ψ])])O(Ket[(SubMinus[ψ])]))+((Subscript[c, +]^( * ))(SubMinus[c])(Bra[(SubPlus[ψ])])O(Ket[(SubMinus[ψ])])(e^(i ((SubPlus[ω])-(SubMinus[ω]))t))+c.c.).

The first term is constant in time.

Only the second term has time dependence, and oscillates with a unique frequency

Ω=(SubPlus[ω])-(SubMinus[ω])
=2(Sqrt[((((Δ/2)))^2)+(g^2)(n+1)])
=(Sqrt[(Δ^2)+4(g^2)(n+1)]).

Vacuum Rabi oscillations: Ω remains finite even if n=0 => Rabi oscillation can occur in vacuum (typically inside a high-quality optical cavity).

Coherent State

Single-Mode Photon

A=(1/(Sqrt[2]))(ω^(-1/2))e (a (e^(i k·r))+(SuperDagger[a])(e^(-i k·r))),
E=(i/(Sqrt[2]))(ω^(1/2))e (a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r))),
B=(i/(Sqrt[2]))(ω^(-1/2))k×e (a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r))).

a, (SuperDagger[a]) - photon annihilation, creation operators, satisfying

a(Ket[vac])=0.

Definition

its eigenvalues α∈ℂ can be complex,

its eigenstates with different eigenvalues may not be orthogonal, i.e. (BraKet[(Subscript[α, 1]), (Subscript[α, 2])])≠δ((Subscript[α, 1])-(Subscript[α, 2])).

Nevertheless, we do assume that (Ket[α]) is normalized, i.e. (BraKet[α, α])=1.

(Bra[α])a(Ket[α])=α,
(Bra[α])(SuperDagger[a])(Ket[α])=(SuperStar[α]),
(Bra[α])(a^n)(Ket[α])=(α^n),
(((Bra[α])((SuperDagger[a])))^n)(Ket[α])=((((SuperStar[α])))^n),
(Bra[α])(SuperDagger[a])a(Ket[α])=(SuperStar[α])α=((|α|)^2),
(Bra[α])a (SuperDagger[a])(Ket[α])=(Bra[α])((SuperDagger[a])a+𝟙)(Ket[α])=((|α|)^2)+1.

Physical Properties

α=|α|(e^(i φ)).

Graphics[{{AbsoluteThickness[1],Gray,Circle[{0,0},1],Line[{{0,0},{0.6,0.8}}],Dashed,Line[{{0.6,0},{0.6,0.8}}],Line[{{0,0.8},{0.6,0.8}}]},
{Arrowheads[0.08],Arrow[1.3{-#,#}]&/@{{1,0},{0,1}}},{AbsolutePointSize[6],Point[{0.6,0.8}]},{Circle[{0,0},0.2,{0,ArcTan[0.6,0.8]}],Text["\!\( α\)",{0.6,0.8},{-1,-1}],Text["φ",{0.3,0.1}],Text["\!\(\[LeftBracketingBar]α\[RightBracketingBar]\)",{0.3,0.4},{0.5,-0.5}]},Text["\!\( Re α\)",{1.3,0},{-1,0}],Text["\!\( Im α\)",{0,1.3},{0,-1}],Text["ℂ",{-1,-1}]},ImageSize->{Automatic,140}]

Linear properties in fields:

Derive Eq. () using Eq. ().

(Bra[α])A(Ket[α])=(1/(Sqrt[2]))(ω^(-1/2))e (Bra[α])(a (e^(i k·r))+(SuperDagger[a])(e^(-i k·r)))(Ket[α])
=(1/(Sqrt[2]))(ω^(-1/2))e (α (e^(i k·r))+(SuperStar[α])(e^(-i k·r)))
=(1/(Sqrt[2]))(ω^(-1/2))e |α|((e^(i (k·r+φ)))+(e^(-i (k·r+φ))))
=(Sqrt[2])(ω^(-1/2))e |α|cos(k·r+φ)

(Bra[α])E(Ket[α])=(i/(Sqrt[2]))(ω^(1/2))e (Bra[α])(a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r)))(Ket[α])
=(i/(Sqrt[2]))(ω^(1/2))e (α (e^(i k·r))-(SuperStar[α])(e^(-i k·r)))
=(i/(Sqrt[2]))(ω^(1/2))e  |α|((e^(i (k·r+φ)))-(e^(-i (k·r+φ))))
=-(Sqrt[2])(ω^(1/2))e |α|sin(k·r+φ),

(Bra[α])B(Ket[α])=(i/(Sqrt[2]))(ω^(-1/2))k×e (Bra[α])(a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r)))(Ket[α])
=(i/(Sqrt[2]))(ω^(-1/2))k×e (α (e^(i k·r))-(SuperStar[α])(e^(-i k·r)))
=(i/(Sqrt[2]))(ω^(-1/2))k×e  |α|((e^(i (k·r+φ)))-(e^(-i (k·r+φ))))
=-(Sqrt[2])(ω^(-1/2))k×e |α|sin(k·r+φ),

The coherent state (Ket[α]) of a photon mode (of wave vector k and polarization e) describes a snapshot of electromagnetic wave in the space with

|α| - wave amplitude,

φ - phase of the wave.

Quadratic properties in fields:

(Bra[α])(SuperDagger[E])·E(Ket[α])=(Bra[α])(SuperDagger[B])·B(Ket[α])=2ω ((|α|)^2)(sin^2)(k·r+φ)+(ω/2).

Derive Eq. () using Eq. ().

(Bra[α])(SuperDagger[E])·E(Ket[α])=(ω/2) (Bra[α])((SuperDagger[a]) (e^(-i k·r))-a (e^(i k·r)))(a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r)))(Ket[α])
=(ω/2) (Bra[α])((SuperDagger[a]) a -(SuperDagger[a]) (SuperDagger[a])(e^(-2i k·r))-a a (e^(2i k·r))+a (SuperDagger[a]))(Ket[α])
=(ω/2) (Bra[α])((SuperDagger[a]) a -(SuperDagger[a]) (SuperDagger[a])(e^(-2i k·r))-a a (e^(2i k·r))+(SuperDagger[a])a+𝟙)(Ket[α])
=(ω/2) (-(SuperStar[α]) (SuperStar[α])(e^(-2i k·r))-α α (e^(2i k·r))+2(SuperStar[α])α)+(ω/2)
=(ω/2) (((i (α (e^(i k·r))-(SuperStar[α]) (e^(-i k·r)))))^2)+(ω/2)
=(ω/2) (((i |α|((e^(i (k·r+φ)))-(e^(-i (k·r+φ))))))^2)+(ω/2)
=(ω/2) (((-2|α|sin(k·r+φ)))^2)+(ω/2)
=2ω ((|α|)^2)(sin^2)(k·r+φ)+(ω/2)

(Bra[α])(SuperDagger[B])·B(Ket[α])=((k^2)/(2 ω)) (Bra[α])((SuperDagger[a]) (e^(-i k·r))-a (e^(i k·r)))(a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r)))(Ket[α])
=(ω/2) (Bra[α])((SuperDagger[a]) (e^(-i k·r))-a (e^(i k·r)))(a (e^(i k·r))-(SuperDagger[a])(e^(-i k·r)))(Ket[α])
=2ω ((|α|)^2)(sin^2)(k·r+φ)+(ω/2).

Therefore, for single-mode photon coherent state,

var E=(Bra[α])(SuperDagger[E])·E(Ket[α])-(Bra[α])(SuperDagger[E])(Ket[α])·(Bra[α])E(Ket[α])=(ω/2),
var B=(Bra[α])(SuperDagger[B])·B(Ket[α])-(Bra[α])(SuperDagger[B])(Ket[α])·(Bra[α])B(Ket[α])=(ω/2).

var E var B≥(((ω/2))^2).

Fock State Representation

Verify Eq. () by showing that (Ket[α]) constructed this way satisfies the definition Eq. ().

a(Ket[α])=(e^(-(1/2)((|α|)^2)))((∑_(n=1)^∞)((α^n)/(Sqrt[n!]))a(Ket[n])+a(Ket[0]))
=(e^(-(1/2)((|α|)^2)))((∑_(n=1)^∞)((α^n)/(Sqrt[n!]))(Sqrt[n])(Ket[n-1])+0)
=(e^(-(1/2)((|α|)^2)))(∑_(n=1)^∞)((α^n)/(Sqrt[(n-1)!]))(Ket[n-1]),

a(Ket[α])=(e^(-(1/2)((|α|)^2)))(∑_(m=0)^∞)((α^(m+1))/(Sqrt[m!]))(Ket[m])
=α(Ket[α]).

In particular, the vacuum state (Ket[vac]):=(Ket[n=0]) is also a coherent state with α=0, and admits minimal uncertainty.

Based on Eq. (), the probability to observe n photons in the coherent state (Ket[α]) is given by

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The mean photon number is determined by the expectation value of the photon number operator (OverHat[n])=(SuperDagger[a])a,

Verify Eq. ().

By Eq. () and Eq. (),

(Bra[α])(OverHat[n])(Ket[α])=(Bra[α])(SuperDagger[a])a(Ket[α])
=(Bra[α])(SuperStar[α])α(Ket[α])
=((|α|)^2)(BraKet[α, α])
=((|α|)^2).

(Subscript[p, α])(n)=((Subscript[⟨n⟩, α]^n)/(n!))(e^(-(Subscript[⟨n⟩, α]))),

Time Evolution

H=ω((OverHat[n])+(1/2)).

U(t)=(e^(-i H t/ℏ))=(e^(-((i ω t)/2) ))(e^(-i ω t (OverHat[n]))).

U(t)(Ket[α])=(e^(-((i ω t)/2) ))(Ket[α (e^(-i ω t))]):=(e^(-((i ω t)/2) ))(Ket[α(t)]).

Show Eq. ().

U(t)(Ket[α])=((e^(-((i ω t)/2) ))(e^(-i ω t (OverHat[n]))))((e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(Ket[n]))
=(e^(-((i ω t)/2) ))(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(e^(-i ω t (OverHat[n])))(Ket[n])
=(e^(-((i ω t)/2) ))(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((((α (e^(-i ω t))))^n)/(Sqrt[n!]))(Ket[n])
=(e^(-((i ω t)/2) ))(e^(-(1/2)((|α (e^(-i ω t))|)^2)))(∑_(n=0)^∞)((((α (e^(-i ω t))))^n)/(Sqrt[n!]))(Ket[n]).

the amplitude |α| remains the same,

the phase φ→φ-ω t will rotate with time t by the angular frequency ω.

(Bra[α(t)])E(Ket[α(t)])=-(((2ω))^(1/2))e |α|sin(k·r-ω t),
(Bra[α(t)])B(Ket[α(t)])=-(((2/ω))^(1/2))k×e |α|sin(k·r-ω t),

They minimize the quantum fluctuation in both phase and amplitude on top of the classical (average) behavior of wave, saturating their uncertainty bound.

Large |α| => large average number of photons (Subscript[⟨n⟩, α])=((|α|)^2) in the coherent state => a macroscopic occupation of the same photon mode with quantum coherence, — making coherent states an ideal description of laser light (intense and coherent light).

Superradiant Light

Tavis-Cummings Model

N - total number of atoms, each indexed by i=1,2,…,N.

(Subscript[ω, 0]) - atom excitation energy (energy level splitting, transition frequency).

ω - photon energy (mode frequency),

g - light-matter coupling strength.

({{ , atomic ((2^N)-dim),  , photonic (∞-dim)}, {ℋ=, (((span{(Ket[g]),(Ket[e])}))^(⊗N)), ⊗, span{(Ket[0]),(Ket[1]),(Ket[2]),…}}})

Goal: find the ground state (lowest energy state) of H.

Challenge: exact diagonalization is computationally difficult, given the huge (infinite) Hilbert space dimension.

U(1) Symmetry

Symmetry ⇔ Conservation Law: The excitation number conservation generates a U(1) symmetry, corresponding to the unitary operator (for any given U(1) rotation angle θ)

Under the U(1) symmetry transformation,

a->(SuperDagger[U(θ)])a U(θ)=(e^(i θ))a,
(SuperDagger[a])->(SuperDagger[U(θ)])(SuperDagger[a]) U(θ)=(e^(-i θ))(SuperDagger[a]),
(Subscript[σ, i]^±)->(SuperDagger[U(θ)])(Subscript[σ, i]^±) U(θ)=(e^(∓i θ))(Subscript[σ, i]^±).

Check Eq. ().

a (Subscript[N, exc])=((Subscript[N, exc])+𝟙)a,
(SuperDagger[a]) (Subscript[N, exc])=((Subscript[N, exc])-𝟙)(SuperDagger[a]),
(Subscript[σ, i]^±) (Subscript[N, exc])=((Subscript[N, exc])∓𝟙)(Subscript[σ, i]^±).

For example,

a (Subscript[N, exc])=a ((SuperDagger[a])a+(∑_i)(((Subscript[σ, i]^z)+𝟙)/2))
=a (SuperDagger[a])a+a(∑_i)(((Subscript[σ, i]^z)+𝟙)/2)
=((SuperDagger[a])a+𝟙)a+(∑_i)(((Subscript[σ, i]^z)+𝟙)/2)a
=((SuperDagger[a])a+𝟙+(∑_i)(((Subscript[σ, i]^z)+𝟙)/2))a
=((Subscript[N, exc])+𝟙)a.

Similar for (SuperDagger[a])

(SuperDagger[a]) (Subscript[N, exc])=(SuperDagger[a]) ((SuperDagger[a])a+(∑_i)(((Subscript[σ, i]^z)+𝟙)/2))
=(SuperDagger[a]) (SuperDagger[a])a+(SuperDagger[a])(∑_i)(((Subscript[σ, i]^z)+𝟙)/2)
=(SuperDagger[a])(a (SuperDagger[a])-𝟙)+(∑_i)(((Subscript[σ, i]^z)+𝟙)/2)(SuperDagger[a])
=((SuperDagger[a])a-𝟙+(∑_i)(((Subscript[σ, i]^z)+𝟙)/2))(SuperDagger[a])
=((Subscript[N, exc])-𝟙)(SuperDagger[a]).

For any particular qubit, we observe that

(SuperPlus[σ])(σ^z)=(Ket[e])(Bra[g])((Ket[e])(Bra[e])-(Ket[g])(Bra[g]))=-(Ket[e])(Bra[g])=-(SuperPlus[σ]),
(σ^z)(SuperPlus[σ])=((Ket[e])(Bra[e])-(Ket[g])(Bra[g]))(Ket[e])(Bra[g])=(Ket[e])(Bra[g])=(SuperPlus[σ]),

therefore,

(SuperPlus[σ])(σ^z)=(σ^z)(SuperPlus[σ])-2(SuperPlus[σ])=((σ^z)-2𝟙)(SuperPlus[σ]).

Similarly,

(SuperMinus[σ])(σ^z)=(σ^z)(SuperMinus[σ])+2(SuperMinus[σ])=((σ^z)+2𝟙)(SuperMinus[σ]).

Apply these conclusions to every qubit,

(Subscript[σ, i]^±)(Subscript[σ, i]^z)=((Subscript[σ, i]^z)∓2𝟙)(Subscript[σ, i]^±),

which implies

(Subscript[σ, i]^±)(Subscript[N, exc])=(Subscript[σ, i]^±)((SuperDagger[a])a+(∑_j)(((Subscript[σ, j]^z)+𝟙)/2))
=(Subscript[σ, i]^±)(SuperDagger[a])a+(Subscript[σ, i]^±)(∑_(j≠i))(((Subscript[σ, j]^z)+𝟙)/2)+(Subscript[σ, i]^±)(((Subscript[σ, i]^z)+𝟙)/2)
=(SuperDagger[a])a (Subscript[σ, i]^±)+(∑_(j≠i))(((Subscript[σ, j]^z)+𝟙)/2)(Subscript[σ, i]^±)+((((Subscript[σ, i]^z)∓2𝟙)+𝟙)/2)(Subscript[σ, i]^±)
=((SuperDagger[a])a +(∑_j)(((Subscript[σ, j]^z)+𝟙)/2)∓𝟙)(Subscript[σ, i]^±)
=((Subscript[N, exc])∓𝟙)(Subscript[σ, i]^±)

U(SuperDagger[(θ)])a U(θ)=(e^(-i θ (Subscript[N, exc])))a (e^(i θ (Subscript[N, exc])))
=(e^(-i θ (Subscript[N, exc])))(e^(i θ ((Subscript[N, exc])+𝟙)))a
=(e^(i θ))a,

U(SuperDagger[(θ)])(SuperDagger[a]) U(θ)=(e^(-i θ (Subscript[N, exc])))(SuperDagger[a]) (e^(i θ (Subscript[N, exc])))
=(e^(-i θ (Subscript[N, exc])))(e^(i θ ((Subscript[N, exc])-𝟙)))(SuperDagger[a])
=(e^(-i θ))a,

U(SuperDagger[(θ)])(Subscript[σ, i]^±) U(θ)=(e^(-i θ (Subscript[N, exc])))(Subscript[σ, i]^±) (e^(i θ (Subscript[N, exc])))
=(e^(-i θ (Subscript[N, exc])))(e^(i θ ((Subscript[N, exc])∓𝟙)))(Subscript[σ, i]^±)
=(e^(∓i θ))(Subscript[σ, i]^±).

Therefore, the Tavis-Cummings Hamiltonian H in Eq. () is invariant under the symmetry transformation, i.e. ∀θ:[H,U(θ)]=0.

Mean-Field Approach

Variational Ansatz: Propose a trial (variational) state that disentangle the atomic and photonic degrees of freedom.

Photons: assumed to be in a coherent state with complex parameter α

(Subscript[(Ket[α]), photon])=(e^(-(1/2)((|α|)^2)))(∑_(n=0)^∞)((α^n)/(Sqrt[n!]))(Ket[n]).

Under U(1) symmetry transformation: α->(e^(i θ))α.

Atoms: assumed to be identical product state of

(Subscript[(Ket[ψ(α)]), atom])=(Subscript[ψ, e])(α)(Ket[e])+(Subscript[ψ, g])(α)(Ket[g]),

which could also depend on the parameter α.

Objective: Minimize the expectation value of the Tavis-Cummings Hamiltonian H with respect to the variational state.

Hope: the minimal energy state will be a good approximation of the true ground state within the variational subspace.

Mean-Field Energy

Photonic Expectation: The photons are in a coherent state (Ket[α]),

(Bra[α])a(Ket[α])=α,
(Bra[α])(SuperDagger[a])(Ket[α])=(SuperStar[α]),
(Bra[α])(SuperDagger[a])a(Ket[α])=((|α|)^2).

Taking the expectation value of the photon part of H in Eq. (),

(Bra[α])H(Ket[α])=((Subscript[ω, 0])/2)(∑_(i=1)^N)(Subscript[σ, i]^z)+ω (((|α|)^2)+(1/2))-g(∑_(i=1)^N)((Subscript[σ, i]^(-))(SuperStar[α])+(Subscript[σ, i]^(+))α)
=ω (((|α|)^2)+(1/2))+(∑_(i=1)^N)(Subscript[H, i])(α),

which has decoupled into an overall photon energy plus a sum of N identical effective atomic Hamiltonians (Subscript[H, i])(α).

Atomic Expectation: (Subscript[H, i])(α) is the effective Hamiltonian for the ith atom on the photon coherent state background

(Subscript[H, i])(α)=((Subscript[ω, 0])/2)(Subscript[σ, i]^z)-g ((Subscript[σ, i]^(-))(SuperStar[α])+(Subscript[σ, i]^(+))α),

which can be represented as a 2×2 matrix in the {(Ket[e]),(Ket[g])} basis,

(Subscript[H, i])(α)≏(({{(Subscript[ω, 0])/2, -g α}, {-g (SuperStar[α]), -(Subscript[ω, 0])/2}})),

whose minimal energy expectation value is given by the lowest energy eigenvalue:

(Bra[ψ(α)])(Subscript[H, i])(α)(Ket[ψ(α)])=-(Sqrt[((Subscript[ω, 0]^2)/4)+(g^2)((|α|)^2)]).

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Mean-Field Solutions

((∂(Subscript[E, MF])(α))/(∂α))=(2ω-N (((g^2)(((Subscript[ω, 0]^2)/4)+(g^2)((|α|)^2)))^(-1/2)))(SuperStar[α])=0.

Trivial solution: α=0 (always valid)

The mean-field energy reaches

(Subscript[E, MF])(0)=(ω/2)-N((Subscript[ω, 0])/2).

The variational state becomes

(Ket[Ψ(0)])=(Subscript[(Ket[g]), atom]^(⊗N))⊗(Subscript[(Ket[0]), photon]),

describing: all atoms in ground states & photon vacuum state.

Non-trivial solution: around the circle of

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which are valid only if

The mean-field energy (always lower than (Subscript[E, MF])(0) as long as Eq. () holds)

(Subscript[E, MF])((Subscript[α, ★]))=(ω/2)-(1/4)((((g^2)(N^2))/ω)+((ω (Subscript[ω, 0]^2))/(g^2))).

The variation state is

(Ket[Ψ((Subscript[α, ★]))])=(Subscript[(Ket[ψ((Subscript[α, ★]))]), atom]^(⊗N))⊗(Subscript[(Ket[(Subscript[α, ★])]), photon]).

In the large-N limit (N≫1): photon in a coherent state (Ket[(Subscript[α, ★])]) with the average photon number (light intensity)

⟨n⟩=((|(Subscript[α, ★])|)^2)∼(N^2),

and each atom in an equal-weight superposition of (Ket[g]) and (Ket[e]) with their relative phase locked to (Subscript[α, ★])/|(Subscript[α, ★])|

(Ket[ψ((Subscript[α, ★]))])≃(1/(Sqrt[2]))((Ket[g])+((Subscript[α, ★])/(|(Subscript[α, ★])|))(Ket[e])).

Spontaneous Symmetry Breaking: Any choice of (Subscript[α, ★])=|(Subscript[α, ★])|(e^(i θ)) breaks the U(1) symmetry spontaneously, i.e. H respects the U(1) symmetry, but its (approximate) ground state (Ket[Ψ((Subscript[α, ★]))]) does not.

Superradiant Phase

Cooperative Radiation: Many atoms radiate coherently, such that the emitted light intensity add constructively.

Strong Intensity: Macroscopic photon occupation with ⟨n⟩∼(N^2) (in contrast to the linear N-scaling for independent spontaneous emission),

Phase Coherence: Emitted photons are phase-locked, and the collective light-matter interaction stabilizes the phase of (Subscript[α, ★]), resulting in the spontaneous breaking of U(1) symmetry.

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