Theoretical Quantum Optics and Quantum Information (SS 2017) Exercise II (discussion: 17.05.17)

1. Coherent states

(a) Eigenstates of ˆb† Show that an eigenstate of ˆb† cannot exist. Hint: Expand a putative eigenstate of ˆb† in the basis of Fock states |ni

∞ X |βi = |ni hn|βi . n=0 Then you will see that the creation operator produces a new superposition of occupations |ni with different range of occupancy than |βi.

(b) Action of ˆb† Show that  ∂ α∗  ˆb† |αi = + |αi ∂α 2

where |αi is a coherent state, i.e. ˆb |αi = α |αi. Use the Wirtinger derivative ∂ 1  ∂ ∂  = − i . ∂(x + iy) 2 ∂x ∂y

Note that the chain rule of differentiation does not hold in general for matrix functions!

(c) Superposition of coherent states Calculate the probability distribution of the photon number and the mean photon number for the state

|ψ±i ∝ (|αi ± |−αi), (1)

which is a superposition of two coherent states. Start by normalizing |ψ±i. Hint: The resulting prefactor can be simplified by the use of hyperbolic func- tions. 2. Two coupled harmonic oscillators Consider a system that consists of two coupled harmonic oscillators with frequen-

cies ω1 and ω2. The first oscillator corresponds to a mode of an electromagnetic field inside a cavity, which couples via radiation pressure to a mechanical oscilla- tor, as shown in the figure. The parameter χ characterizes the coupling strength

between them. The dynamics of such a system is governed by the standard Hamiltonian in optomechanics

2 ˆ X ˆ†ˆ ˆ†ˆ ˆ† ˆ
H = ~ ωibi bi − ~χb1b1 b2 + b2 . i=1 ˆ† ˆ Here, bi and bi, i ∈ {1, 2} denote the bosonic creation and annihilation operator of the corresponding harmonic oscillator.

(a) Show that the unitary displacement transformation

Dˆ†(βˆ)Hˆ Dˆ(βˆ) =: Hˆ 0 (2)

ˆ χ ˆ†ˆ decouples the two harmonic oscillators for β = b b1. Here, the generalized ω2 1 ˆ ˆ ˆˆ† ˆ†ˆ
ˆ displacement operator is defined as D(β) := exp βb2 −β b2 . Note that β is ˆ ˆ†ˆ a hermitian operator, that fulfills [β, b1b1] = 0. ˆ† ˆ ˆ ˆ ˆ ˆ ˆ Hint: Verify first, that D (β)b2D(β) = b2 + β.

(b) The displaced Hamiltonian Hˆ 0 obeys the following eigenequation

ˆ 0 H |En1,n2 i = En1,n2 |En1,n2 i .

ˆ ˆ
Find the eigenenergies En1,n2 and eigenstates D β |En1,n2 i of the non- displaced Hamiltonian Hˆ .

(c) With the eigenstates

ˆ ˆ
ˆ
ˆ
D β |En1,n2 i = D χn1/ω2 |n1i ⊗ |n2i ≡ D χn1/ω2 |n1, n2i (3) and the eigenenergies

2 χ 2 En1,n2 = ~ω1n1 + ~ω2n2 − ~ n1 (4) ω2 one can study the dynamics of the system. Use the Schr¨odinger equation

∂ i |ψ(t)i = Hˆ |ψ(t)i ~∂t to calculate the time evolution of the state |ψ0i = |n1i⊗|αi ≡ |n1, αi, where the first harmonic oscillator is initially prepared in a Fock state |n1i and the second one in a coherent state |αi by expanding |ψ(t)i in the eigenbasis of Hˆ . Show that (up to a global phase)

−iω2t |ψ(t)i = |n1i ⊗ η(t) + αe with η(t) := χn1 (1 − e−iω2t). ω2 Hint: Use the expansion of a coherent state in the Fock basis |γi = 2 ∞ − |γ| γk † e 2 P √ |ki with γ ∈ C. Also use that Dˆ (µ) |νi = Dˆ(−µ) |νi = |ν − µi k! k=0 with µ, ν ∈ C, up to a global phase. 3. Squeezed States

(a) Squeeze Operator

By using the Baker-Hausdorff formula, 1 1 1 eABe−A = B + [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + ··· , 1! 2! 3! show that Sˆ†(ε)ˆbSˆ(ε) = ˆb cosh r − ˆb†e2iφ sinh r

Sˆ†(ε)ˆb†Sˆ(ε) = ˆb† cosh r − ˆbe−2iφ sinh r and

Sˆ†(ε)f(ˆb, ˆb†)Sˆ(ε) = f(ˆb cosh r − ˆb†e2iφ sinh r , ˆb† cosh r − ˆbe−2iφ sinh r), (5)

where 1   Sˆ(ε) = exp ε∗ˆb2 − ε(ˆb†)2 , ε = re2iφ 2 and f(ˆx, yˆ) is a function which can be expanded into a power series.

(b) Squeeze and Displacement operator (bonus) The order of the Dˆ and Sˆ operators in the definition of a squeezed state in Eq. (36) (see lecture) does matter, since

Dˆ(α)Sˆ(ε) = Sˆ(ε)Dˆ(γ).

Express the parameter γ in terms of α and ε. Hint: Use the transformation given in Eq. (5).

(c) Properties of squeezed states (bonus) Calculate the mean photon number

n¯ = hnˆi = hα, ε| nˆ |α, εi = hα, ε| ˆb†ˆb |α, εi

and the variance of the photon probability distribution

(∆n)2 = hα, ε| nˆ2 |α, εi − (¯n)2

for the squeezed state |α, εi.