Theoretical Quantum Optics and Quantum Information (SS 2017) Exercise II (discussion: 17.05.17) 1. Coherent states (a) Eigenstates of ^by Show that an eigenstate of ^by cannot exist. Hint: Expand a putative eigenstate of ^by in the basis of Fock states jni 1 X jβi = jni hnjβi : n=0 Then you will see that the creation operator produces a new superposition of occupations jni with different range of occupancy than jβi. (b) Action of ^by Show that @ α∗ ^by jαi = + jαi @α 2 where jαi is a coherent state, i.e. ^b jαi = α jαi. 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 j ±i / (jαi ± |−αi); (1) which is a superposition of two coherent states. Start by normalizing j ±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 !1 and !2. 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 ^ X ^y^ ^y^ ^y ^ H = ~ !ibi bi − ~χb1b1 b2 + b2 : i=1 ^y ^ Here, bi and bi, i 2 f1; 2g denote the bosonic creation and annihilation operator of the corresponding harmonic oscillator. (a) Show that the unitary displacement transformation D^y(β^)H^ D^(β^) =: H^ 0 (2) ^ χ ^y^ decouples the two harmonic oscillators for β = b b1. Here, the generalized !2 1 ^ ^ ^^y ^y^ ^ displacement operator is defined as D(β) := exp βb2 −β b2 . Note that β is ^ ^y^ a hermitian operator, that fulfills [β; b1b1] = 0. ^y ^ ^ ^ ^ ^ ^ Hint: Verify first, that D (β)b2D(β) = b2 + β. (b) The displaced Hamiltonian H^ 0 obeys the following eigenequation ^ 0 H jEn1;n2 i = En1;n2 jEn1;n2 i : ^ ^ Find the eigenenergies En1;n2 and eigenstates D β jEn1;n2 i of the non- displaced Hamiltonian H^ . (c) With the eigenstates ^ ^ ^ ^ D β jEn1;n2 i = D χn1=!2 jn1i ⊗ jn2i ≡ D χn1=!2 jn1; n2i (3) and the eigenenergies 2 χ 2 En1;n2 = ~!1n1 + ~!2n2 − ~ n1 (4) !2 one can study the dynamics of the system. Use the Schr¨odinger equation @ i j (t)i = H^ j (t)i ~@t to calculate the time evolution of the state j 0i = jn1i⊗jαi ≡ jn1; αi, where the first harmonic oscillator is initially prepared in a Fock state jn1i and the second one in a coherent state jαi by expanding j (t)i in the eigenbasis of H^ . Show that (up to a global phase) −i!2t j (t)i = jn1i ⊗ η(t) + αe with η(t) := χn1 (1 − e−i!2t). !2 Hint: Use the expansion of a coherent state in the Fock basis jγi = 2 1 − jγj γk y e 2 P p jki with γ 2 C. Also use that D^ (µ) jνi = D^(−µ) jνi = jν − µi k! k=0 with µ, ν 2 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 S^y(")^bS^(") = ^b cosh r − ^bye2iφ sinh r S^y(")^byS^(") = ^by cosh r − ^be−2iφ sinh r and S^y(")f(^b; ^by)S^(") = f(^b cosh r − ^bye2iφ sinh r ; ^by cosh r − ^be−2iφ sinh r); (5) where 1 S^(") = exp "∗^b2 − "(^by)2 ;" = re2iφ 2 and f(^x; y^) is a function which can be expanded into a power series. (b) Squeeze and Displacement operator (bonus) The order of the D^ and S^ operators in the definition of a squeezed state in Eq. (36) (see lecture) does matter, since D^(α)S^(") = S^(")D^(γ): Express the parameter γ in terms of α and ". Hint: Use the transformation given in Eq. (5). (c) Properties of squeezed states (bonus) Calculate the mean photon number n¯ = hn^i = hα; "j n^ jα; "i = hα; "j ^by^b jα; "i and the variance of the photon probability distribution (∆n)2 = hα; "j n^2 jα; "i − (¯n)2 for the squeezed state jα; "i..