Homework 3 Contact: jangi@ethz.ch Due date: December 5, 2014 Nano Optics, Fall Semester 2014 Photonics Laboratory, ETH Zürich www.photonics.ethz.ch 1 Coherent Control [22 pts.] In the first part of this exercise, we will discuss the time evolution of the two level system under coherent excitation. In the second part of this exercise, we will remind ourselves of how to treat optical elements such as beam splitters and free propagation with the transfer matrix formalism and apply this formalism the the Mach Zehnder interferometer. In the third part, we will derive the analogous transfer matrices for the Ramsey interferometer and we will look at the similarities between light in a Mach Zehnder interferometer and coherent manipulations of a two-level system. 1.1 State vector vs Bloch vector [8 pts.] The purpose of this exercise is to understand the relationship between [ ] cos(θ/2)e i(φ/2) s x sin(θ) cos(φ) ψ = sin(θ/2)e +i(φ/2) and s = s y = sin(θ) sin(φ), (1) s z cos(θ) the state vector and the Bloch vector, respectively, which are two equivalent descriptions of the two level system. In the lecture we saw that the Bloch equation ṡ = R s, (2) is the equation of motion for the Bloch vector s. It describes a rotation of the Bloch vector s around the rotation axes R = [Ω R cos(φ R ), Ω R sin(φ R ), δ] T at frequency Ω = R. We also found that the equation of motion of the state vector ψ is given by the Schrödinger equation i d dt ψ = ĤRF ψ, (3) where the Hamiltonian is given by Ĥ RF = Ω 2 n R σ, (4) with n R = Ω 1 [Ω R cos(φ R ), Ω R sin(φ R ), δ] T and σ = [σ x, σ y, σ z ] T. The Pauli matrices are defined as [ ] [ ] [ ] 0 1 0 i 1 0 σ x =, σ 1 0 y =, σ i 0 z =. 0 1 1.1.1 State vector equations of motion [3 pts.] 1. Show that the state vector y(t) = M R (Ωt) y 0, (5) 1
[ ] cos(ϕ) sin(ϕ) where the ket y(t) = [y 1, y 2 ] T is a shorthand for a vector of length two and M R (ϕ) = sin(ϕ) cos(ϕ) is the rotation matrix, solves the differential equation 2. Show that where U xy = d dt y = iωσ y y. (6) x = U xy y, (7) [ ] 1 0 is a unitary transformation, i.e. U 0 i xy U xy = 1, solves the differential equation d dt x = iωσ x x (8) 3. Which laser parameters ( Intensity (Ω R ), phase φ R and detuning δ) do I have to choose to obtain a Schrödinger equation (3) of the form (6) and (8), respectively? 1.1.2 Pauli matrices [5 pts.] 1. Show that the operation of the Pauli x and y-matrices on a state vector ψ σ x ψ and σ y ψ. correspond to π - rotations of the Bloch vector s around the x-axis and y-axes, respectively. Hint: Use definitions (1) and calculate the rotation of s around the x and y-axes. 2. The action of the matrix e iα 0 [ e iα/2 0 ] 0 e iα/2 Remember: on a state vector ψ corresponds to a rotation of the Bloch vector by an angle α around the z-axis. Here, α 0 is a global phase and as such doesn t have any meaning. Find the values α and α 0 to turn the above matrix into U xy and σ z, respectively. sin(x) = cos(x π/2) sin(x) = cos(x + π) e ±iπ/2 = ±i sin(x) = sin(x + π/2) cos(x) = cos(x + π/2) 1.2 Mach Zehnder interferometer [9 pts.] This exercises serves as a reminder of how to treat optical elements such as beam splitters and free propagation with the transfer matrix formalism. We will apply the formalism to find the transfer function of a Mach Zehnder interferometer. 2
(a) beam splitter (b) beam propagation A B (c) Mach Zehnder Interferometer A B C D D1 Figure 1: (a) Beam splitter (b) Light propagation (c) Mach Zehnder interferometer D2 1. Consider the following optical element, known as a beam splitter (Fig. 1). The beam splitter has two input ports and two output ports. Assuming that the complex reflection (r) and transmission coefficients (τ) for the two incoming waves are the same, show that the beam splitter transfer matrix can be expressed as [ ] U BS (θ) = e iφ 0 cos(θ/2) i sin(θ/2), (9) i sin(θ/2) cos(θ/2) where θ = 2 arccos( τ ) and φ 0 is an arbitrary global phase. Hint: Write the inputs and outputs as ψ = [a, b] T and use the unitarity condition U BS U BS = 1 2. Consider two plane waves propagating from A to B along different paths (the red and blue path in Fig. 1b). Upon propagation along a path of length L, a plane wave acquires a phase shift of φ prop = 2π λ L, where λ is the wavelength. What is the phase shifter matrix U φ if the path length difference between A and B is L A L B = L? 3. The transfer matrix for the Mach Zehnder interferometer is given by U M = U BS U φu BS (10) 3
What is the intensity at the detectors D1 and D2 for a single beam incident at the first beam splitter? Plot the detector signal as a function of path difference L/λ in the range L/λ = [0... 5]. (Both beam splitters are 50:50) 4. Assume now that L = λ. For each of the zones A, B, C, D give the state vector ψ A, ψ B, etc. and the corresponding Stokes vectors s A, s B. (Optional plot the Stokes vectors on the Bloch sphere) 1.3 Ramsey Interferometer [5 pts.] In this part of the exercise, we will show that the transfer function of the Ramsey sequence (π/2-pulse, free propagation, π/2-pulse) acting on the two-level system state vector has an analogy in the transfer matrix of the Mach Zehnder interferometer from the previous exercise. Ramsey interferometry is used to measure transition frequencies of atoms and has found its application in atomic clocks and in the S.I. definition of the second. 1. Show that the Hamiltonian Ĥz = δ 2 ˆσ z acts as a phase shifter with relative phase shift φ = δt. Hint: Solve the Schrödinger Equation (3) for the Hamiltonian Ĥz. 2. Show that the Hamiltonian Ĥx = Ω R 2 ˆσ x acts as a beam splitter with transmission coefficient ( ) T = τ 2 = cos 2 ΩR 2 t. (11) Hint: Use the results from the previous exercises. Especially (5), (7) and (9). 3. What is the experimental protocol to realize the equivalent Ramsey interferometer of the above Mach Zehnder interferometer? Assume atoms with dipole moment µ = 2 10 29 C/m and transition frequency ω 0 = 2.4 10 15 Hz and a laser with field strength of 4139 V/m and frequency ω = ω 0 + 2π 100 khz. Assume that the first laser pulse starts at t = 0. Plot the field strength of the laser over time. 2 Optical Tweezers [14 pts.] In 1971, Arthur Ashkin investigated dielectric particles exposed to a tightly focused laser beam. He discovered that particles could be trapped an manipulated with the laser beam. By now, this technique (called optical tweezers) of manipulating tiny objects has found widespread applications in biology and physics. x F z Figure 2: Optical force acting on a sub-wavelength particle in the focus of a tightly focused Gaussian beam. 4
In this exercise we consider a dielectric particle subject to a time-harmonic Gaussian beam with angular frequency ω. The particle is much smaller than the wavelength λ. It s polarisability is given by α = α + iα. The field of the Gaussian beam induces a dipole p = αe(r), where r is the location of the particle. The average force acting on the particle given by F(r) = i 1 { } 2 Re p i E i (r), where i [x, y, z]. Using p = αe, the force can be written as F(r) = α 2 i { } Re Ei (r) E i (r) + α 2 i { } Im Ei (r) E i (r) = F grad + F scatt. The first term, which can be expressed as F grad = (α /4) (E E), is referred to as the gradient force. From this expression we see that F grad is the gradient of the potential (E E). Consequently, F grad is conservative, i.e. F grad = 0. The second term, F scatt, is referred to as the scattering force. It can not be expressed as the gradient of a potential. Consequently, F scatt is nonconservative, i.e. F scatt 0. Consider a Gaussian beam in vacuum polarized along the x-axis and focused under an angle θ = 53. The wavelength is λ = 1064 nm and the field strength at the focus is E 0 = 2 10 7 V/m. The particle s radius is a = 150 nm and it s dielectric constant is ε = 2.1. The real part of the polarizability is given by α = 4πε 0 a 3 ε 1 ε + 2 and the imaginary part of the polarizability is given by α = k3 6πε 0 α 2. The origin of our coordinate system is centered at the focus of the Gaussian beam (c.f. Fig. 2). 1. Calculate the power of the Gaussian beam 2. Give an expression for the Gradient force (no numbers) and plot the gradient force F grad along the x-axis in the range x = [ λ.. λ] und F (x) grad = [ 40 pn.. 40 pn]. 3. Expand F (x) grad (x, 0, 0) (x-component of the gradient force along the x-axis) in x and determine the coefficient κ x of the linear term. For small displacements from the focus, the gradient force can now be written as F xgrad κ x x, i.e. the particle behaves as if it was attached to the focus by a spring with spring constant κ x. 4. The equation of motion for a particle moving along the x-axis is mẍ + κ x x = 0. Assume that the specific density of the particle is ρ = 2.5 g/cm 3 and calculate the oscillation frequency Ω of the particle along the x-axis. 5. Give an expression (no numbers) and plot the gradient force F grad along the z-axis in the range z = [ λ.. λ] and F (z) grad = [ 20 pn.. 20 pn]. Calculate the spring constant κ z. Along which direction, x or z, is the motion of the particle more confined? 5
6. The scattering force F scatt pushes the particle away from the focus along the optical axis (z). For the values given above, plot F scatt along the z-axis in the range z = [ λ.. λ] and F (z) scatt = [ 20 pn.. 20 pn]. You can do this numerically. Optional: Find an analytical expression for F (z) scatt. 7. Plot the total force F along the z-axis and calculate (numerically) the displacement z of the equilibrium position due to the scattering force in the range z = [ λ.. λ] and F (z) = [ 10 pn.. 30 pn]. Note that for large particles, the scattering force becomes much larger than the gradient force. Therefore, the size of the particle is an important parameter for optical tweezers. 6