Photon Physics: EIT and Slow light

Transcription

1 1 Photon Physics: EIT and Slow light Dries van Oosten Nanophotonics Section Debye Institute for NanoMaterials Science

2 2 Bloch equations revisited Consider an Hamiltonian H, with basis functions { i }. The density matrix for the system is given by ρ = i p i i i Two extreme cases: pure state, the system populates an eigenstate ψ of H ρ = ψ ψ incoherent mixture, for instance ( ) ρgg 0 ρ = 0 ρ ee

3 The two-level system Start with atomic levels g and e They have energies ωg and ω e, with ω eg = ω e ω g The Hamiltonian of the system is thus H 0 = ω g g g + ω e e e We want to know the response to an external field E = ˆɛE 0 cos ωt The polarisation of the atom is p = qx How do we calculate the polarization? 3

4 4 The two-level system We first look at the coupling p = e qx E g We rewrite the operator qx E as H 1 = ( g g + e e ) qx E ( g g + e e ) This yields H 1 = g g qx E e e + e e qx E g g We can write e qx E g = µ eg E 0 cos ωt

5 5 The two-level system We define the Rabi Ω frequency as in week 12 Ω = 1 µ ege 0 The total Hamiltonian then becomes H = ω g g g + ω e e e Ω ( e g + g e ) cos ωt First, we take the RWA H = ω g g g + ω e e e 1 2 Ω ( e g e iωt + g e e iωt)

6 6 The two-level system Next, we absorb e iωt into e H = ω g g g + (ω e ω) e e 1 Ω ( e g + g e ) 2 Finally, we shift the energy down by (ωg ω) H = ω g g + ω eg e e 1 Ω ( e g + g e ) 2 If we define basis vectors (1, 0) = g and (0, 1) = e ( ) ω Ω/2 Ĥ = Ω/2 ω eg We call this the dressed state picture

7 7 The two-level system Make the matrix look more symmetric, define δ = ω ωeg ( δ Ω ) Ĥ = 2 Ω δ Let us denote an eigenstates of Ĥ by ψ = cg g + c e e What is the polarization of this state? p = ψ qx ψ = c g c e e qx g + c g c e g qx e Which, in the RWA, can be written as p = µ eg ( cg c e + c g c e )

8 8 The two-level system What is the density matrix for ψ = cg g + c e e ( ) ρgg ρ ρ = ge = ρ eg ρ ee ( cg 2 cg c e c g ce c e 2 ) Thus the polarization is p = µ eg ( cg c e + c g c e ) = µeg (ρ eg + ρ ge ) How do we determine the density matrix?

9 9 Bloch equations revisited Write the Hamiltonian using Pauli spin matrices The Liouville equation H = 2 (δσ z Ωσ x ) ρ = i [ρ, H] + L where L describes the damping (Lindblad operator) σ z = ( L = γ 2 (2σ +ρσ σ σ + ρ ρσ σ + ) ) ( 0 1, σ x = 1 0 ) ( 0 0, σ + = 1 0 ) ( 0 1, σ = 0 0 ),

10 Bloch equations revisited in Mathematica First, we define the Pauli spin matrices: they operator on vectors g>=(1,0) and e>=(0,1) In[1]:= g : 1, 0 e : 0, 1 Σz : 1, 0, 0, 1 Σp : 0, 0, 1, 0 Σm : 0, 1, 0, 0 Σx : Σp Σm MatrixForm Σz MatrixForm Σp MatrixForm Σm MatrixForm Σx Out[7]//MatrixForm= Out[8]//MatrixForm= Out[9]//MatrixForm= Out[10]//MatrixForm= Do the lowering and raising operators work correctly? What is the Hamiltonian? Define the density matrix And write down the Liouville equation In components 10

11 11 Bloch equations revisited in Mathematica First, we define the Pauli spin matrices: they operator on vectors g>=(1,0) and e>=(0,1) Do the lowering and raising operators work correctly? In[11]:= Σp.g e Σm.e g Out[11]= Out[12]= True True What is the Hamiltonian? Define the density matrix And write down the Liouville equation In components

12 12 Bloch equations revisited in Mathematica First, we define the Pauli spin matrices: they operator on vectors g>=(1,0) and e>=(0,1) Do the lowering and raising operators work correctly? What is the Hamiltonian? In[13]:= H : 1 Σz Σx 2 MatrixForm H Out[14]//MatrixForm= Define the density matrix In[33]:= Ρ t_ : Ρgg t, Ρeg t, Ρge t, Ρee t MatrixForm Ρ t Out[34]//MatrixForm= Ρgg t Ρeg t Ρge t Ρee t And write down the Liouville equation In components

13 13 Bloch equations revisited in Mathematica First, we define the Pauli spin matrices: they operator on vectors g>=(1,0) and e>=(0,1) Do the lowering and raising operators work correctly? What is the Hamiltonian? Define the density matrix And write down the Liouville equation In[35]:= Ρdot : FullSimplify H.Ρ t Ρ t.h Γ 2 Σm.Ρ t.σp Ρ t.σp.σm Σp.Σm.Ρ t 2 MatrixForm Ρdot Out[36]//MatrixForm= Γ Ρee t 1 Ρeg t Ρge t 1 Ρee t Γ 2 Ρeg t Ρgg t Ρee t Γ 2 Ρge t Ρgg t 1 2 Γ Ρee t Ρeg t Ρge t 2 2 In components

14 14 Bloch equations revisited in Mathematica First, we define the Pauli spin matrices: they operator on vectors g>=(1,0) and e>=(0,1) Do the lowering and raising operators work correctly? What is the Hamiltonian? Define the density matrix And write down the Liouville equation In components In[41]:= D Ρ t, t 1, 1 Ρdot 1, 1 D Ρ t, t 2, 2 Ρdot 2, 2 D Ρ t, t 1, 2 Ρdot 1, 2 D Ρ t, t 2, 1 Ρdot 2, 1 Out[41]= Ρgg t Γ Ρee t 1 Ρeg t Ρge t 2 Out[42]= Ρee t 1 2 Γ Ρee t Ρeg t Ρge t 2 Out[43]= Ρeg t 1 Ρee t Γ 2 Ρeg t Ρgg t 2 Out[44]= Ρge t 1 Ρee t Γ 2 Ρge t Ρgg t 2

15 15 Bloch equations revisited in Mathematica Steady state solutions Solve Ρdot 1, 1 0, Ρdot 2, 2 0, Ρdot 1, 2 0, Ρdot 2, 1 0, Ρee t, Ρgg t, Ρeg t, Ρge t Out[60]= Hmm, no solution because it is overdetermined! Solve Ρgg t Ρee t 1, Ρdot 2, 2 0, Ρdot 1, 2 0, Ρdot 2, 1 0, Ρee t, Ρgg t, Ρeg t, Ρge t 2 Out[61]= Ρee t Γ That is a Lorentzian with γ γ 2 + 2Ω 2 Power broadening! Before we look at the result, an alternative approach!

16 16 The two-level system We use a trick to put the damping in the Hamiltonian. Introduce δ = δ + iγ/2 ( ) δ Ω Ĥ = 2 Ω δ Now, we just determine the eigenvectors of this matrix

17 17 The two-level system With γ 0, define tan 2θ = Ω/ δ ω± = ± Ω 2 + δ 2 /2 Eigenvectors ê = (cos θ, sin θ) and ê + = ( sin θ, cos θ) These are not truly eigenstates and eigenenergies, as they contain damping As in Week 12, the polarization is now p = qx = 2µ eg ρ ge = 2µ eg cos θ sin θ

18 18 The two-level system Can also be written p = ε0 αe, thus α = 2µ eg ε 0 E 0 cos θ sin θ α has the unit of volume The susceptibility of a dilute gas of these atoms is with N the density of atoms χ = Nα

19 19 The two-level system polarizability (m 3 ) 1e e detuning ( ) cross section (m 2 ) polarizability α and scattering cross section σ sc green lines: numerical eigenvalues red lines: optical bloch equations Analytical eigenvalues also give the same result

20 20 The three-level system Now, we have levels g, e and m ωe > ω m, ω g We want to know χ for this system with a strong coupling between e and m H = ω g g g + ω e e e + ω m m m 1 2 Ω ( p e g e iω pt + g e e iωpt) 1 2 Ω ( c e m e iω ct + m e e iωct)

21 21 The three-level system g e iω pt g, ω g = ω g + ω p m e iω ct m, ω m = ω m + ω c ωeg = ω e ω g, ω em = ω e ω m H = (ω p ω eg ) g g + (ω c ω em ) m m 1 2 Ω p ( e g + g e ) 1 2 Ω c ( e m + m e ) Introduce two detunings δp = ω p ω eg and δ c = ω c ω em H = δ p g g + δ c m m 1 2 Ω p ( e g + g e ) 1 2 Ω c ( e m + m e )

22 The three-level system Basis g = (1, 0, 0), e = (0, 1, 0) and m = (0, 0, 1) H = δ p Ω p /2 0 Ω p /2 iγ/2 Ω c /2 0 Ω c /2 δ c Where we assume that only level e decays An analytical solution is clearly cumbersome, but finding a numerical solution is easy 22

23 23 The three-level system Electro-magnetically induced transparency Ωp is weak, δ c γ absorption cross-section( ¾ 0 ) c =0 c = =4 c = =2 c = c =4 Vertical dashed lines indicate ±Ωc /2 Transparancy window is Ωc wide detuning ( )

24 24 The three-level system Electro-magnetically induced transparency Ωp is weak, δ c γ, Ω c = γ What about dispersion? n(10 6 ) Index of refraction n(ω) = 1 + χ(ω) Ωc = γ Highly dispersive in the EIT window! detuning ( )

25 25 The three-level system Electro-magnetically induced transparency Group velocity vg = ω/ k n g (10 3 ) Group index n g = c 1 + ω χ v g 2 ω detuning ( ) absorption cross-section( ¾ 0 ) ng Inside the transparency window! Quite impressive (1 km/s) 1 µs pulse 1.2 mm long But there is more next week!

26 1 Photon Physics: EIT and Slow light Dries van Oosten Nanophotonics Section Debye Institute for NanoMaterials Science

27 2 This week Slow light in atomic and nanophotonic structures What are we going to discuss today? Light stopping/storage in a three-level system Photonic bandstructures Slow light in photonic crystal waveguides

28 3 Last week Slow light in the three-level system Group velocity vg = ω/ k Group index n g = c 1 + ω χ v g 2 ω n g (10 3 ) absorption cross-section( ¾ 0 ) ng Inside the transparency window! Quite impressive (1 km/s) 1 µs pulse 1.2 mm long detuning ( )

29 4 Light stopping/storage in a three-level system Experiments 5P 1/ 2, F=1 Ωc 5S 1/ 2, F=2 -> e> Ωs +> transmission (%) applied magnetic field (mg) laser AOM iris Pockels cell 87Rb cell in oven, solenoid & shields λ/ 4 plate λ/ 4 plate photodetector polarizing beam splitter photodetector

30 5 Light stopping/storage in a three-level system Experiments 50 0 a) I τ = 50 µs II Experimental cycle: First, zero the magnetic field normalized signal intensity (%) b) c) I I τ = 100 µs II τ = 200 µs II Switch on AOM Use Pockels cell to create probe pulse Switch off AOM Wait Switch on AOM time (µs)

31 6 Photonic crystal waveguides The wave equation Dielectric function varies in space, the wave equation reads 2 E(x) + ε(x)k0 2 E(x) = 0 Now assume the dielectric function is periodic in space ε(x) = ε (1 + ε cos k L z) And write the electric field as E = ˆxE 0 f (z) Then the wave equation becomes 2 f (z) + εk 0 2 (1 + ε cos k L z) f (z) = 0

32 7 Photonic crystal waveguides Bloch theorem Bloch theorem With In other words and thus f k (z) = e ikz u k (z) u k (z) = u k (z + L) u k (z) = u nk e ink Lz n= f k (z) = u nk e i(k+nk L)z n=

33 8 Photonic crystal waveguides The mode equation Plug this into the wave equation 2 f (z) + εk 2 0 (1 + ε cos k L z) f (z) = 0 To find, after some algebra that for each n and k u nk { εk 2 0 [k + nk L ] 2} ε εk2 0 {u n 1k + u n+1k } = 0 Diagonalizing this band diagonal matrix will yield k 0 as function of n and k. Problem, the matrix depends on the eigenvalue!

34 9 Photonic crystal waveguides The mode equation Start from the wave equation for H 1 ε(x) H = k2 0 H Has the eigenvalues k0 2 isolated on the right-hand side. Great! Using the same trick as before, we find k 2 0 u nk = n [ε n n ] 1 (k + nk L ) ( k + n k L ) un k Where ε n n = εδ n,n + ( ) δ n+1,n + δ n,n +1

35 10 Photonic crystal waveguides Solving the mode equation in Python/Numpy i m p o r t numpy def i n v e p s i l o n (em, ed,n ) : epsilon=em numpy. identity (2 N+1, dtype=float)+\ ed numpy. identity (2 N+2, dtype=float )[1:,: 1]+\ ed numpy. identity (2 N+2, dtype=float ) [ : 1,1:] r e t u r n numpy. l i n a l g. i n v ( e p s i l o n ) d e f h a m i l ( i n v e p s, k, kl,n ) : q=k+kl numpy. arange( N,N+1) q1=numpy. outer (q, numpy. ones (2 N+1)) return inv eps q1 numpy. transpose ( q1 ) d e f e i g e n ( i n v e p s, ks, kl,n ) : r e t u r n numpy. a r r a y ( [ numpy. s o r t ( numpy. l i n a l g. e i g ( h a m i l ( i n v e p s, k, kl,n ) ) [ 0 ] ) f o r k i n ks ] ) d e f e i g e n n o s c a n (em, ed, k, kl,n ) : i n v e p s=i n v e p s i l o n (em, ed,n) r e t u r n numpy. s o r t ( numpy. l i n a l g. e i g ( h a m i l ( i n v e p s, k, kl,n ) ) [ 0 ] ) d e f p e r t u r b (em, ed, k, kl ) : o f f d i a g= (ed /em 2) k (k kl ) m a t r i x=numpy. a r r a y ( [ [ k 2/em, o f f d i a g ], [ o f f d i a g, ( k kl) 2/em ] ] ) r e t u r n numpy. s o r t ( numpy. l i n a l g. e i g ( m a t r i x ) [ 0 ] ) Try Python, it s awesome and free!

36 Photonic crystal waveguides Solving the mode equation in Python/Numpy k 0 (¼=a) k (¼=a) In practice, we use the MIT Photonic Bands program (does essentially the same thing) 11

37 12 Photonic crystal waveguides Solving the mode equation using perturbation theory k 0 (¼=a) k (¼=a) Gap seems ok, but mid gap is offset

38 13 Photonic crystal waveguides Solving the mode equation using perturbation theory k 0 (¼=a) k (¼=a) For smaller dielectric constant, the match is great!

39 14 Photonic crystal waveguides In practice Photonic band gap causes total reflection Waveguide mode is caught between two photonic crystals Periodicity also causes band structure of the defect mode

40 15 Photonic crystal waveguides Group velocity odd even!(2¼c=a) v g =c k(2¼=a) k(2¼=a) Only slow modes below the light-line are useful Even mode is the slowest And has the smallest group velocity dispersion

41 16 Photonic crystal waveguides Near-field microscope λ/2 400 nm to detector from laser photonic crystal waveguide thickness 200 nm pitch 450 nm hole size 250 nm Aperture probe sub-wavelength resolution Interferometer phase sensitivity

42 Photonic crystal waveguides Real space images I Perform a fourier transform on the images yields wavevector k I Repeat for many optical frequencies ω yields dispersion 17

43 Photonic crystal waveguides Dispersion measurements 18

44 Photonic crystal waveguides Fast-slow interface I Take a piece of fast waveguide (vg = c/10) I Connect it to a piece of slow waveguide (vg = c/50) x/d 6 a 0 λ=1525nm x/d -6 6 b λ= nm y/d I Field enhancement, factor 5 Why the field enhancement? 19