ABSTRACT

SIMULATION OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND ABSORPTION

by Thomas Jenkins

We investigated Electromagnetically Induced Transparency and Absorption in Lambda, Vee, and degenerate two level system systems. By using the Quantum Toolbox in Python (QuTiP) we were able to simulate the coupling and probe absorption spectra by calculating steady state density matrix and finding the expectation values of the dipole transitions. We also used quantum trajectories to model the evolution of each of the quantum states which proved to be helpful in analyzing the absorption spectra seen in the degenerate two level system. SIMULATION OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND ABSORPTION

A Thesis

Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science Department of Physics by Thomas Jenkins Miami University Oxford, Ohio 2013

Advisor: (Dr. Perry Rice)

Reader: (Dr. Samir Bali)

Reader: (Dr. James Clemens) Contents

1 Introduction 1 1.1 Motivation ...... 1 1.2 Three Level Systems ...... 2 1.3 Classical Analog of EIT ...... 3

2 Dark States 7

3 Theoretical Background 10 3.1 Open Systems and the Density Operator ...... 10 3.2 The Master Equation with Dissipation ...... 12 3.3 Quantum Trajectories ...... 15 3.3.1 Two Level Atom in a Field ...... 18

4 EIT and EIA in Three Level Systems 20 4.1 Λ Structures ...... 20 4.1.1 Non-Degenerate Systems ...... 20 4.1.2 Degenerate Systems ...... 23 4.2 V Structures ...... 26

5 Degenerate Two Level Systems 30 5.1 Circularly Polarized Probe Beam ...... 31 5.2 Linearly Polarized Probe Beam ...... 48

6 Conclusion 63 6.1 Summary ...... 63 6.2 Future Work ...... 64

A Classical EIT Code 65

B Code for EIT in Non Degenerate Λ System 67

C Code for EIT in Degenerate Λ System 71

D Code for EIT and EIA in Degenerate V systems 76

ii E Code for Six Level System 82

References 92

iii List of Figures

1.1 Transition Diagram for a Non Degenerate Three Level Atom . . . . . 2 1.2 Diagram of Coupled RLC Circuits ...... 4 1.3 Power Absorption of Coupled Oscillator ...... 6

3.1 Schematic for Constructing a Quantum Trajectory ...... 17 3.2 Quantum Trajectories for Two Level Atom ...... 19

4.1 EIT in a Non Degenerate Λ Structure ...... 21 4.2 Index of Refraction and Absorption in Λ Structure ...... 22 4.3 Transition Diagram for Degenerate Λ Structure ...... 23 4.4 EIT in a Degenerate Λ system ...... 24 4.5 Quantum Trajectories for Degenerate Λ Structure ...... 24 4.7 3D plot of EIT in Degenerate Λ Structure ...... 26 4.8 3D plot of EIT in a Degenerate V Structure ...... 27 4.9 EIA in Degenerate V Structure ...... 27 4.10 Quantum trajectories for EIA in Degenerate V Structure ...... 28

5.1 Transition Diagram for a Six Level Atom ...... 30 5.3 3D Plots of Absorption Spectra in Six Level Atom with σ− Probe Beam 33 5.11 Quantum Trajectories for Six Level Atom with σ− Probe Beam . . . 39 5.19 3D Plots of Absorption Spectra in Six Level Atom with π Probe Beam 49 5.31 Quantum Trajectories for Six Level Atom with π Probe Beam . . . . 55

iv ACKNOWLEDGEMENTS

First and foremost, I cannot thank my family enough for giving me the support that has allowed me to reach this point. Without them I would not be where I am today. Chase will always be the king. I would also like to thank my advisor, Dr. Perry Rice. There is no doubt that Dr. Rice has helped me to grow as a physics student during my time at Miami University, and it is because of his guidance and patience that I was able to complete this project. I always enjoyed our meetings because of his personality and sense of humor. I knew that Perry was a good guy when I discovered that he was a Browns fan. Thank you for everything. Finally I would like to thank Drs. James Clemens and Samir Bali for taking the time to go through this thesis and for giving me helpful feedback. I feel it is also necessary to thank Samir for setting up weekly volleyball matches, without them I would have never gotten revenge on him for his EM course.

v Chapter 1

Introduction

1.1 Motivation

Electromagnetically Induced Transparency (EIT) and Electromagnetically Induced Absorption (EIA) are phenomena that are well known in the field of quantum optics. EIT is the product of quantum interference and is characterized by an atomic system becoming effectively transparent to a laser, meaning that there is no absorption at certain frequencies. Along with this we EIT results in a rapid change of the refractive index which produces a low group velocity, meaning that the speed of light can be slowed down. Because of this effect EIT is useful for quantum information processing [12,13]. EIA, the counterpart to EIT, is also the effect of quantum interference and it is characterized by increased absorption. There are several different atomic systems in which we can produce EIT, ranging from systems with three quantum states to more complicated spectra seen in degenerate two level atomic structures. Typically the master equation must be solved in order to model EIT and EIA, however the density matrix theory that is used to solve the the master equation averages away information that we may be interested in such as spontaneous emission events. By doing this we effectively hide what we call quantum jumps. We will do this by finding the steady state solution to the master equation so that we can model EIT and EIA, but we are also going to use what is known as quantum trajectory theory so that we model the evolution of each of the quantum states in the atomic structures. A single quantum trajectory allows us to see how the atoms oscillate through the various states between each quantum jump. Many trajectories averaged together give us an accurate portrayal of how the population evolves with time. This will prove especially useful in degenerate two level systems because knowing which states the population is in will help us understand what type of interference may occur, and subsequently why we see the complex absorption spectra that are exhibited.

1 1.2 Three Level Systems

There are various 3 level atom structures that can exhibit EIT and EIA effects under the right circumstances. In this section we will be investigating Λ and V structures in particular. The Λ structure, which resembles the greek letter Λ, is characterized by having a single excited state and two ground states, as seen in Fig. 1.1a. There are electro- magnetic fields (in our case, lasers) that interact with the three states of the material. One beam probes the |g1i → |ei transition while the second couples the |g2i → |ei transition. The |g1i → |g2i transition is forbidden. It is important to understand that EIT and EIA arise from quantum interference. In the Λ structure, both of the ground states are being driven to a single excited state. When an atom becomes excited, one cannot determine which path the atom took to get into that state which results in interference similar to that found in Young’s double slit experiment. The probability amplitude for |g1i → |ei destructively interferes with the probability amplitude for |g1i → |ei and effectively renders the medium transparent to the probe beam. What this means is that the atoms can no longer absorb photons and that eventually all of the atoms will become trapped in the ground states. Even if atoms were in the excited state to begin with, they would decay via spontaneous emission and would have no means of becoming excited again. When the atoms become trapped in the ground states, they are in what is known as a dark state [1].

a.) b.) Figure 1.1: Transition diagram for a a.) Λ and b)V atomic structure. The sponta- neous emission frequencies are denoted by γ and Γ while Ωp and Ωc represent the rabi frequencies of the probe and coupling laser beams. ∆ and φ are the detunings in the system.

EIT is not constrained to Λ structures, for instance it can also be produced in an atoms with V structures. A V structure (shown in Fig 1.1b) is similar to a Λ structure, it has three quantum states with two lasers driving two transitions, the difference is that it has two excited states with a single ground state. By having two lasers driving the atoms from the ground state to the two ground states V structures can also exhibit EIT. In this case however, the destructive interference comes from

2 the atoms decaying from each of the excited states to the single ground state. Along with EIT, V systems can also exhibit Electromagnetically Induced absorption (EIA) if the lasers are used drive the atoms from the excited states to the ground state. While EIT results in no absorption of photons, EIA is characterized by constructive quantum interference which leads to an increase of absorption. In EIA, the atoms become trapped in the excited states in what is known as a bright state. The atoms cannot go into the ground state via stimulated emission. However, it should be noted that the atoms can still decay into the ground state through spontaneous emission. In V structures we produce either both EIT and EIA based on whether we drive the atoms into the ground or excited states. This is not true for Λ systems. In Λ systems we can produce EIT by driving the atoms into the excited state, however if we were to drive the atoms into the ground states we would not get EIA, we would just see all of the atoms ending up in the ground states because there would not be any interference.

1.3 Classical Analog of EIT

We will be taking a semiclassical approach to studying EIT, that is, we will treat the atoms quantum mechanically while treating the lasers classically. It turns out that we can model EIT with a classical system, something we now investigate in order to help us understand the underlying physics involved in EIT. A Λ structure can be looked at as two two-level systems that are coupled together, which is why we can use the coupled RLC circuits in Fig. 1.2 to simulate the effects that we see in a Λ atomic structure. The inductor and capacitors in the first mesh model the atom while the inductor and capacitors in the second mesh plays the role of the coupling transition. The resistors represent the decay in the system. The switch in the second mesh can be opened or closed, analogous to the coupling laser being turned on and off. The capacitor C represents the coupling between the two meshes and determines the Rabi frequency associated with the coupling transition. The voltage source simulates the probe field.

3 Figure 1.2: We can simulate the effects of EIT with coupled RLC circuits. Credit for this figure is given to [9]

We wish to analyze the frequency dependence of the power transferred from Vs to the first mesh. To do this we first write the equations the two currents, which are

2 2 q¨1(t) + γ1q˙1(t) + ω1 q1(t) − Ωr q2(t) = 0 (1.1)

2 2 Vs(t) q¨2(t) + γ2q˙2(t) + ω2 q2(t) − Ωr q1(t) = (1.2) L2

Ri 2 1 2 1 where γi = , ω = and Ω = . Ce1 is the series combination of C and C1. Li i LiCe1 r L2C For simplicity we will consider the case where C1 = C2 and L1 = L2 = L, resulting in each of the ground states having the same energy. Our goal is to find the power absorbed in the first mesh by the probe force, so we seek a solution for q2(t). We make the assumption that the solution is of the form

−iωst q2(t) = Ne (1.3)

where N is a constant. This results in

2 2 −iωst (ω − ωs − iγ1ωs)Vse q2(t) = 2 2 2 2 4 (1.4) L [(ω − ωs − iγ2ωs)(ω − ωs − iγ1ωs) − Ωr] the power absorbed by the probe force is given by

−iωst P (t) = Vse q˙(t) (1.5)

we plug in equation 1.4 to find the power absorbed during one period of oscillation by the probe force is

2 2 2 2πV ωs(ω − ωs − iγ1ωs) P2(ωs) = − 2 2 2 2 4 (1.6) L [(ω − ωs − iγ2ωs)(ω − ωs − iγ1ωs) − Ωr] This absorption is plotted in Fig. 1.3. We see that as the coupling capacitor C is

4 reduced the Rabi frequency of the coupling/pump transition is increased, and the transparency in the plots become more pronounced. This is similar to a Λ atomic structure in that there is usually a weak probe laser beam and a strong coupling beam. The interference for the coupled RLC circuit is between the power from the voltage source to R2L2Ce2 and from the power from the second mesh to R2L2Ce2. Fig. 1.3 shows that even though we are dealing with a totally classical system, the interference is analogous to that of EIT.

5 −7 −2 Figure 1.3: Power absorbed by the probe force. γ1 = 10 , γ2 = 4 ∗ 10 in frequency V units. L = .1 and the values of Ωr are a.)0, b.)0.1, c.)0.2, and d.)0.4 in same frequency units. We see the transparency effects become noticeable as the Rabi frequency is increased

6 Chapter 2

Dark States

The Hamiltonian of a three level atom that is interacting with two lasers like the one in Fig. 1.1a is ˆ ˆ ˆ H = H0 + H1 (2.1) where ˆ H0 = ~ωg1 |g1ihg1| + ~ωg2 |g2ihg2| + ~ωe|eihe| (2.2) Hˆ = −~ Ω e−iφp e−iωpt|eihg | + Ω e−iφc e−iωct|eihg |
+ H.c. (2.3) 1 2 p 1 c 2

−iφp −iφc And Ωpe and Ωce are the Rabi frequencies that are associated with the field modes coupled to the atomic transitions. The state vector of a three level atom is given by

|ψi = Cg1 |g1i + Cg2 |g2i + Ce|ei (2.4) and by taking the time derivative we get ˙ ˙ ˙ ˙ |ψi = Cg1 |g1i + Cg2 |g2i + Ce|ei (2.5)

Our goal is to derive expressions for how the probability amplitudes change in time. To solve for these we will need to use the Schrodinger equation i |ψ˙i = − Hˆ |ψi (2.6) ~ We use our Hamiltonian to act on the state vector and set it equal to Eqn. 2.5. Then by acting on both sides with hg1| and using the relation hi|ji = δij we get i C˙ = −iω C + Ω eiφp eiωptC (2.7) g1 g1 g1 2 p e

7 Similarly, we can get i C˙ = −iω C + Ω eiφc eiωctC (2.8) g2 i i 2 c e i i C˙ = −iω C + Ω e−iφc e−iωctC + Ω e−iφp e−iωptC (2.9) e e e 2 c g2 2 p g1 We are interested in the phase difference of the lasers, not their individual frequencies. To do this we move to a rotating frame by defining

−iωg1 t Cg1 = Dg1 e (2.10)

−iωg2 t Cg2 = Dg2 e (2.11) −iωet Ce = Dee (2.12)

Taking the time derivative of Eqn. 2.10 gives us

˙ ˙ −iωg1 t −iωg1 t Cg1 = Dg1 e − iωg1 Dg1 e (2.13)

Now we plug into Eqn. 2.7

  −iω t −iω t i iφ iω t −iω t D˙ − iω D e g1 = −iω D e g1 + Ω e p e p D e e (2.14) g1 g1 g1 g1 g1 2 p e Finally we can solve this for i D˙ = Ω D (2.15) g1 2 p e Using the same method we arrive with i D˙ = (Ω D + Ω D ) (2.16) e 2 p g1 c g2 i D˙ = Ω D (2.17) g2 2 c e For simplicity, we have set ∆ = φ = 0, and we now write these equations in matrix form  D˙   0 Ω 0   D  g1 i p g1 D˙ = Ω 0 Ω D (2.18)  e  2  p c   e  ˙ 0 Ω 0 D Dg2 c g2 The 3x3 matrix on the right hand side has three eigenvalues, and there is one in particular that we are interested in. For λ = 0, the corresponding eigenstate is   −Ωc Ddark =  0  (2.19) Ωp

8 This vector represents the dark state, and by looking at it we see that as the state evolves the probability of being in the excited state is zero. Here the population is always in a superposition of the ground states which means that there is no absorption, even in the presence of a laser.

9 Chapter 3

Theoretical Background

3.1 Open Systems and the Density Operator

When studying introductory quantum mechanics, the state vector |ψi is sufficient in describing a quantum system. This is because the system being studied is a closed system which means that the system is isolated from the environment[3]. The total energy of the system is conserved, or in other words, there is no dissipation. In quantum optics we investigate systems that are coupled to an outside reservoir, which are also known as open systems. The reservoir in which the system is coupled to is all of the field modes that surround the system. So for instance, when an atom absorbs or emits light it becomes coupled to the environment. Because of this it is necessary abandon the state vector for a more powerful tool to describe the quantum state; for this we use what is known as the density matrix. For an ensemble in which each member can be described by the same state vector the density matrix is ρ = |ψihψ| (3.1) When the density matrix can be described this way, it is said to be a pure state. However, when a system interacts with the surrounding field modes, information is lost to the environment [4]. There are a vast amount of field modes, but we generally do not care about their details, we only care about the details of the system. When an atom interacts with the environment we do not care which field mode it is with, we only care about the state the atom ends up in. Because of this we use the mixed density operator when describing an open systems, which is given by X ρ = Pi|ψiihψi| (3.2) i

Where Pi is the probability of the system being in the eigenstate |ψii. The density matrix gives us important information about the ensemble. The diagonals of the density matrix give the populations, or in other words the probability of being in a

10 state |ψii, while the off diagonals give the coherence between states [5]. We know that the expectation value for an operator is given by

ˆ X ˆ hOi = Pihψi|O|ψii (3.3) i However, it can also be written in terms of the density matrix which is seen by taking the trace ˆ X ˆ T r(Oρ) = hψi|Oρ|ψii i ! X ˆ X = hψi|O Pj|ψjihψj| |ψii i j X X ˆ = Pjhψi|O|ψjihψj|ψii i j X ˆ = Pihψi|O|ψii (3.4) i Because the state of the system changes in time when it interacts with the reservoir, it is useful to derive the expression for the time evolution of the density operator

X  ˙ ˙  ρ˙ = Pi |ψiihψi| + |ψiihψi| i   X i ˆ i ˆ = Pi − H|ψiihψi| + |ψiihψi| H i ~ ~ i h i = − H,ˆ ρ (3.5) ~ This equation describes systems that are in mixed states, however we wish to treat open systems with dissipation into an environment using the density matrix. Our first step in deriving this equation includes writing the Hamiltonian as ˆ ˆ ˆ ˆ H = HS + HR + HSR (3.6)

Where HS describes the evolution of the system, HR is the Hamiltonian for the reservoir and HSR describes the interaction between the system and the reservoir. The full density operator χ describes both the system and the reservoir, but because we are only interested in the dynamics of the system we can eliminate the reservoir description from the density matrix by taking the partial trace over the reservoir, giving us ρ(t) = trRχ(t) (3.7) Where before we were using χ(t) as the density operator that described both the

11 system and the reservoir. Now we see that we can write the expectation value as

ˆ  ˆ   ˆ   ˆ  hOi = trSR Oχ = trS OtrRχ = trS Oρ (3.8)

because Oˆ only acts upon the state of the system. Using the same method as we did to get Eqn. 3.5, we arrive with the following expression for the time evolution of the reduced density matrix

i h ˆ i ρ˙ = − HS, ρ (3.9) ~ This equation is known as the master equation, and we see that it does not have any reservoir or interaction terms. We now need to derive dissipation terms to add to this equation.

3.2 The Master Equation with Dissipation

To derive an expression that includes dissipation terms, we are going to follow the approach taken by Carmichael in [6]. We begin by writing Eqn. 3.5 in the interaction picture where we separate the system and reservoir Hamiltonians from the interaction Hamiltonian. We define

ˆ ˆ ˆ ˆ χ˜ = ei(Hs+HR)t/~χe−i(Hs+HR)t/~ (3.10)

Whose time derivative is

i ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i(Hs+HR)t/~ −i(Hs+HR)t/~ i(Hs+HR)t/~ −i(Hs+HR)t/~ χ˜˙ = (Hs + HR)e χe + e χe˙ ~ i ˆ ˆ ˆ ˆ i(Hs+HR)t/~ ˆ ˆ −i(Hs+HR)t/~ − e χ(Hs + HR)e (3.11) ~ By plugging in Eqns. 3.5 and 3.10 we arrive with

i h i i ˆ ˆ h i ˆ ˆ ˆ ˆ i(Hs+HR)t/~ ˆ ˆ ˆ −i(Hs+HR)t/~ χ˜˙ = Hs + HR, χ˜ − e Hs + HR + HSR, χ e (3.12) ~ ~

Because HSR is the Hamiltonian that involves the interactions between the system and reservoir it is the only Hamiltonian that is time dependant in the interaction picture, meaning that most of the terms in Eqn. 3.12 drop out, leaving us with

i h ˜ˆ i χ˜˙ = − HSR, χ˜ (3.13) ~

12 where we have defined

ˆ ˆ ˆ ˆ ˜ˆ i(Hs+HR)t/~ ˆ −i(Hs+HR)t/~ HSR ≡ e HSRe (3.14)

We see that by moving to the interaction picture that we have now arrived with an equation of motion that only has the HSR term from the total Hamiltonian. This is significant because it simplifies our derivation while still allowing us to find an equation of motion that does include dissipation. The reason we can still reach our goal is because in open systems the dissipation comes from the interaction of system with the field modes. Our next step is to integrate Eqn. 3.13, yielding

Z t h i i 0 0 ˜ˆ 0 χ˜(t) =χ ˜(0) + dt χ˜(t ), HSR(t ) (3.15) ~ 0 Which we substitute back into Eqn. 3.13 to get

h i Z t hh i i ˙ i ˜ˆ 1 0 0 ˜ˆ 0 ˆ χ˜ = − HSR(t), χ˜(0) − 2 dt χ˜(t ), HSR(t ) , HSR(t) (3.16) ~ ~ 0 We will assume that at t = 0 that there is no coupling between S and R, allowing us to write χ(0) = ρ(0)R0 (3.17)

Where R0 is the initial reservoir density operator. With this in mind, we note that

h ˆ ˆ ˆ ˆ i i(Hs+HR)t/~ −i(Hs+HR)t/~ trR [˜χ(t)] = trR e ρ(t)e

ˆ h ˆ ˆ i ˆ iHst/~ iHRt/~ −iHRt/~ −iHst/~ = trR [˜χ(t)] = trRe e ρ(t)e e (3.18) where the cyclic property of the trace allows us to arrive with

ˆ ˆ iHst/~ −iHst/~ trR [˜χ(t)] = e ρ(t)e =ρ ˜(t) (3.19)

Which means that the master equation can now be written as

h i Z t nhh i io ˙ i ˜ˆ 1 0 ˜ˆ 0 ˆ 0 ρ˜ = − trR χ(0), HSR + 2 trR χ˜(t ), HSR(t ) , HSR(t) dt (3.20) ~ ~ 0

h ˜ˆ i For simplicity we assume trR R0(r), HSR = 0, leaving us with

Z t nhh i io ˙ 1 0 ˜ˆ 0 ˆ 0 ρ˜ = 2 trR χ˜(t ), HSR(t ) , HSR(t) dt (3.21) ~ 0 We briefly pause to think back about the system and the reservoir that are interacting. The reservoir in general has many more degrees of freedom than the system. For

13 example in a 3 level atom such as that in Fig. 1.1a, the system contains only a couple of degrees of freedom (the state of the 3 level atom) while the reservoir contains all of the field modes surrounding the atom. As a result the reservoir has a profound effect on the system, while the system hardly effects the reservoir. It is because of this that the relaxation time of the reservoir is very short in time compared to that of the system, indicating that the reservoir has ”no memory” of past interactions. This is important because it means that the dissipative process is irreversible, anything that leaks out into the reservoir is lost. Anything that happened in the past does not matter, the evolution of the system depends solely on its present state. With these things in mind we can make two approximations to allow us to complete our derivation. Because the reservoir is unaffected by the reservoir, we can approximate thatχ ˜ only depends upon the dynamics of the reduced density operator.

χ˜(t) =ρ ˜(t)R0 + O(HSR) (3.22)

Where R0 is stationary. This is what is know as the Born approximation. Because the reservoir has no memory of past interactions, we can make our second approximation in which we write the reduced density operator as

ρ˜(t0) → ρ˜(t) (3.23)

This approximation is known as the Markov approximation. Together Eqns. 3.22 and 3.23 are known as the Born-Markov approximation and they are used to modify Eqn. 3.21, giving us the master equation in the Born-Markov approximation:

Z t nhh i io ˙ 1 ˜ˆ 0 ˆ 0 ρ˜ = 2 trR ρ˜(t)R0, HSR(t ) , HSR(t) dt (3.24) ~ 0 This is the result that we were looking for, with it we can switch back to the Schrodinger picture and add it to Eqn. 3.9, giving us

i h i ˆ ˆ ˆ −iHst/~ iHst/~ ρ˙ = − HS, ρ + e ρe˜˙ (3.25) ~ As an example we will see what this equation will look like for the radiative decay of a two level atom. We begin by defining 1 Hˆ ≡ ω σ S 2~ 0 z ˆ X † HR ≡ ~ωjrj rj j X ∗ † Hˆ ≡ (κ r σ + κ~ r~ σ ) (3.26) SR ~ ~k,p ~k,p − k,p k,p + ~k,p

14 where σz is the pauli operator. The annihilation and creation operators for the reser- † ~ voir are given by r and r , and the frequency of the jth field mode is ωj. In HSR the k term represents the wavevector of the field mode while p denotes the polarization. σ+ and σ− represent the absorption and emission of a photon, respectively. We assume the reservoir to be in thermal equilibrium at room temperature, allowing us to write the reservoir density operator as

† ω r r ~ j j j  ~ωj  Y − k T − k T R0 ≡ e B 1 − e B (3.27) j

The combination of Eqns. 3.26 and 3.27 with 3.25 results in i γ ρ˙(t) = − ω [σ , ρ] + (¯n + 1)[2σ ρσ − σ σ ρ − ρσ σ ] 2 0 z 2 − + + − + − γ + n¯[2σ ρ(t)σ − σ σ ρ − ρσ σ ] (3.28) 2 + − − + − + where γ is the spontaneous emission rate andn ¯ is the thermal photon number of the reservoir that the atom couples to. It should be noted that at the optical wavelengths that are usually considered in cavity QED experiments,n ¯ is negligible and can be ignored. This equation is said to be in Lindblad form, a form which generally looks like i ˆ X ˆ ˆ† ˆ† ˆ ˆ† ˆ ρ˙ = − [Hs, ρ] + Γi(2OiρOi − Oi Oiρ − ρOi Oi) (3.29) ~ i Where the Oˆ terms are collapse operators. By defining the superoperator L, we rewrite Eqn. 3.29 as ρ˙ = Lρ (3.30)

3.3 Quantum Trajectories

The master equation describes the evolution of an open system and we can use the density matrix to solve this equation. In quantum optics experiments there are events such as spontaneous and stimulated emission that we may be interested in. However in Eqn. 3.7, detail of these events is averaged away in the trace. There is an alternate method known as Quantum Trajectory Theory that can be used to describe an open system. This method allows for us to write the the density matrix as an ensemble of pure state wavefunctions [7]. Each of these state vectors is known as a quantum trajectory, and they can be viewed as one possible history of the evolution of the system [4]. Using these trajectories we can create what is known as a scattering record, denoted as REC. One could create a scattering record in a lab by surrounding the interaction region with detectors and keeping track of when

15 a detector clicks. This scattering record allows us to rewrite the density matrix as X ρ = PREC |ψREC ihψREC | (3.31) REC By taking the trace of both sides we see that X PREC = 1 (3.32) REC Or in other words, the various probabilities must sum to unity. We need to construct a density matrix that both solves the master equation and gives the probabilities PREC that correctly account for what an observer sees in the lab. In order to use quantum trajectories to solve the master equation, we break Eqn. 3.30 into two parts

ρ˙ = (L − Lγ) ρ + Lγρ

= LBρ + Lγρ (3.33)

Where Lγ is the jump term. Each operator in Lγ are collapse operators, they are the dissipation terms that cause the wavefunction to collapse. For now we choose Lγ to ˆ ˆ† include the dissipation terms of the form OρO . The other term LB represents the evolution of the system between jumps. Once we have chosen the jump operator we can write

i h ˆ i h ˆ i LBρ = − Hs, ρ + HD, ρ (3.34) ~ + ˆ Where we have defined HD to include the dissipation terms that are not in Lγ. With this we can construct a Hamiltonian to govern the evolution between jumps ˆ ˆ ˆ H = Hs + i~HD (3.35) Note that the Hamiltonian is non-Hermitian. We now shift our focus to the last term of Eqn. 3.33. As we said this is the jump term and is written as

L = 2ΓOρˆ Oˆ† γ √ √ = 2ΓOˆ|ψihψ|Oˆ† 2Γ (3.36)

In our case the dissipation terms√ come from spontaneous emission so the collapse operator would take the form of 2Γˆσ−, which represents a photon being emitted and the atom decaying into a ground state. Now that we have the collapse operator, we need to look at the probability of the

16 wavefunction collapsing. The probability of a photodection in a time step is given by

† 2ΓhψREC (t)|O O|ψREC (t)idt Pcoll(t) = (3.37) hψREC (t)|ψREC (t)i The denominator is to normalize the probability and is necessary because the Hamil- tonian is non-Hermitian. The process in which a photon is emitted is completely random so we compare this probability to a random number generated by a com- puter that has a value between 0 and 1. If the probability is greater than the random number, it signifies a collapse of the wavefunction, if the probability is less the the random number then the state continues to evolve. We can run individual trajectories for many lifetimes, and we must run many trajectories to create a model since the processes that we are modeling are random. Figure 3.1 gives a pictorial representation of the process that goes into creating a single trajectory.

Figure 3.1: Basic process involved for constructing quantum trajectories. This pro- cess must be repeated many times with a small dt value in order to create a single trajectory.

17 3.3.1 Two Level Atom in a Field For a better understanding of how the quantum trajectory process works we investi- gate a two level atom in a field. This type of atom has only two states, an excited state and a ground state. Atoms can absorb photons from the field and become ex- cited, and can also decay back into |gi via spontaneous emission. The probability amplitudes evolve like so: iΩ γ C˙ = − C e−i∆t − C (3.38) e 2 g 2 e iΩ C˙ = − C ei∆t (3.39) g 2 e Our first step is to set t = 0 and to choose an initial state, and then calculate the probability of a collapse by

† + − Pcoll = hJ Jidt = γhσ σ idt = γPedt (3.40)

A random number r between 0 and 1 is then generated and compared to Pcoll ˙ If Pcoll < r Then: Ce = Ce + Cedt ˙ Cg = Ce + Cgdt

Else: Ce = 0 Cg = 1

As the evolution of the state is non Hermitian, the state must then be normalized, therefore

Ce Ce = (3.41) p 2 2 |Ce| + |Cg|

Cg Cg = (3.42) p 2 2 |Ce| + |Cg|

Then t is increased by a small time step dt and the process starts all over again, however this time we begin the process with the t value and probability amplitudes that we just calculated. This is repeated many times and is how the scattering record of the system is constructed. Fig. 3.2a shows a single trajectory for this two level system when the field is on resonance. It shows how the state evolves and it allows us to see when the state collapses into the ground state. As mentioned earlier, the processes that we are modeling are random so we must run many quantum trajectories and average them together to get an accurate idea of what is going on. Fig. 3.2b is 1000 trajectories averaged together, and as one might expect in a two level system, the atoms are in a superposition of the ground and excited state.

18 Figure 3.2: Single trajectory and 1000 trajectories averaged together for a two level atom. In the single trajectory we see Rabi oscillations in between each collapse. 1000 trajectories are averaged together to give us an accurate idea of how the state evolves

19 Chapter 4

EIT and EIA in Three Level Systems

Now that the necessary mathematical background has been laid out we can begin with our analysis of EIT and EIA. The program we used for this was the Quantum Toolbox in Python (QuTiP). QuTiP allowed us to carry out calculations for quantum trajectories, steady state values, and expectation values by using the master equation.

4.1 Λ Structures

4.1.1 Non-Degenerate Systems The Λ structure seen in Fig. 1.1a is a non-degenerate system, meaning that the two ground states do not share the same energy level. In order to use QuTiP we need to define the system Hamiltonian and the collapse operators. The total Hamiltonian is ˆ H = ∆ (|eihe| − |g1ihg1|) + φ(|eihe| − |g2ihg2|)

+Ωp(|g1ihe| + |eihg1|) + Ωc(|g2ihe| + |eihg2|)

+Γ(|g1ihe| − |eihg1|) + γ(|g2ihe| − |eihg2|) (4.1)

Therefore system Hamiltonian is given by ˆ Hs = ∆ (|eihe| − |g1ihg1|) + φ(|eihe| − |g2ihg2|)

+Ωp(|g1ihe| + |eihg1|) + Ωc(|g2ihe| + |eihg2|) (4.2) and because an atom in the excited state can end up in one of two different ground states, there are two collapse operators: √ Jˆ = γ|g ihe| (4.3) 1 √ 2 ˆ J2 = Γ|g1ihe| (4.4)

20 In QuTiP, the steady function calculates the steady state density matrix which is useful because the diagonals of this matrix give the populations of each of the states. We used this to view the excited state population as the probe beam was scanned, the results are shown in the plot below.

Figure 4.1: Excited state population hσ+σ−i as a function of ∆/t. Important values are φ = 0 and γ = 5, Γ = 0.1, Ωp = 0.1γ in the same frequency units. The dashed green spectra are a result of the coupling beam being turned off. We do not see a dip in the green spectra because there is no interference when there is only one beam. The blue spectra are produced when we set a.) Ωc = .2γ, b.) Ωc = .5γ, c.) Ωc = 1γ, d.) Ωc = 2γ. As the coupling gets stronger we see that the effects of EIT become more pronounced

When both of the beams are on resonance the population dips to zero. This dip, which is the result of destructive quantum interference between the two transitions,

21 is the defining characteristic of EIT. When we turn the coupling beam off, the plots are lorentzian as normal absorption has a lorentzian profile. For more information about this system, the expectation value was calculated for the weak dipole transition as the probe beam was again scanned, resulting in Fig. 4.2. The reason why this was plotted was because the expectation values give us valuable information about the system. The expectation values are complex with the real components giving the index of refraction while the imaginary components correspond to absorption [8]. Once again we see that there is no absorption when the probe beam is on resonance, which agrees with what is seen in Fig. 4.1.

Figure 4.2: By plotting the expectation values of σe1g1 we are able to see the index of refraction (solid line) and absorption (dashed line). Ωp = .1γ and Ωc = 2γ

22 4.1.2 Degenerate Systems We are now going investigate a degenerate Λ system, where both of the ground states both have the same energy level. In this case a magnetic field is used to break the degeneracy to produce the structure seen in Fig. 4.3. The magnetic field causes the magnetic substates m = -1 and m = +1 to split from the m = 0 state, giving us the structure that we need to produce EIT. When modeling this system we will set ∆ = φ = 0, however, we will be scanning the magnetic field which will change the detuning by a known amount −µB, where µ is the magnetic dipole moment [2].

Figure 4.3: Zeeman splitting results in the substates |g1i and |g2i. This diagram shows a J=1 → J=0 transition, where J is the total angular momentum. The atomic transitions are excited by σ+ and σ− and they represent the left and right handed circularly polarized beams respectively.

The Rabi frequencies between the Zeeman levels are proportional to the Clebsch- Gordan coefficients [10], meaning the Rabi frequencies between the sublevels can be written as Ωge = cgeΩ (4.5)

Where the cge terms are the Clebsch-Gordan coefficients. For simplicity we set ∆ = φ = 0 and then write the Hamiltonian for a degenerate Λ system as ˆ H = µB(me1 |e1ihe1| + mg1 |g1ihg1| + mg2 |g2ihg2|) +

cg1e1 Ωp(|g1ihe1| + |e1ihg1|) + cg2e1 Ωc(|g2ihe| + |e1ihg2|) + 2 2 γ|cg1e1 | (|g1ihe1| − |e1ihg1|) + γ|cg2e1 | (|g2ihe1| − |e1ihg2|) (4.6)

As the magnetic field was scanned, the steady state values of the density matrices were calculated, allowing for the Pe values to be plotted.

23 Figure 4.4: EIT in a degenerate Λ system. Ωc = 3γ,Ωp = 1.5γ.

For more information we used the mcsolve function so that we could analyze the evolution of the state vector with the use of quantum trajectories. We set B = 0 as that was the point of interest.

Figure 4.5: Single quantum trajectory for degenerate Λ system. We arbitrarily chose |ψ0i = |ei and allowed the state to evolve for 20 spontaneous emission lifetimes when B = 0

24 Figure 4.6: 1000 quantum trajectories averaged together. We see the population decays into the ground states and become transparent to the two lasers

We see that after about 13 spontaneous emission lifetimes the population of the excited state goes to 0. Atoms in the excited state decay into the ground states and can no longer absorb energy from the two lasers. The atoms have been sentenced to live their life in a dark state, and it is obvious that they are not excited about it. EIT is the result of quantum interference, because of this the intensities of the laser beams have a major influence on the shape of the plots that you see for the steady state populations. In a lab one can easily change these intensities, therefore it was necessary to show what happens to the population of the excited state as the one of the Rabi frequencies was changed. Of course, both of the beams can be be changed, but we chose to keep Ωp constant and looked at several values for Ωc as we scanned the magnetic field. Fig. 4.7 shows that as expected that once again, increasing the coupling beam leads to the transparency becoming more pronounced.

25 Figure 4.7: EIT for different values of Ωc. We set Ωp = 1.5γ and see that the stronger the coupling beam is, the greater the transparency is.

4.2 V Structures

We now shift our focus to V atomic structures. We will be looking at the degenerate case, therefore all of the same concepts that were applied in the degenerate case for the Λ system will also be applied here. The system is similar to that of Fig. 4.3, except now we will have one ground state with excited states that will have their degeneracy broken by a magnetic field. Because of this the system Hamiltonian takes on a familiar form. ˆ H = µB(mg1 |g1ihg1| + me1 |e1ihe1| + me2 |e2ihe2|) +

cg1e1 Ωp(|g1ihe1| + |e1ihg1|) + cg1e2 Ωc(|g2ihe2| + |e2ihg2|) 2 2 γ|cg1e1 | (|g1ihe1| − |e1ihg1|) + γ|cg1e2 | (|g1ihe2| − |e2ihg1|) (4.7) we have both of the lasers driving the atoms from the ground state to each of the excited states, and we expect to see destructive interference because the atoms in the excited states decay into the same ground state.. We again use the steady function to tell us about the excited states. Now that there are two excited states, the probabilities of being in those two individual states must be summed up in order to give us the total probability of being in the excited state.

26 Figure 4.8: Destructive interference in a V system. Ωc = 3γ

We see dips at B = 0 as expected, and now we want to study what happens when we have the two lasers driving the atoms from the excited states to the single ground state. Because of the symmetry in the transition probabilities the Hs part of Eqn. 4.7 will stay the same, however the jump operators need to be modified. This is easily done by taking the Hermitian conjugate of the collapse operators that we had before. Invoking this changes leads to Fig. 4.9. We also used quantum trajectories so that we could follow the population for 20 spontaneous lifetimes.

Figure 4.9: Constructive interference in a V system

27 With both of the lasers driving to the same ground state we see that on resonance that the atoms cannot be stimulated by either of the lasers to emit a photon, resulting in the entire population being in the excited state.

Figure 4.10: Single trajectory for a degenerate V structure in the case where both lasers are driving atoms into the ground state. Ωc = 1.5γ Ωp = 3γ and B = 0

28 Figure 4.11: 1000 Trajectories averaged together. We see that almost all of the population ends up in the excited states, a result of the quantum interference which makes it so the atoms cannot end up in the ground state via stimulated emission. However, atoms can decay into the ground state via spontaneous emission which is why we see still a small amount of the population in |g1i

29 Chapter 5

Degenerate Two Level Systems

Now that we have analyzed Λ and V structures, we now shift our focus to a degenerate two level system. That is, a structure where each of the excited state sublevels possess the same energy along with each of the ground states having the same energy. We will create profiles of the probe and coupling absorption spectra of the J = 1/2 → J = 3/2 transition. As seen in Fig. 5.1, we will be looking at the cases where there is a σ+ polarized coupling beam and probe beam which will be either σ− or π polarized.

Figure 5.1: Energy level diagram for J = 1/2 → J = 3/2 transition interacting with σ+ polarized coupling beam and a σ− probe beam

30 Figure 5.2: Energy level diagram for J = 1/2 → J = 3/2 transition interacting with σ+ polarized coupling beam and a π polarized probe beam

While this structure is more complex than the three level structures, the Hamil- tonian is constructed the same way as before. It is given by ˆ H = µB[mg1 |g1ihg1| + mg2 |g2ihg2| + me1 |e1ihe1| + me2 |e2ihe2| + me3 |e3ihe3| +

me4 |e4ihe4|] + Ω2 [cg1e1 (|g1ihe1| + |e1ihg1|) + cg2e2 (|g2ihe2| + |e2ihg2|)] +

Ω1 [cg1e2 (|g1ihe2| + |e2ihg1|) + cg2e3 (|g2ihe3| + |e3ihg2|)] + 2 Ω0 [cg1e3 (|g1ihe3| + |e3ihg1|) + cg2e4 (|g2ihe4| + |e4ihg2|)] + γ|cg1e1 | (|g1ihe1| − 2 2 |e1ihg1|) + γ|cg1e2 | (|g1ihe2| − |e2ihg1|) + γ|cg2e2 | (|g2ihe2| − |e2ihg2|) + 2 2 γ|cg2e3 | (|g2ihe3| − |e3ihg2|) + γ|cg2e4 | (|g2ihe4| − |e4ihg2|) (5.1)

− Note that there are minor changes in notation. In equation 5.1, Ω2 represents the σ polarized probe beam, Ω0 represents the π polarized probe beam, leaving Ω1 as the frequency of the coupling beam. This is the general Hamiltonian that will be used for this system, however when studying the case where the probe is σ− polarized we set Ω0 to zero. When we looking at the case where the probe is π polarized we set Ω2 to zero. Because there are six total transitions there will be six collapse operators, each with the same form as those seen for the Λ and V structures.

5.1 Circularly Polarized Probe Beam

To find the total probe and coupling absorption, we must first find the absorption of each of the individual transitions. For the σ− polarized probe beam, we find the expectation values of the dipole for the |g1i → |e1i and |g2i → |e2i transitions. The imaginary components of these expectation values reveal the absorption for each of those transitions. Each of these absorption spectra are added together to give us the

31 total probe absorption. The same method is used to find the coupling absorption, except of course this time the absorption for the |g1i → |e3i and |g2i → |e4i transi- tions are added together. Figures 5.2-5.5 shows the absorption spectra of the dipole transitions. In these figures we have chosen Ω1 to equal 0.5γ and then scanned the magnetic field. The values for the z axis are different values of the σ− polarized probe so that we can see what happens as the probe value is changed.

32 Figure 5.3: Absorption spectra for |g1i → |e1i transition. We see it change from EIT to EIA as the Ωp increases

33 Figure 5.4: Absorption for the |g2i → |e2i transition. We see the absorption change from EIA with transparency at B = 0 to almost no absorption once Ωp reaches a value that is about 2γ, a result of the strong probe beam forcing almost all of the population into either the |g1i and |e1i states

34 Figure 5.5: Front view of the total probe absorption. We note that it changes from EIT to EIA as Ωp increases

Figure 5.6: Back view of the total probe absorption.

35 Figure 5.7: Absorption for the |g1i → |e3i transition. We see that the transparency becomes more pronounced as Ωp increases

Figure 5.8: Absorption for the |g2i → |e4i transition. We see the most absorption for small values of Ωp, something we might expect as more of the population will be in either |g2i or |e4i when the coupling beam is stronger than the probe beam. As the intensity of the probe beam is increased we see EIT effects.

36 Figure 5.9: Front view of the total coupling absorption, it is the addition of the absorption spectra of the |g1i → |e3i and |g2i → |e4i dipole transitions. We see the change from EIA to EIT as Ωp increases

Figure 5.10: Back view of the total coupling absorption

37 When looking at the total probe and coupling absorption spectra, we see that something very interesting happens. As Ωp is increased we see the shape of the probe absorption go from EIT to EIA. A similar reversal is prevalent in coupling absorption as it starts as enhanced absorption but then changes to EIT. Here is where quantum trajectories become especially useful, with them we can see the populations for dif- ferent Ωp values to help us see what is going on. We will investigate what happens for the Ωp values where we see change in the absorption. First we choose Ωp = 0.2γ, then we will look at Ωp = 0.75γ and Ωp = 3γ. For each of these values we will look at µB = 0.

38 Figure 5.11: Single quantum trajectory with Ωc = 0.5γ and Ωp = 0.2γ. Between collapses we see Rabi oscillations. The caret symbols represent where a collapse occurs and the color of the caret tells us which collapse operator contributed to the collapse. Because the coupling beam is more than twice as strong as the probe beam we might expect most of the atoms to end up in the |g2i and |e4i states after some time. The single trajectory verifies this thought because we see that after the ensemble collapses into |g2i, all the other collapses are from |e4i to |g2i. Of course this is only a single trajectory, the process we are modeling is random so we must construct many trajectories and average them together to get an accurate portrayal of the population for each of the different states.

39 Figure 5.12: 1000 trajectories with Ωc = 0.5γ and Ωp = 0.2γ. We see that the stronger coupling beam leads to most of the population ending up in either |g2i or |e4i

40 Figure 5.13: Single trajectory with Ωc = 0.5γ and Ωp = 0.75γ. We see that the for the most part that the atoms are oscillating in |g1i or |e1i

41 Figure 5.14: 1000 trajectories with Ωc = 0.5γ and Ωp = 0.75γ. Now that the probe beam is stronger than the coupling beam we see the population shift to the left side of the atomic structure.

42 Figure 5.15: Single trajectory with Ωc = 0.5γ and Ωp = 3γ. The ensemble does not spend any time in |g2i, |e2i, or |e4i. With the probe beam significantly stronger than the coupling beam, no atoms are able to make it to the right side of the structure. Less than 1% of the atoms end up in |e3i, with this being the case the collapse will almost always be from |e1i to |g1i

43 1000 trajectories for Ωc = 0.5γ and Ωp = 3γ

Figure 5.16: 1000 trajectories with Ωc = 0.5γ and Ωp = 3γ. As we expect, almost all of the population is in |e1i or |g1i

44 Looking at Fig. 5.6, we can see that the majority of the population is found in the two ground states when Ωp = 0.2γ. Because the |g1i → |e2i and |g2i → |e3i transitions are not allowed, it is almost as if we are looking at two V structures. The atoms in each ground state can absorb photons and end up in one of the two excited states, and because of this one might expect to see absorption that has the same profile as that seen in Fig. 4.8. In fact we do see that the total probe absorption does have the dip that is seen in EIT, however, the coupling absorption is exhibiting EIA. To explain this we should realize that we cannot just describe the six level atom by simply as saying that it is two V structures. The obvious difference is that in a V structure, atoms in the excited states decay back down to the single ground state. In our model, atoms in the either |e2i or |e3i can decay into either of the two ground states, meaning that atoms from three different excited states can decay into a single ground state at the same time. This is significant because EIT and EIA is predicated on quantum interference, interference that will not be the same as it was in a three level atom. It is also necessary to note that based on the Clebsch-Gordan coefficients, the decay rates from |e2i → |g1i and |e3i → |g2i are twice that of |e2i → |g2i and |e3i → |g1i. With this in mind we can go back and look at Fig. 5.5 and analyze it once more. For our simulation we chose the atoms to start in |e1i and we see them decay into |g1i, from here the majority of the atoms end up in |e3i because the Rabi frequency of the coupling beam is more than twice that of the probe beam. Now because of the decay rates, more atoms decay into |g2i than |g1i. To avoid redundancy we look at what happens after the atoms end up in |g2i. Again, because of the difference in the Rabi frequencies most of the atoms that are excited end up in |e4i. Some atoms do end up in |e3i, where they can decay into the ground states and where the process starts again. The main thing to take away is that most of the population ends up on the right side of the structure, that is, most of the atoms can eventually be found in either |g2i or |e4i. With a better understanding of what happens we take another look at the different absorption spectra. For each dipole transition except for the |g2i → |e4i transition we see EIT, which agrees with what we see in the populations for |e1i, |e2i, and |e3i found in Fig. 5.6. We can also now understand why we see EIA in the |g2i → |e4i transition, we know that because of difference in the Rabi frequencies that most of the population is going to end up on the right side of the structure. What is important though is that whatever atoms are excited into |e2i are likely to end up in |g1i and then |e3i and while the atoms are spending time in these states there are still the other atoms in |g2i that are being excited by the stronger coupling beam. It is because of the atoms being on the other side of the structure along with the decay from |e3i → |g2i while the atoms are being excited into |e4i that leads to the quantum interference that results in us seeing EIA in the |g2i → |e4i dipole transition. The absorption in this transition is so great that even with EIT in the |g2i → |e2i transition we still have overall EIA for the total coupling absorption. This also helps us to see why the

45 probe switches from EIT to EIA and the coupling absorption switches from EIA to EIT. When Ωp becomes larger than Ωc it stands to reason that the same thing will happen as when Ωp = 0.2γ, with the exception that everything now becomes reversed. Instead of the population being swept to the right side of the diagram, it is instead swept to the left side. When Ωp = 0.75γ the coupling absorption shows EIT for the same reason that the probe absorption did when Ωp = 0.2γ, just as the probe absorption now gives us EIA for the same reason that the coupling absorption did when Ωp = 0.2γ. By looking at Fig. 5.7, we see agreement with this argument by noticing that the populations are essentially reversed. The last thing that we notice when looking at the coupling and probe absorption spectra is that when Ωp gets significantly larger than that of Ωc, both spectra have EIT characteristics. Fig. 5.8 shows that almost all of the population is in |e1i and |g1i when Ωp = 3γ, something we might expect since not much of the population can end up somewhere in the right side of the structure due to the relatively very weak Ωc. With not many atoms making the |g1i → |e3i transition, we do not have the same type of interference that we had when Ωp = 0.2γ or Ωp = 0.75γ. In other words there is not going to be a dipole transition with an absorption that shows EIA. For the most part we can say that the six level atom becomes mostly a V structure that is experiencing EIT, this is not exactly true because still whatever ends up in |e3i can decay into |g2i, be excited into |e2i and then decay back into |g1i. However, since very little of the population does this we can for the most part hypothesize that the the interference is very similar to that of an ordinary V structure. With very little of the population being in the right side of the atom, we see the absorption of the |g1i → |e1i and |g1i → |e3i transitions dominate the overall probe and coupling absorption spectra. Just to show just how similar the interference is between the 6 level atom and the V structure, we took Eqn. 4.7 and plugged in Ωp = 3γ and Ωc = 0.5γ and plotted the absorption of the two dipole transitions and compared them to the absorption of the |g1i → |e1i and |g1i → |e3i transitions in the six level atom, resulting in Figs. 5.9 and 5.10. As expected, the spectra were very similar.

46 Figure 5.17 Figure 5.9: Absorption spectra for dipole transitions in a degenerate 3 level V struc- ture.

47 Figure 5.18 Figure 5.10: Absorption spectra for |g1i → |e1i and |g1i → |e3i transitions in a six level atom. We see that when Ωp becomes significantly larger than Ωc the absorption spectra has a striking resemblance to those in a V structure. This is because the strong probe beam does not allow for many atoms to end up in the other states.

5.2 Linearly Polarized Probe Beam

Now that we know what happens in when the probe beam is circularly polarized, we are now going to investigate the case where it is linearly polarized. We will take the same approach as before, with some slightly different parameters. This time we set Ωc = 1.0γ and then look at the change in absorption as Ωp changes. Now that the probe beam is linearly polarized, the total probe absorption becomes the sum of the |g1i → |e2i and |g2i → |e3i transitions. We then used quantum trajectories to plot the populations when Ωp = .5γ, 1.5γ, 3γ, all when B = 0.

48 Figure 5.19: Front view of absorption spectra for |g1i → |e1i dipole transition. We see the for low Ωp values that there is a dip within a dip at B = 0, we will call this transparency within transparency (TWT). As the probe gets stronger, we see that it appears to change to EIT

Figure 5.20: Back view of absorption spectra for |g1i → |e1i dipole transition Figure 5.2: Absorption spectra for a.) |g1i → |e1i and b.)|g2i → |e2i dipole transitions.

49 Figure 5.21: Front view of absorption spectra for |g2i → |e3i dipole transition. We see absorption within the transparency (AWT) at B = 0 which changes into some degree of TWT as Ωp increases

Figure 5.22: Back view of absorption spectra for |g2i → |e3i dipole transition

50 Figure 5.23: Front view of total probe absorption. We see it changes from TWT to AWT.

Figure 5.24: Back view of total probe absorption

51 Figure 5.25: Front view of absorption spectra for |g1i → |e3i dipole transition. We see it change from AWT to EIT

Figure 5.26: Back view of absorption spectra for |g1i → |e3i dipole transition

52 Figure 5.27: Front view of absorption spectra for |g2i → |e4i dipole transition. It starts out with regular absorption and then it changes to EIT as less atoms become absorbed when Ωp is increased.

Figure 5.28: Back view of absorption spectra for |g2i → |e4i dipole transition

53 Figure 5.29: Front view of total coupling absorption. It starts out with an EIA peak at line center, as Ωp is increased the peak becomes a dip

Figure 5.30: Back view of total coupling absorption

54 Figure 5.31: Single Trajectory with Ωp = 1γ and Ωp = 0.5γ. With the stronger coupling beam we see that most of the collapses are into |g2i as the population is making it’s way to the right side of the atomic structure

Figure 5.32: 1000 Trajectories with Ωc = 1γ and Ωp = 0.5γ. As expected, when the trajectories are averaged together we see that most of the population ends up in either |g2i or |e4i

55 Figure 5.33: Single Trajectory with Ωc = 1γ and Ωp = 1.5γ. With the probe beam now stronger than the coupling beam one might think that the most of the population would end up in the left side of the structure, this however is not the case. We see the ensemble oscillating throughout all of the states except for |e1i, which cannot be reached since the probe beam is linearly polarized.When an atom gets into |g2i there is no laser

56 Figure 5.34: 1000 Trajectories with Ωc = 1γ and Ωp = 1.5γ. We see that the popula- tion becomes spread out throughout the states that have allowed dipole transitions. The reason why we do not see most of the population in |e2i and |g1i is because the linearly polarized probe beam makes it more difficult for atoms that end up on the right side of the diagram to make it back to the left side

57 Figure 5.35: Single Trajectory with Ωc = 1γ and Ωp = 3γ. Again we see the ensemble oscillating throughout all of the states that have allowed dipole transitions, a result of the probe beam being linearly polarized.

Figure 5.36: 1000 Trajectories with Ωc = 1γ and Ωp = 3γ averaged together. We see note that by comparing this plot to when Ωp = 1.5γ that less of the population can make it into |e4i

58 Again we see that the absorption spectra change significantly as Ωp increases, and this time what is going on seems to be slightly more complicated than when the probe beam was circularly polarized. When we look at Fig. 5.2 we notice that within the six level atom we can actually see two different V structures as the atoms in each ground state and end up in two different excited states. Along with two V structures we also notice a Λ structure as the atoms in both ground states can be excited into |e3i. The reason we point this out is because perhaps with the decay process as it is in the six level atom it might stand to reason that absorption that we are seeing are some sort of combination of a V and Λ structure. While the transitions for the π probe are different than that of a σ− probe, the concepts that we use to analyze the absorption is the same. We first look at the case where Ωp < Ωc by looking at the trajectories we used to model the population when Ωp = .5γ. Looking at Fig. 5.29 we see that most of the population is found in either |g2i or |e4i, something we expect as the stronger coupling beam is circularly polarized. When atoms end up in |g2i they can either be excited into |e3i or |e4i. Those atoms which make the |g2i → |e3i transition do not always decay back into |g2i as they can decay into |g1i as well. This means that some of the atoms are found over on the left side of the structure while a much larger portion of the population is undergoing the |g2i → |e4i transition. This results in an almost lorentzian profile for the absorption for the |g2i → |e4i transition since there is not much quantum interference. Meanwhile the atoms that are not on the far right of the atomic structure experience what can be described as a combination of the effects of the individual V and Λ structures, which is why we see strange absorption profiles for these transitions. One can refer to Fig. 5.32 for a 2D representation of the different absorption spectra to get a better view of these effects. The absorption for the |g1i → |e2i has a dip, and in the middle of this dip there is actually a second dip. We will refer to this as transparency within transparency (TWT). The TWT is enough to make the total probe absorption exhibit TWT. The absorption for |g1i → |e3i transition shows absorption within transparency (AWT) which results in the total coupling absorption exhibiting an EIA peak at line center.

59 Figure 5.37: 2D Plot of Absorption Spectra in Six Level Atom with π Probe Beam. Ωc = 1γ and Ωp = .5γ

Next we want to look at the case where Ωp > Ωc, for this we analyze what happens when Ωp = 1.5γ. One might expect that now that the probe beam is stronger than the coupling beam that the majority of the population would be found in either |g1i or |e2i. However, Fig. 5.29 shows that only about half of the population is in these two states even with the probe being 50% more intense than the coupling beam. In the case where Ωp < Ωc we saw that when atoms were excited into |e4i they could only decay into |g2i, which is why most of the population was found in these two states. The same thing does not happen for Ωp > Ωc. With the probe beam being linearly polarized the atoms cannot make it to the very far left side of the atomic structure, in other words, they cannot end up in |e1i. Instead, the leftmost excited state that the atoms can be found in is |e2i, a state where of course if the atoms were to decay they could end up in either ground state. The reason this is important because if an atom were to decay into |g2i, the only way for it to get back to the left side of the diagram is if it were to end up in |e3i and then decay back into |g1i. This is the reason why the populations in Fig. 5.29 are more spread out than first predicted. With these populations spread out the way they are, all of the absorption spectra are going to exhibit a combination of V and Λ structures, which is what we see in Fig. 5.33. We notice that now that Ωp > Ωc we see a change in the absorption at B = 0, where we saw a peak in Fig. 5.32 we see a dip in Fig. 5.33 and vice versa.

60 Figure 5.38: 2D Plot of Absorption Spectra in Six Level Atom with π Probe Beam. Ωc = 1γ and Ωp = 1.5γ

Finally we look at the case where the pump is much greater than the coupling beam. When we looked at this case for the σ− polarized probe we saw that only about 1% of the population was not found in |g1i or |e1i. For the π probe we do not see such a large percentage shift to the left side of the structure because as we discussed when Ωp = 1.5γ, once atoms get into the right side of the diagram it’s harder for them to make it back into |g1i and |e2i so we see the population is more spread out than when we had a circularly polarized beam. It is because of this that we once again see absorption spectra that can be explained as a combination of V and Λ EIT and EIA effects.

61 Figure 5.39: 2D Plot of Absorption Spectra in Six Level Atom with π Probe Beam. Ωc = 1γ and Ωp = 3γ

62 Chapter 6

Conclusion

6.1 Summary

In this thesis we have investigated EIT and EIA in several atomic structures with the use of the steady state solution to the master equation and also with quantum trajectories. In the first chapter we showed that even though we are modeling EIT semiclassically, the interference which causes EIT is analogous to that in coupled RLC circuits. As EIT is characterized by no absorption, the entire population be- comes trapped in a dark state. This is shown mathematically in Chapter 2 for a non degenerate Λ structure. In open systems such as the ones we were studying, the system becomes coupled to the environment due to emission of photons. Because of this it was necessary to define the density matrix in Chapter 3. We also derived the master equation so that we could show how the density operator evolves in time, and then we were able to add dissipation terms to this equation with the use of the Born-Markov approximation. Once we made this approximation we were able to complete an example so that we could see the final form of the master equation. Finally we were able to introduce quantum trajectory theory and explain how splitting up the superoperator can allow us to construct a scattering record which can be used to solve the master equation. Chapters 4 and 5 both dealt with modeling EIT and EIA. Chapter 4 was more of an introduction to the methods that we were going to use in Chapter 5, as Λ and V structures are less complex than degenerate two level systems. We were able able to use the steady state density matrix to model the absorption of the various systems; for the 6 level atom if there was a peak at B = 0 when Ωp > Ωc then there would be a dip at Ωp < Ωc, and vice versa. We then made use of quantum trajectories to help us understand the absorption spectra that was exhibited.

63 6.2 Future Work

The future work lies in modeling structures that are even more complex than the ones done in this thesis. For example in [11], the absorption spectra are shown 133 for the Fg = 4 → Fe = 5 transition D2 line of Cs, and for that the flip in the total probe and coupling spectra are very noticeable when the change is made from Ωp > Ωc to Ωp < Ωc. In a system like this it would be very helpful to use quantum trajectories as they would tell which states the population ends up in for the different laser intensities. This would be important because it would tell which dipole transitions have the greatest contributions to the quantum interference, giving a better understanding of the system that is being modeled.

64 Appendix A

Classical EIT Code

#Code used to produce Figure 1.3 #Classical EIT from numpy import * from pylab import * k1 = 2.0 k2 = 1 gamma1 = 4.0e-2 gamma2 = 1.0e-7 F = 0.1 m = 1 omega_r = [0, 0.1, 0.2, 0.3] omega = 1.05613 steps = 1000 detuning = linspace(-0.15, 0.15, steps) p = [zeros(steps) for i in range(4)] for i in range(steps): o_s = omega + detuning[i] for k in range(4): p[k][i] = (-2*pi*1j*F**2*o_s*(omega**2 - o_s**2 - 1j*gamma2*o_s) / (m * ((omega**2 - o_s**2 - 1j*gamma1*o_s)*(omega**2 - o_s**2 - 1j*gamma2*o_s) - omega_r[k]**4)))

f = figure(figsize=(6,7)) subplots_adjust(hspace=0.1)

65 # subplot 1 (top) ax1 = subplot(411) ax1.plot(detuning,p[0],’b’) ax1.text(-.14,1.8,’a.)’) ylim([0,2]) xlim([-.15,.15]) ylabel(r’$P_2$’)

# subplot 2 ax2=subplot(412,sharex=ax1) #share x-axis of subplot 1 ax2.plot(detuning,p[1],’b’) ylim([0,2]) xlim([-.15,.15]) ylabel(r’$P_2$’) # subplot 3 ax3=subplot(413,sharex=ax1) #share x-axis of subplot 1 ax3.plot(detuning,p[2],’b’) ylim([0,2]) xlim([-.15,.15]) ylabel(r’$P_2$’) # subplot 4 ax4=subplot(414,sharex=ax1) #share x-axis of subplot 1 ax4.plot(detuning,p[3],’b’) ylim([0,2]) xlim([-.15,.15]) ylabel(r’$P_2$’) xticklabels = ax1.get_xticklabels()+ax2.get_xticklabels()+ax3.get_xticklabels() setp(xticklabels, visible=False) ax1.xaxis.set_major_locator(MaxNLocator(4)) xlabel(’$\delta=\omega_s-\omega$’) ax1.text(-.14,1.8,’a.)’) ax2.text(-.14,1.8,’b.)’) ax3.text(-.14,1.8,’c.)’) ax4.text(-.14,1.8,’d.)’) show()

66 Appendix B

Code for EIT in Non Degenerate Λ System

#Code for Figure 4.1 from qutip import * from pylab import * from numpy import *

#Non degenerate Lambda steady state function ustate = basis(3,0) excited = basis(3,1) ground = basis(3,2) nsteps=100 delta_min=-3 delta_max=3 xmin=delta_min xmax=delta_max d=(delta_max-delta_min)*(nsteps)**(-1) omega1=.1 omega2=2 phi=0 gamma_eu=.1 #decay rates gamma_eg=5 sigma_ee = tensor(excited * excited.dag()) sigma_uu = tensor(ustate * ustate.dag()) sigma_gg = tensor(ground * ground.dag()) sigma_ue = tensor(ustate * excited.dag())

67 sigma_ge = tensor(ground * excited.dag()) clist=[] dlist=[]

#collapse operators c1=sqrt(gamma_eg)*sigma_ge c2=sqrt(gamma_eu)*sigma_ue c1dc1=c1.dag()*c1 c2dc2=c2.dag()*c2 fig=figure(figsize=(6.1,5)) fig.subplots_adjust(wspace=0) fig.subplots_adjust(hspace=0) ax1 = fig.add_subplot(111) ax1.set_ylabel(r’$P_e$’) for i in range (0,nsteps+1): #system hamiltonian H=(-delta_min)*(sigma_ee-sigma_gg)+(phi)*(sigma_ee-sigma_uu)+ omega1*(sigma_ge+sigma_ge.dag())+omega2*(sigma_ue+sigma_ue.dag()) #Liouvillian L=liouvillian(H,[c1,c2]) rhoss=steady(L) prob_excited=rhoss[1,1] clist.append(prob_excited) dlist.append(delta_min) delta_min=delta_min+d ax1.plot(dlist,clist) locator_params(axis = ’y’, nbins = 6) show() plot(dlist,clist)

68 #Code used to produce Fig. 4.2 from qutip import * from pylab import * from numpy import * import matplotlib.pyplot as plt

#Absorption and Index of Refraction for Non-degenerate 3 level lambda system ustate = basis(3,0) excited = basis(3,1) ground = basis(3,2) nsteps=100 delta_min=-3 xstart=delta_min delta_max=3 d=(delta_max-delta_min)*(nsteps)**(-1) omega1=.1 omega2=2 phi=0 gamma_eu=.1 #decay rates gamma_eg=5 sigma_ee = tensor(excited * excited.dag()) sigma_uu = tensor(ustate * ustate.dag()) sigma_gg = tensor(ground * ground.dag()) sigma_ue = tensor(ustate * excited.dag()) sigma_ge = tensor(ground * excited.dag()) dlist=[] elist=[]

#collapse operators c1=sqrt(gamma_eg)*sigma_ge c2=sqrt(gamma_eu)*sigma_ue c1dc1=c1.dag()*c1 c2dc2=c2.dag()*c2 for i in range (0,nsteps+1): #system hamiltonian H=(-delta_min)*(sigma_ee-sigma_gg)+(phi)*(sigma_ee-sigma_uu)+

69 omega1*(sigma_ge+sigma_ge.dag())+omega2*(sigma_ue+sigma_ue.dag()) #Liouvillian L=liouvillian(H,[c1,c2]) rhoss=steady(L) n1=expect(sigma_ge.dag(),rhoss) dlist.append(delta_min) elist.append(n1) delta_min=delta_min+d real=[i.real for i in elist] imag=[i.imag for i in elist] plot(dlist,real) plot(dlist,imag,’--’) ylabel(r’$\langle{\sigma_{e_1g_1}}\rangle$’, fontsize=27) xlim(xstart, delta_max) xlabel(r’$\Delta/t$’, fontsize=27) locator_params(axis = ’y’, nbins = 6) xticks(fontsize=27) yticks(fontsize=27) legend((’Real’,’Imaginary’),’upper right’) plt.show()

70 Appendix C

Code for EIT in Degenerate Λ System

#Code used to produce Figs. 4.5 and 4.6 from qutip import * from pylab import * from numpy import *

#Quantum Trajectories for Degenerate Lambda System nsteps=10 e1=basis(3,0) g1=basis(3,1) g2=basis(3,2)

#dipole transitions sigma_g1e1=tensor(g1*e1.dag()) sigma_g2e1=tensor(g2*e1.dag()) sigma_g1g1=tensor(g1*g1.dag()) sigma_e1e1=tensor(e1*e1.dag()) sigma_g2g2=tensor(g2*g2.dag())

#Rabi frequencies (right, left polarized) omega1=.01 #pump/coupling beam ratio=2 omega2=1.5 #probe beam omegamax=3 ratiomax=2

71 uB=-4 reset=uB uB_max=4 r=(uB_max-uB)*(nsteps)**(-1)

#clebsch gordan coefficients e1g1=1 e1g2=sqrt(1.0/3.0)

#spontaneous emisssion gamma_e1g1=e1g1**2 gamma_e1g2=e1g2**2

#collapse operators c1=sqrt(gamma_e1g1)*sigma_g1e1 c2=sqrt(gamma_e1g2)*sigma_g2e1 collapse=[c1,c2] xlist=[] list1=[] for i in range (nsteps+1): #interaction hamiltonian H_i=e1g1*omega2*(sigma_g1e1.dag()+sigma_g1e1)+ e1g2*omega1*(sigma_g2e1.dag()+sigma_g2e1) H_b=(uB)*(-sigma_g1g1+0*sigma_e1e1+sigma_g2g2) H=H_i+H_b #liouvillian L=liouvillian(H,collapse) rhoss=steady(L) pe1=rhoss[0,0] list1.append(pe1) xlist.append(uB) uB=uB+r ax=plot(xlist,list1) ylabel(r’$P_e$’, fontsize=25) yticks(fontsize=20) xticks(fontsize=20) xlabel(’$\mu{B}$’, fontsize=25) xlim(reset,uB_max)

72 show() from qutip import * from pylab import * from numpy import *

#Code used to produce Fig. 4.7 nsteps=100 zsteps=nsteps e1=basis(3,0) g1=basis(3,1) g2=basis(3,2)

#dipole transitions sigma_g1e1=tensor(g1*e1.dag()) sigma_g2e1=tensor(g2*e1.dag()) sigma_g1g1=tensor(g1*g1.dag()) sigma_e1e1=tensor(e1*e1.dag()) sigma_g2g2=tensor(g2*g2.dag())

#Rabi frequencies (right, left polarized) ratio=.5 omega2=3 #probe beam omega1=omega2*ratio #pump/coupling beam ratio=.5 omega2=3 #probe beam omegamax=3 ratiomax=3 uB=-5 reset=uB uB_max=5 r=(uB_max-uB)*(nsteps)**(-1) d=(omegamax-omega2)*(zsteps)**(-1) rr=(ratiomax-ratio)*(zsteps)**(-1)

#clebsch gordan coefficients e1g1=1 e1g2=sqrt(1.0/3.0)

#spontaneous emisssion

73 gamma_e1g1=e1g1**2 gamma_e1g2=e1g2**2

#collapse operators c1=sqrt(gamma_e1g1)*sigma_g1e1 c2=sqrt(gamma_e1g2)*sigma_g2e1 collapse=[c1,c2] x=linspace(uB,uB_max,nsteps+1) y=linspace(ratio,ratiomax,nsteps+1) list1=[] zlist1=zeros(shape=(nsteps+1,nsteps+1))

for j in range (zsteps+1): for i in range (nsteps+1): #interaction hamiltonian H_i=e1g1*omega2*(sigma_g1e1.dag()+sigma_g1e1)+ e1g2*omega1*(sigma_g2e1.dag()+sigma_g2e1) H_b=(uB)*(-sigma_g1g1+0*sigma_e1e1+sigma_g2g2) H=H_i+H_b #liouvillian L=liouvillian(H,collapse) rhoss=steady(L) pe1=rhoss[0,0] list1.append(pe1) uB=uB+r zlist1[j]=list1 list1=[] uB=reset ratio=ratio+rr omega1=omega2*ratio x,y=meshgrid(x,y) fig= plt.figure() ax = fig.add_subplot(111, projection =’3d’) surf=ax.plot_surface(x, y, zlist1, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax.set_ylabel(r’$\Omega_{p}/\Omega_{c}$’, fontsize=27) ax.set_zlabel(r’$P_e$’, fontsize=27)

74 ax.set_xlabel(’$\mu{B}$’, fontsize=27) show()

75 Appendix D

Code for EIT and EIA in Degenerate V systems

#Code for Figs. 4.8 and 4.9 #degenerate vee structure. #take hermitian conjugate of operators to go from EIT to EIA from qutip import * from pylab import * from numpy import * nsteps=100 zsteps=3 e1=basis(3,0) e2=basis(3,1) g1=basis(3,2)

#dipole transitions sigma_g1e1=tensor(g1*e1.dag()) sigma_g1e2=tensor(g1*e2.dag()) sigma_g1g1=tensor(g1*g1.dag()) sigma_e1e1=tensor(e1*e1.dag()) sigma_e2e2=tensor(e2*e2.dag()) sigma_g1g1=tensor(g1*g1.dag())

#Rabi frequencies (right, left, linearly polarized) omega1=1.5 #pump/coupling beam ratio=2 omega2=.1 #probe beam omegamax=3

76 ratiomax=2 uB=-4 reset=uB uB_max=4 r=(uB_max-uB)*(nsteps)**(-1) d=(omegamax-omega2)*(zsteps)**(-1)

#clebsch gordan coefficients g1e1=1 g1e2=sqrt(1.0/3.0)

#spontaneous emisssion gamma_e1g1=g1e1**2 gamma_e2g1=g1e2**2

#collapse operators c1=sqrt(gamma_e1g1)*sigma_g1e1 c2=sqrt(gamma_e2g1)*sigma_g1e2 collapse=[c1,c2] xlist=[] list1=[] list2=[] fig=figure() ax = fig.add_subplot(121, projection =’3d’) ax1 = fig.add_subplot(122, projection =’3d’) fig=figure() ax2 = fig.add_subplot(111, projection =’3d’) for j in range (zsteps+1): for i in range (nsteps+1): #interaction hamiltonian H_i=g1e1*omega2*(sigma_g1e1.dag()+sigma_g1e1)+ g1e2*omega1*(sigma_g1e2.dag()+sigma_g1e2) H_b=(uB)*(0*sigma_g1g1+sigma_e2e2-sigma_e1e1) H=H_i+H_b #liouvillian L=liouvillian(H,collapse) rhoss=steady(L) pe1=expect(sigma_g1e1.dag(), rhoss).imag pe2=expect(sigma_g1e2.dag(),rhoss).imag

77 list1.append(pe1) list2.append(pe2) xlist.append(uB) uB=uB+r ylist1=list1 ylist2=list2 ylist3=linspace(0,0,nsteps+1)+list1+list2 z=[omega2]*(nsteps+1) ax.plot3D(xlist,z,ylist1) ax1.plot3D(xlist,z,ylist2) ax2.plot3D(xlist,z,ylist3) list1=[] list2=[] ylist=[] xlist=[] uB=reset omega2=omega2+d ax.set_ylabel(r’$\Omega_{probe}/\gamma$’, fontsize=27) ax.set_zlabel(r’$P_{e1}$’, fontsize=27) ax.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) ax1.set_ylabel(r’$\Omega_{probe}/\gamma$’, fontsize=27) ax1.set_zlabel(r’$P_{e2}$’, fontsize=27) ax1.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) ax2.set_ylabel(r’$\Omega_{probe}/\gamma$’, fontsize=27) ax2.set_zlabel(r’$P_e$’, fontsize=27) ax2.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) show()

78 #Code for Fig. 4.10 and 4.11 #Quantum trajectories for degenerate vee system. #take hermitian conjugate of operators to go from EIT to EIA from qutip import * from pylab import * from numpy import * e1=basis(3,0) e2=basis(3,1) g1=basis(3,2) psi0=e1

#dipole transitions #dipole transitions sigma_g1e1=tensor(g1*e1.dag()) sigma_g1e2=tensor(g1*e2.dag()) sigma_g1g1=tensor(g1*g1.dag()) sigma_e1e1=tensor(e1*e1.dag()) sigma_e2e2=tensor(e2*e2.dag()) sigma_g1g1=tensor(g1*g1.dag())

#Rabi frequencies (right, left polarized) omega1=1.5 #pump/coupling beam ratio=2 omega2=3 #probe beam omegamax=3 ratiomax=2 uB=0 reset=uB uB_max=4

#clebsch gordan coefficients g1e1=1 g1e2=sqrt(1.0/3.0)

#spontaneous emisssion gamma_e1g1=g1e1**2 gamma_e2g1=g1e2**2

79 #collapse operators c1=sqrt(gamma_e1g1)*sigma_g1e1.dag() c2=sqrt(gamma_e2g1)*sigma_g1e2.dag() collapse=[c1,c2]

#expectation values e1=c1*c1.dag()/gamma_e1g1 e2=c2*c2.dag()/gamma_e2g1 g1=c1.dag()*c1/gamma_e1g1 expect_values=[e1,e2,g1] ntraj=1 tlist=linspace(0,20,200)

#hamiltonian H_i=g1e1*omega2*(sigma_g1e1.dag()+sigma_g1e1)+ g1e2*omega1*(sigma_g1e2.dag()+sigma_g1e2) H_b=(uB)*(0*sigma_g1g1+sigma_e2e2-sigma_e1e1) H=H_i+H_b #liouvillian L=liouvillian(H,collapse) data=mcsolve(H,psi0,tlist,collapse,expect_values,ntraj) fig=figure(figsize=(6,7)) fig.subplots_adjust(wspace=.5) ax1 = fig.add_subplot(221) ax2 = fig.add_subplot(222) ax3 = fig.add_subplot(212) ax1.plot(tlist,data.expect[0]) ax2.plot(tlist,data.expect[1]) ax3.plot(tlist,data.expect[2]) ax1.set_ylabel(r’$|C_{e1}|^2$’) ax1.set_xlabel(’$\gamma{t}$’) ax2.set_ylabel(r’$|C_{e2}|^2$’) ax2.set_xlabel(’$\gamma{t}$’)

80 ax3.set_ylabel(r’$|C_{g1}|^2$’) ax3.set_xlabel(’$\gamma{t}$’) show()

81 Appendix E

Code for Six Level System

#3D plots for six level system from qutip import * from pylab import * from numpy import * import matplotlib.pyplot as plt

#surface plot for 6 level atom e1=basis(6,0) e2=basis(6,1) e3=basis(6,2) e4=basis(6,3) g1=basis(6,4) g2=basis(6,5)

#dipole transitions sigma_g1e1=tensor(g1*e1.dag()) sigma_g1e2=tensor(g1*e2.dag()) sigma_g1e3=tensor(g1*e3.dag()) sigma_g2e2=tensor(g2*e2.dag()) sigma_g2e3=tensor(g2*e3.dag()) sigma_g2e4=tensor(g2*e4.dag()) sigma_g1g1=tensor(g1*g1.dag()) sigma_g2g2=tensor(g2*g2.dag()) sigma_e1e1=tensor(e1*e1.dag()) sigma_e2e2=tensor(e2*e2.dag()) sigma_e3e3=tensor(e3*e3.dag())

82 sigma_e4e4=tensor(e4*e4.dag()) nsteps=100 zsteps=nsteps

#Rabi frequencies (right, left, linearly polarized) ratio=0.1 ratiostart=ratio ratiomax=5 omega1=0.5 #pump/coupling beam omega2=0 #probe beam ystart=omega2 omega0=0 #pi polarized probe omegamax=5 d=(omegamax-omega2)*(zsteps)**(-1) r=(ratiomax-ratio)*(zsteps)**(-1) uB=-4 uB_max=4 reset=uB u=(uB_max-uB)*(nsteps)**(-1)

#clebsch gordan coefficients g1e1=1 g1e2=sqrt(2.0/3.0) g1e3=sqrt(1.0/3.0) g2e2=sqrt(1.0/3.0) g2e3=sqrt(2.0/3.0) g2e4=1

#spontaneous emission gamma_e1g1=g1e1**2 gamma_e2g1=g1e2**2 gamma_e2g2=g2e2**2 gamma_e3g1=g1e3**2 gamma_e3g2=g2e3**2 gamma_e4g2=g2e4**2

#collapse operators

83 c1=sqrt(gamma_e1g1)*sigma_g1e1 c2=sqrt(gamma_e2g1)*sigma_g1e2 c3=sqrt(gamma_e3g1)*sigma_g1e3 c4=sqrt(gamma_e2g2)*sigma_g2e2 c5=sqrt(gamma_e3g2)*sigma_g2e3 c6=sqrt(gamma_e4g2)*sigma_g2e4 collapse=[c1,c2,c3,c4,c5,c6] xlist=[] list1=[] list2=[] list3=[] list4=[] list5=[] list6=[] zlist1=zeros(shape=(nsteps+1,nsteps+1)) zlist2=zeros(shape=(nsteps+1,nsteps+1)) zlist3=zeros(shape=(nsteps+1,nsteps+1)) zlist4=zeros(shape=(nsteps+1,nsteps+1)) zlist5=zeros(shape=(nsteps+1,nsteps+1)) zlist6=zeros(shape=(nsteps+1,nsteps+1)) for j in range (zsteps+1): for i in range (nsteps+1): #interaction hamiltonian H_i=g1e1*omega2*(sigma_g1e1+sigma_g1e1.dag())+ g2e2*omega2*(sigma_g2e2+sigma_g2e2.dag())+g1e2*omega0*(sigma_g1e2+ sigma_g1e2.dag())+g2e3*omega0*(sigma_g2e3+sigma_g2e3.dag())+ g1e3*omega1*(sigma_g1e3+sigma_g1e3.dag())+ g2e4*omega1*(sigma_g2e4+sigma_g2e4.dag()) #magnetic field-atom interaction hamiltonian H_b=(uB/2)*(-sigma_g1g1+sigma_g2g2+sigma_e3e3+ 3*sigma_e4e4-sigma_e2e2-3*sigma_e1e1) #total hamiltonian H=H_i+H_b #liouvillian L=liouvillian(H,collapse) rhoss=steady(L) p_e1=expect(sigma_g1e1.dag(),rhoss).imag p_e2=expect(sigma_g2e2.dag(),rhoss).imag p_e3=expect(sigma_g1e3.dag(),rhoss).imag

84 p_e4=expect(sigma_g2e4.dag(),rhoss).imag list1.append(p_e1) list2.append(p_e2) list3.append(p_e3) list4.append(p_e4) uB=uB+u zlist1[j]=list1 zlist2[j]=list2 zlist3[j]=list3 zlist4[j]=list4 uB=reset omega2=omega2+d list1=[] list2=[] list3=[] list4=[] print(j) x=linspace(uB,uB_max,nsteps+1) y=linspace(ystart,omegamax,nsteps+1) x,y=meshgrid(x,y) z_coupling=zlist3+zlist4 z_probe=zlist1+zlist2 fig= plt.figure() ax1 = fig.add_subplot(111, projection =’3d’) surf=ax1.plot_surface(x, y, zlist1, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax1.set_ylabel(r’$\Omega_{p}/\gamma$’, fontsize=27) ax1.set_zlabel(r’$Im(\langle\sigma_{g_1e_1}\rangle)$’, fontsize=27) ax1.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) fig= plt.figure() ax2 = fig.add_subplot(111, projection =’3d’) surf=ax2.plot_surface(x, y, zlist2, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax2.set_ylabel(r’$\Omega_{p}/\gamma$’, fontsize=27) ax2.set_zlabel(r’$Im(\langle\sigma_{g_2e_2}\rangle)$’, fontsize=27) ax2.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) fig= plt.figure() ax3 = fig.add_subplot(111, projection =’3d’)

85 surf=ax3.plot_surface(x, y, zlist3, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax3.set_ylabel(r’$\Omega_{p}/\gamma$’, fontsize=27) ax3.set_zlabel(r’$Im(\langle\sigma_{g_1e_3}\rangle)$’, fontsize=27) ax3.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) fig= plt.figure() ax4 = fig.add_subplot(111, projection =’3d’) surf=ax4.plot_surface(x, y, zlist4, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax4.set_ylabel(r’$\Omega_{p}/\gamma$’, fontsize=27) ax4.set_zlabel(r’$Im(\langle\sigma_{g_2e_4}\rangle)$’, fontsize=27) ax4.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) fig=plt.figure() ax8 = fig.add_subplot(111, projection =’3d’) surf=ax8.plot_surface(x, y, z_coupling, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax8.set_ylabel(r’$\Omega_{p}/\gamma$’, fontsize=27) ax8.set_zlabel(r’$Coupling Absorption$’, fontsize=27) ax8.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) fig= plt.figure() ax9 = fig.add_subplot(111, projection =’3d’) surf=ax9.plot_surface(x, y, z_probe, rstride=1, cstride=1, cmap=cm.jet, linewidth=0, antialiased=False) ax9.set_ylabel(r’$\Omega_{p}/\gamma$’, fontsize=27) ax9.set_zlabel(r’$Probe Absorption$’, fontsize=27) ax9.set_xlabel(’$\mu{B}/\gamma$’, fontsize=27) show()

86 #Trajectories for 6 level system from qutip import * from pylab import * from numpy import * import matplotlib.pyplot as plt

#quantum trajectories for six level atom ntraj=1 nsteps=1000 e1=basis(6,0) e2=basis(6,1) e3=basis(6,2) e4=basis(6,3) g1=basis(6,4) g2=basis(6,5) psi0=e1

#dipole transitions sigma_g1e1=tensor(g1*e1.dag()) sigma_g1e2=tensor(g1*e2.dag()) sigma_g1e3=tensor(g1*e3.dag()) sigma_g2e2=tensor(g2*e2.dag()) sigma_g2e3=tensor(g2*e3.dag()) sigma_g2e4=tensor(g2*e4.dag()) sigma_g1g1=tensor(g1*g1.dag()) sigma_g2g2=tensor(g2*g2.dag()) sigma_e1e1=tensor(e1*e1.dag()) sigma_e2e2=tensor(e2*e2.dag()) sigma_e3e3=tensor(e3*e3.dag()) sigma_e4e4=tensor(e4*e4.dag())

#Rabi frequencies (right, left, linearly polarized) ratio=3 omega2=0 #probe beam omega1=.5 #pump/coupling beam omega0=0 #pi probe uB=0

87 #clebsch gordan coefficients g1e1=1 g1e2=sqrt(2.0/3.0) g1e3=sqrt(1.0/3.0) g2e2=sqrt(1.0/3.0) g2e3=sqrt(2.0/3.0) g2e4=1

#spontaneous emission gamma_e1g1=g1e1**2 gamma_e2g1=g1e2**2 gamma_e2g2=g2e2**2 gamma_e3g1=g1e3**2 gamma_e3g2=g2e3**2 gamma_e4g2=g2e4**2

#collapse operators c1=sqrt(gamma_e1g1)*sigma_g1e1 c2=sqrt(gamma_e2g1)*sigma_g1e2 c3=sqrt(gamma_e3g1)*sigma_g1e3 c4=sqrt(gamma_e2g2)*sigma_g2e2 c5=sqrt(gamma_e3g2)*sigma_g2e3 c6=sqrt(gamma_e4g2)*sigma_g2e4 collapse=[c1,c2,c3,c4,c5,c6]

#operators num1=sigma_g1e1*sigma_g1e1.dag() num2=sigma_g1e2*sigma_g1e2.dag() num3=sigma_g1e3*sigma_g1e3.dag() num4=sigma_g2e2*sigma_g2e2.dag() num5=sigma_g2e3*sigma_g2e3.dag() num6=sigma_g2e4*sigma_g2e4.dag() numb=[num1,num2,num3,num4,num5,num6]

#expectation values e1=c1.dag()*c1 e2=c2.dag()*c2 e3=c3.dag()*c3 e4=c4.dag()*c4 e5=c5.dag()*c5 e6=c6.dag()*c6

88 g1=c1*c1.dag() g2=c6*c6.dag() expect_values=[e1,e2,e3,e4,e5,e6,g1,g2] tlist=linspace(0,20,nsteps)

#interaction hamiltonian H_i=g1e1*omega2*(sigma_g1e1+sigma_g1e1.dag())+ g2e2*omega2*(sigma_g2e2+sigma_g2e2.dag())+ g1e2*omega0*(sigma_g1e2+sigma_g1e2.dag())+ g2e3*omega0*(sigma_g2e3+sigma_g2e3.dag())+ g1e3*omega1*(sigma_g1e3+sigma_g1e3.dag())+ g2e4*omega1*(sigma_g2e4+sigma_g2e4.dag()) #magnetic field-atom interaction hamiltonian H_b=(uB/2.0)*(-sigma_g1g1+sigma_g2g2+ sigma_e3e3+3*sigma_e4e4-sigma_e2e2-3*sigma_e1e1) #total hamiltonian H=H_i+H_b #un trajecto data=mcsolve(H,psi0,tlist,collapse,expect_values,ntraj,num) p_e1=data.expect[0] p_e2=data.expect[1]+data.expect[3] p_e3=data.expect[2]+data.expect[4] p_e4=data.expect[5] p_g1=data.expect[6] p_g2=data.expect[7] times=data.col_times[0] which=data.col_which[0] columns=len(times) y1=[max(p_e1)]*columns y2=[max(p_e2)]*columns y3=[max(p_e3)]*columns y4=[max(p_e4)]*columns y5=[max(p_g1)]*columns y6=[max(p_g2)]*columns fig=plt.figure()

89 ax1=fig.add_subplot(611) ax1.plot(tlist,p_e1, linewidth=2) ylabel(r’$|C_{e1}|^2$’, size=27) setp(ax1.get_xticklabels(), visible=False) locator_params(nbins=4)

#there is probably a much easier way to do this for i in range (columns): if which[i] == 0: x=times[i] y=y1[i] ax1.scatter(x,y, s=100,marker=7,c=’b’) ax1.set_ylim([0, max(p_e1+.1)]) ax1.set_xlim([0, max(tlist)]) ax2=fig.add_subplot(612, sharex=ax1) ax2.plot(tlist,p_e2, linewidth=2) ylabel(r’$|C_{e2}|^2$’, size=27) setp(ax2.get_xticklabels(), visible=False) locator_params(nbins=4) for i in range (columns): if which[i] == 1: x=times[i] y=y2[i] ax2.scatter(x,y, s=100,marker=7,c=’g’) if which[i] == 3: x=times[i] y=y2[i] ax2.scatter(x,y, s=100,marker=7,c=’y’) ax2.set_ylim([0, max(p_e2+.1)]) ax2.set_xlim([0, max(tlist)]) ax3=fig.add_subplot(613,sharex=ax1) ax3.plot(tlist,p_e3, linewidth=2) ylabel(r’$|C_{e3}|^2$’, size=27) setp(ax3.get_xticklabels(), visible=False) locator_params(nbins=4) for i in range (columns): if which[i] == 2:

90 x=times[i] y=y3[i] ax3.scatter(x,y, s=100,marker=7,c=’r’) if which[i] == 4: x=times[i] y=y3[i] ax3.scatter(x,y, s=100,marker=7,c=’c’) ax3.set_ylim([0, max(p_e3+.1)]) ax3.set_xlim([0, max(tlist)]) ax4=fig.add_subplot(614,sharex=ax1) ax4.plot(tlist,p_e4, linewidth=2) ylabel(r’$|C_{e4}|^2$’, size=27) setp(ax4.get_xticklabels(), visible=False) locator_params(nbins=4) for i in range (columns): if which[i] == 5: x=times[i] y=y4[i] ax4.scatter(x,y, s=100,marker=7,c=’m’)

ax4.set_ylim([0, max(p_e4+.1)]) ax4.set_xlim([0, max(tlist)]) ax5=fig.add_subplot(615,sharex=ax1) ax5.plot(tlist,p_g1, linewidth=2) ylabel(r’$|C_{g1}|^2$’, size=27) setp(ax5.get_xticklabels(), visible=False) locator_params(nbins=4) ax5.set_xlim([0, max(tlist)]) ax6=fig.add_subplot(616, sharex=ax1) ax6.plot(tlist,p_g2, linewidth=2) xlabel(r’$\gamma{t}$’, size=27) ylabel(r’$|C_{g2}|^2$’, size=27) locator_params(nbins=4) ax6.set_xlim([0, max(tlist)]) show()

91 References

1. Scully, Marlan O., and Muhammad Suhail Zubairy. Quantum Optics. Cam- bridge: Cambridge UP, 1997. Print.

2. Jason Barkeloo. Investigation of Electromagnetically Induced Transparency and Absorption in Warm Rb Vapor by Application of a Magnetic Field and Co-propagating Single Linearly Polarized Light Beam. Master’s thesis, Miami University, 2012

3. Charles H. Baldwin. Cavity qed with center of mass tunneling. Master’s thesis, Miami University, 2011.

4. van Dorsselaer, F. E.; Nienhuis, G. PhD TUTORIAL: Quantum trajectories. Journal of Optics B: Quantum and Semiclassical Optics, Volume 2, Issue 4, pp. R25-R33 (2000).

5. Kurt Jacobs and Daniel A. Steck, A Straightforward Introduction to Continuous Quantum Measurement, Contemporary Physics 47, 279 (2006)

6. Carmichael, Howard. Statistical Methods in Quantum Optics: Master Equa- tions and Fokker-planck Equations. New York: Springer, 1999. Print.

7. Carmichael, Howard. Statistical Methods in Quantum Optics 2: Non-classical Fields. Berlin: Springer, 2008. Print.

8. Milonni, Peter W. Fast Light, Slow Light, and Left-handed Light. Bristol: Institute of Physics, 2005. Print.

9. Garrido Alzar, C. L., M. A. G. Martinez, and P. Nussenzveig. ”Classical Analog of Electromagnetically Induced Transparency.” American Journal of Physics 70.1 (2002): 37. Print.

10. Kwon, M.; Kim, K.; Park, H.D.; Kim, J.B.; Moon, H.S.. Journal of Physics B: Atomic, Molecular and Optical Physics, 14 August 2001, 34(15):2951-2961

11. Zigdon, T., A. D. Wilson-Gordon, and H. Friedmann. ”Absorption Spectra for Strong Pump and Probe in Atomic Beam of Cesium Atoms.” Physical Review A 80.3 (2009): n. pag. Print.

92 12. Beausoleil, R. G., W. J. Munro, D. A. Rodrigues, and T. P. Spiller. ”Appli- cations of Electromagnetically Induced Transparency to Quantum Information Processing.” Journal of Modern Optics 51.16-18 (2004): 2441-448. Print.

13. Andr, A., M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin. ”Quantum Control of Light Using Electromagnetically Induced Transparency.” Journal of Physics B: Atomic, Molecular and Optical Physics 38.9 (2005): S589- 604. Print.

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