Electromagnetically Induced Transparency: The Zeeman Method. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College

Transcription

1 Electromagnetically Induced Transparency: The Zeeman Method A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of Arts W. Ace Furman May 016

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3 Approved for the Division Physics) Lucas Illing

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5 Acknowledgements Before I begin, I would like to mention a handful of people without whom this thesis and my career at Reed could never have been completed: Lucas Illing, whose guidance and seemingly bottomless knowledge got me through even the most difficult of thesis crises; Jay Ewing, for the technical knowhow, for the avid reassurance, and for a sleek apparatus; John Essick, Joel Franklin, Darrell Schroeter, and David Griffiths, who were always available to mull over confusing topics and usually solve all of my problems; Bob Ormond, firstly, for all the anecdotes and, secondly, for the electronics; Johnny Powell and Noah Muldavin, for reminding me to hang loose; Gary and Frankie Furman, who ensured my sanity during the thesis process and also throughout the course of my life; Zubenelgenubi Scott and Indy Liu, for being the best officemates in the whole subbasement Dominion anyone?); Alex Deich and Naomi Gendler, for teaching me what friends are for; Colleen Werkheiser, without whom I would not have known where to even start; Seth Gross, for making Mondays the best day of the week, which, by the way, is no trivial matter; Taylor Holdaway and Kai Addae, for helping me procrastinate just the right amount #Slurpees; Spencer Fussy and Carly Goldblatt, for accepting me the way I am; Jack Taylor, for undisclosed reasons ; Munyo Frey, whose continuous encouragement and loving outlook pulled me through the final stretch; and, Emilia and Inés Furman and Arrow and Vega Henson, who remind me everyday that the universe is a beautiful place. Thank you to all.

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7 Table of Contents Introduction Chapter 1: Theory Derivation of the EIT Hamiltonian Bare-Atom Hamiltonian Applied Fields Rotating Wave Approximation Time and Phase Independent Frame Dressed State Picture Dressed States The Dark State and The Bright State Higher-Level Systems Absorption in EIT Density Operator Von Neumann Equation Von Neumann Equation in EIT Complex Susceptibility And Its Implications Chapter : Experiment Rubidium and Zeeman Splitting Preliminary Setup Diode Laser Saturated Absorption Spectroscopy Primary EIT Setup Components of Polarization Zeeman Apparatus Procedural Overview Chapter 3: Results Data Analysis Possible Improvements F = 1 F = 0 Transition of the D Line D 1 Line Laser Locking Higher Resolution Detection and Real-Time Scanning

8 Conclusion References

9 List of Figures 0.1 A typical absorption spectrum, and an EIT absorption spectrum Λ-type structure All three level structures capable of demonstrating EIT Λ-type level structure with applied field energies ω p and ω c ) and their energy detunings from resonance p and c ) Energy diagram of a six-level system Linear susceptibility plotted in arbitrary units as a function of normalized probe frequency The F = 1 F = 1 transitions Schematic of saturated absorption setup Hyperfine spectra of 87 Rb Schematic of the primary EIT setup Section and 3D rendering of the Zeeman apparatus Magnetic field attenuated by mu metal shielding inside the Zeeman apparatus Magnetic field driven by the solenoid inside the Zeeman apparatus Experimental EIT absorption of 87 Rb Simulation of absorption and dispersion as the decoherence parameter γ varies

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11 Abstract Electromagnetically induced transparency EIT) a quantum phenomenon wherein a material undergoes optical and dispersive modifications under the influence of resonant electromagnetic fields was observed in warm rubidium vapor using the Zeeman method. Specifically, the absorption of a probe laser through the atomic medium in a magnetic field was measured, and a transparency window was observed around a typically absorbent atomic resonance, consistent with EIT.

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13 Introduction Electromagnetically induced transparency EIT) is a quantum phenomenon that makes use of resonant electromagnetic fields to alter the optical properties of an atomic medium. Under typical conditions that is, in the presence of a single nearresonant field atoms will absorb energy from the surrounding field and excite into higher energy states. The absorption spectrum, as a function of field frequency, follows a Lorentzian curve peaked around the natural frequency of the resonant atomic transition, as shown in Fig. 0.1a. In EIT, two fields are introduced, each resonant with a distinct atomic transition. This alters the absorption spectrum to include a window of frequencies, for which the medium exhibits high transparency, as shown in Fig. 0.1b. Hence, we get the name electromagnetically induced transparency. In addition, the dispersion properties are also affected, giving rise to many useful applications such as effective nonlinear frequency conversion, slow light and nonlinear mixing. EIT was first theorized in 1990 by Harris et al. [1] as a variation on coherent population trapping. The following year, EIT was observed experimentally by Boller et al. [] in strontium vapor and by Field et al. [3] in lead vapor. Since then EIT has been an extremely prevalent topic of research in physics. In 1996, Jain et al. [4] showed its applications in nonlinear mixing, wherein light beams of a certain wavelength are converted to another wavelength with high efficiency. In 1998, Ling et al. [5] created an electromagnetically induced diffraction grating from a gaseous atomic medium. In 1999, Budker et al. [6] used EIT to control the group velocity of light traveling through rubidium vapor. They slowed group velocities of light to 8 m/s. Since two fields must be present to see EIT, it is one of the few examples of optical setups in which light signals interact with other light signals. For example, say one field were to be turned off; the medium would become absorbent again and the remaining field would be obscured. This gives rise to even more applications such as optical switching [7]. The quantum mechanical mechanisms underlying EIT are quite complicated and not easily intuited. One simple way to think of EIT is as a result of destructive interference of transition probabilities. Given the three-level structure in Fig. 0., there exist two paths between state 1 and state 3 excluding longer paths): 1 3 and If the probabilities of each of these transitions occurring in an atom have opposing phases, destructive interference could occur and result in no transitions from 1 to 3 [8]. This would, in turn, lead to a reduction in the absorption of the field resonant with that transition. More rigorous derivations of

14 Introduction a b Figure 0.1: a) A typical absorption spectrum, and b) an EIT absorption spectrum. In both cases, is the laser detuning from atomic resonance. The subscript p in b) signifies that the probe laser is being scanned. Figure 0.: Λ-type structure. three-level systems are discussed in Chapter 1, as well as a novel derivation describing the six-level system used in this experiment. This experiment explores specifically Zeeman EIT, where the transparency window is seen not as a function of the incoming light but as a function of the strength of a magnetic field applied across the atomic medium. This is a result of the Zeeman effect, which allows us to split degenerate energy levels in the atoms, effectively changing the resonant frequencies of the transitions. In essence, instead of changing the frequency of light to match the atomic resonance as in standard EIT we will change the level splittings such that the resonant frequency matches that of the incoming light. Specifics about how Zeeman EIT is observed in this experiment are discussed in Chapter. The results and analysis of the experiment are shown in Chapter 3.

15 Chapter 1 Theory EIT is a quantum phenomenon that cannot be predicted classically. However, a fully quantum mechanical derivation is not strictly necessary. This chapter will make a semiclassical derivation of the Hamiltonian for EIT, which treats atoms as quantum mechanical objects that interact with classical vector fields. This derivation follows closely the works of Weatherall [9] and Erickson [10]. We will then interpret the Hamiltonian in two ways. First, we will take a look at the dressed state basis of the EIT Hamiltonian to see how a dark state is induced by the applied fields. Then, we will use the Hamiltonian to explicitly calculate the absorption and dispersion of such a system. The semiclassical approach is sufficient to see both the mechanisms behind EIT and its effects. The main tradeoff of this derivation is that, while it is much simpler than the fully quantum mechanical derivation, it does not account for spontaneous emission of the atoms. Instead, we will add necessary phenomenological decay terms by hand in Section Derivation of the EIT Hamiltonian EIT is only seen in atoms with specific energetic properties. Three-level energy configuration must match one of the following: Λ-type, V-type, or ladder cascade) type, all shown in Fig The transitions in rubidium that will be utilized in this experiment exhibit a combination of Λ-type and V-type energy level structures. Since Λ-atoms have a very low decay rate from the state, the effect of EIT is much stronger than in the other configurations [10]. For this reason, we will focus on the theory as it applies to Λ-structure; however, the following derivations generalize to any of these energy structures, and further, to Zeeman EIT. Let 1,, and 3 be the names given to the ground state, the metastable state, and the excited state, respectively, of the atom without any fields applied; their energies will be defined in terms of their corresponding frequencies as E n = ω n, 1.1) for n = 1,, 3. While 1 and can both excite into 3, we assume that the transition between 1 and is dipole forbidden. Therefore, the resonant frequencies between

16 4 Chapter 1. Theory Figure 1.1: All three level structures capable of demonstrating EIT, namely, Λ-type left), V-type middle), and ladder type right). relevant energy levels are ω 13 = ω 3 ω 1, ω 3 = ω 3 ω. 1.a) 1.b) Bare-Atom Hamiltonian Before any external field is applied, the energy eigenvalues and their eigenstates are denoted ω n and n, respectively. We can enforce completeness and orthogonality such that n n = 1, 1.3) and n n m = δ nm. 1.4) The bare-atom Hamiltonian H 0 can therefore be expressed in this basis as a matrix with elements ω H 0 = 0 ω ) 0 0 ω 3 Without any external stimulation, this system will tend to settle in the lowest energy 1 state due to spontaneous emission from higher energy states Applied Fields Now consider that two fields are applied so that electrons populate the upper states. We can assume that, other than the three states in the Λ-configuration, all other states are relatively unoccupied and can be neglected. The control field is tuned to the resonant frequency ω c ω 3 with amplitude E c, and the probe field is tuned to the other resonance ω p ω 13 with amplitude E p. A energy diagram of this scheme is shown in Fig. 1.. This yields an electric field E = E p cos ω p t k p r) + E c cos ω c t k c r), 1.6)

17 1.1. Derivation of the EIT Hamiltonian 5 ħ p ħ c ħω p ħω 13 ħω 1 ħω c Figure 1.: Λ-type level structure with applied field energies ω p and ω c ) and their energy detunings from resonance p and c ). where the magnitude of the wavevector k = π for the wavelength λ of the fields. λ Note that for wavelengths much larger than the diameter of an atom, we have λ r; thus, k r is small, so it can be dropped, leaving E = E p cos ω p t) + E c cos ω c t). 1.7) The fields will perturb the Hamiltonian from that of the bare atom such that H = H 0 + H 1, 1.8) where H 1 is the interaction term between the fields and the atom and can be written as H 1 = qe d. 1.9) This assumes that the atom behaves similarly to an electric dipole with charge q and separation vector d. This is a particularly good approximation for hydrogenic atoms like rubidium with only one electron in their outer shell. The dipole will quickly align with the fields, so we can instead write H 1 = qed. 1.10) Now define the dipole moment operator qd and its elements mn n m. With these definitions, the interaction Hamiltonian becomes H 1 = E. 1.11) Note that many of the matrix elements of this Hamiltonian are zero. For example, the dipole elements 1 and 1 must be zero for the 1 transition to be dipole forbidden. Additionally, the diagonal terms must go to zero because of the spherical symmetry of the wavefunction [9]. Therefore, the matrix representation of the Hamiltonian simplifies further to H 1 = E )

18 6 Chapter 1. Theory Rotating Wave Approximation As it is, the interaction term of the Hamiltonian looks simple, but thus far, the electric field has not played its part. In order to clearly see its effect, it is useful to transform into the interaction picture of the unperturbed system. To do that, we use a unitary matrix U defined as the time evolution operator, such that Ut) = e ih0t/ = e iω 1t e iω t e iω 3t. 1.13) Applying this transformation to H 1 yields e iω 13t UH 1 U = E e iω 3t. 1.14) 31 e iω 13t 3 e iω 3t 0 As seen from Eq. 1.7, the electric field is a sum of cosines. These can be written instead as exponentials: E = E p e iω pt + e iωpt) + E c e iω ct + e iωct). 1.15) Plugging this into the transformed Hamiltonian, the nonzero elements become UH 1 U ) 13 = 13 Ep UH 1 U ) 3 = 3 Ep UH 1 U ) 31 = 31 Ep UH 1 U ) 3 = 3 Ep e iω pt + e iωpt) + E c e iω ct + e iωct)) e iω13t, 1.16a) e iω pt + e iωpt) + E c e iω ct + e iωct)) e iω3t, 1.16b) e iω pt + e iωpt) + E c e iω ct + e iωct)) e iω13t, 1.16c) e iω pt + e iωpt) + E c e iω ct + e iωct)) e iω3t. 1.16d) In the rotating wave approximation, it is assumed that the rapidly oscillating terms will average out quickly and therefore can be ignored. Thus, exponential terms with large imaginary arguments drop out. Since ω c ω 3 and ω p ω 13, there will be one exponential term in each matrix element that oscillates slowly enough to survive this approximation. Namely, UH 1 U ) 13 = 1 E p 13 e iωp ω13)t, UH 1 U ) 3 = 1 E c 3 e iωc ω3)t, UH 1 U ) 31 = 1 E p 31 e iω 13 ω p)t, UH 1 U ) 3 = 1 E c 3 e iω 3 ω c)t. 1.17a) 1.17b) 1.17c) 1.17d)

19 1.1. Derivation of the EIT Hamiltonian 7 Transforming back to the Schrödinger picture, the interaction Hamiltonian becomes H 1 = U UH 1 U ) U = E p 13 e iωpt 0 0 E c 3 e iωct E p 31 e iωpt E c 3 e iωct ) The electric dipole operators can be represented in terms of their magnitudes and phases such that 13 = 31 = 13 e iφp, 3 = 3 = 3 e iφc. 1.19a) 1.19b) The Rabi frequencies of this system are defined to be Ω p = E p 13, 1.0a) Ω c = E c b) Plugging these definitions into the interaction Hamiltonian and summing it with the bare-atom Hamiltonian, the full Hamiltonian in the presence of applied fields becomes H = ω 1 0 Ω p e iωpt+φp) 0 ω Ω c e iωct+φc) Ω p e iωpt+φp) Ω c e iωct+φc) ω ) Time and Phase Independent Frame The Hamiltonian in Eq. 1.1 still depends on phases and time, which muddies the interpretation. To clearly see how EIT will come from such a Hamiltonian, we must transform into a new basis the corotating basis. The unitary matrix that will achieve this transformation is defined as Ũt) = e iωpt+φp) e iωct+φc) ) Note that the full Hamiltonian has a different set of eigenstates than the bare atom. Define n to be the eigenstates of H. There are also analogous eigenstates ñ = Ũ n of the corotating Hamiltonian H. For the corotating eigenstates to satisfy Schrödinger s

20 8 Chapter 1. Theory equation, we must have H ñ = i t ñ = i ) Ũ n t ) Ũ = i t n + Ũ n t ) Ũ = i t n + 1 i ŨH n ) = i Ũ t Ũ + ŨHŨ Ũ n H ñ = i Ũ t Ũ + ŨHŨ ) ñ. 1.3) Therefore, the transformation of the Hamiltonian into the corotating frame is H = i Ũ t Ũ + ŨHŨ 1.4) = ω p ω c 0 + ω 1 0 Ω p 0 ω Ω c Ω p Ω c ω 3 H = ω 1 + ω p ) 0 Ω p 0 ω + ω c ) Ω c. 1.5) Ω p Ω c ω 3 By noting that the physical interpretation of a Hamiltonian is unaltered by the addition of a scalar multiple of the identity, the Hamiltonian can be coerced into an even more interpretable form. That is, adding ω 1 + ω p )1 yields H = 0 0 Ω p 0 ω + ω c ω 1 ω p ) Ω c Ω p Ω c ω 3 ω 1 ω p ). 1.6) Finally, defining the laser detunings from resonance as p ω 13 ω p = ω 3 ω 1 ω p and c ω 3 ω c = ω 3 ω ω c, the EIT Hamiltonian takes on its standard form: H = 0 0 Ω p 0 p c ) Ω c. 1.7) Ω p Ω c p

21 1.. Dressed State Picture 9 1. Dressed State Picture The following dressed state analysis draws heavily on the work of Fleischhauer et al. [11], Purves [1], and Marangos [13] Dressed States In order to analyze the eigensystem of the EIT Hamiltonian, we will approximate that p c. This is reasonable because these detunings are small compared to other relevant frequencies in the experiment. Solving the characteristic equation yields the following eigenvalues: det H λ1 = 0, 1.8) λ 0 = 0, 1.9a) λ ω = ) + Ω p + Ω c, 1.9b) λ + ω + = ) + + Ω p + Ω c. 1.9c) The corresponding eigenstates of the dressed system, expressed as linear combinations of the bare-atom states, are 0 = Ω c 1 Ω p, Ω p + Ω c 1.30a) Ω p 1 + Ω c = + 3, + Ω p + Ω c 1.30b) + = Ω p 1 + Ω c + + Ω p + Ω c c) Note that an atom in state 0 cannot excite into state 3 because 0 has no component of the 3 state. For this reason, we call 0 the dark state. When the fields are close to resonance that is, 0 the upper dressed states can be normalized and approximated by ) = 1 Ω p 1 + Ω c + 3, 1.31a) Ω p + Ω c ) + = 1 Ω p 1 + Ω c b) Ω p + Ω c

22 10 Chapter 1. Theory 1.. The Dark State and The Bright State If we assume that the amplitude of the probe beam is small compared to that of the control, then Ω p Ω c. Using this approximation and normalizing, the dark state 0 becomes ) Under these conditions, the dark state is approximately the same as the ground state of the bare atom. Therefore, the dark state is not only decoupled from the other two, but additionally, it is a stationary state of both the bare-atom Hamiltonian and the dressed system. This means that, due to their time-independence, atoms in the ground state will remain there and never excite. Thus, the probe beam will remain unabsorbed. It would seem that atoms that start out in some admixture of the upper states would remain excited. However, this semiclassical treatment of a quantum phenomenon neglects the possibility of spontaneous emission from the upper states into the ground state. In actuality, these states will decay into the ground state and then be trapped there. Spontaneous emission will be discussed further in Section As one final note about dressed states, we wish to gain some intuition into the nature of the non-dark states + and. Using Eqs. 1.31a and 1.31b, we may define two linear combinations of the states a + = 3, 1.33a) b + + = Ω p 1 + Ω c. 1.33b) Ω p + Ω c a and b are not stationary states. In fact, the strong control field acts to oscillate atoms back and forth between these two states. For this reason we shall call b the bright state; it accounts for all transitions to and from the excited state a. Using the same weak probe field approximation as before, we can write b. 1.34) Therefore, in the presence of both electric fields, atoms in state are still able to absorb energy from the control field and excite into state 3. However, 1 decouples from the excited state completely, and we see no absorption of the probe beam Higher-Level Systems The majority of the theory included in this discussion of EIT applies to three-level systems specifically Λ-type energy structures. However, in practice, few systems consist of just three energy levels. In fact, the atomic levels relevant to this experiment consist of six levels, as shown in Fig Most direct methods of calculating absorption like the methods discussed in the following section would be much more difficult for systems with more than three levels. The dressed state analysis, however, is more or less the same. For this

23 1.. Dressed State Picture Figure 1.3: Energy diagram of a six-level system. Solid arrows indicate strong control fields while dashed arrows represent weak probe fields. reason, it is often more convenient to verify that a dark state exists for a given system before going through any computation-heavy analysis. Consider the six-level system described in Fig In this scenario, there are two control beams, both with energy ω c ω 4 ω 35 shown as solid arrows and two probe beams, both with energy ω p ω 15 ω 6 shown as dashed arrows. We again start with a bare-atom Hamiltonian that looks like H 0 = ω ω ω ω ω ω ) After taking the rotating wave approximation, we can express the full Hamiltonian analogous to Eq. 1.1 as H = ω Ω ce iωct+φc) 0 0 ω 0 Ω pe iωpt+φp) 0 Ω ce iωct+φc) 0 0 ω 3 0 Ω pe iωpt+φp) 0 0 Ω pe iωpt+φp) 0 ω Ω ce iωct+φc) 0 Ω pe iωpt+φp) 0 ω Ω ce iωct+φc) ω ) Now, we wish to transform the Hamiltonian into a frame that is time and phase.

24 1 Chapter 1. Theory independent using the unitary matrix Ũt) = e iωpt+φp) e iωct+φc) e iωct+φc) e iωpt+φp) Plugging into Eq. 1.4 yields the corotating Hamiltonian H = ω 1 + ω p ) Ω c 0 0 ω 0 Ω p 0 Ω c 0 0 ω 3 + ω c ) 0 Ω p 0 0 Ω p 0 ω 4 ω c ) 0 0 Ω c 0 Ω p 0 ω Ω c ω 6 ω p ). 1.37). 1.38) We can now assume the relative energy spacings of each cluster of three levels are the same. This implies that we can express all six frequencies in terms of ω and the relevant spacings such that ω 1 = ω δ, ω 4 = ω + ω 0 δ, 1.39a) ω = ω, ω 5 = ω + ω 0, 1.39b) ω 3 = ω + δ, ω 6 = ω + ω 0 + δ, 1.39c) where δ is the spacing between adjacent states within a single cluster, and ω 0 is the energy gap between the two clusters. This yields two resonant frequencies ω 15 = ω 6 = ω 0 + δ, ω 4 = ω 35 = ω 0 δ 1.40a) 1.40b) Since adding a scalar multiple of the identity preserves the spectrum of the matrix, we may add ω 1 + ω p ) 1 and express the frequencies in terms of ω 0 and δ. This yields H = In terms of the detunings Ω c 0 0 δ ω p) 0 Ω p 0 Ω c 0 0 δ+ω c ω p) 0 Ω p 0 0 Ω p 0 ω 0 ω c ω p) 0 0 Ω c 0 Ω p 0 ω 0 +δ ω p) 0 0 Ω c ω 0 +δ ω p). 1.41) p = ω 15 ω p = ω 0 + δ ω p, c = ω 4 ω c = ω 0 δ ω c, 1.4a) 1.4b)

25 1.. Dressed State Picture 13 the Hamiltonian becomes H = Ω c 0 0 p ω 0 ) 0 Ω p 0 Ω c 0 0 p+ c) 0 Ω p 0 0 Ω p 0 p+ c ω 0 ) 0 0 Ω c 0 Ω p 0 p 0 0 Ω c p ω 0 ). 1.43) In the case when the control and probe laser are both on resonance, the diagonal terms simplify, and the Hamiltonian of interest becomes Ω c 0 H = 0 ω 0 0 Ω p 0 Ω c Ω p 0 0 Ω p 0 ω 0 0 0, 1.44) Ω c 0 Ω p Ω c ω 0 with the following eigenstates: 0 Λ = Ω c 1 Ω p 3, Ω p + Ω c 1.45a) 0 V = Ω p 4 Ω c 6, Ω p + Ω c 1.45b) a = b = c = d = Ω p + Ω c + Ω c 4 + Ω p 6, 1.45c) Ω p + Ω c Ω p + Ω c Ω c 4 Ω p 6, 1.45d) Ω p + Ω c Ω p + Ω c 1 + Ω c 3 + Ω p 5, 1.45e) Ω p + Ω c Ω p + Ω c 1 Ω c 3 Ω p f) Ω p + Ω c As expected, we see that one of the eigenstates the one that is conveniently named 0 Λ looks almost exactly like the dark state in the three-level system. As in the Λ-type scheme, when the probe beam amplitude is small, i.e. Ω p Ω c, we find 0 Λ ) This shows that the states 1, 3, and 5 make up a mini-λ scheme inside the greater six-level system and behave similarly to the normal Λ-type structure. Additionally, there is another state of a similar form. That is, under the same approximation, 0 V ) This indicates that there are in fact two dark states of the system: one of the Λ scheme and the other of the V scheme made up of states, 4, and 6. It is important to

26 14 Chapter 1. Theory note that, due to spontaneous emission from the upper states into the lower states, atoms will transition between Λ scheme states and V scheme states; however, they will tend to populate the Λ scheme states because there is a greater probability that an atom will decay into 1 or 3 than into. Therefore, the EIT effects due to the Λ scheme will be much stronger than those of the V scheme. Assuming that both the Λ and V schemes become resonant at the same set of frequencies an appropriate assumption when using the Zeeman method to show EIT, as in this experiment both associated transparency windows will occur concurrently, and appear as one trough in the absorption spectrum. 1.3 Absorption in EIT The dressed state approach is helpful in showing the mechanism behind EIT, but it is not as useful in making physical predictions about the medium. Since this experiment deals not with a single atom, but with a whole ensemble of atoms, the next calculation will relate ideas of the previous sections to measurable macroscopic quantities. In particular, we will calculate the electric susceptibility χ of the medium in the presence of fields. The susceptibility is of interest because its real and imaginary parts determine the two primary effects of EIT: its real part determines the dispersion of the medium while its imaginary part governs the absorption. From this calculation, both of these properties will be predicted as functions of probe frequency. The linearized susceptibility is defined by the relation P = ɛ 0 χe, 1.48) where P is the dielectric polarization and ɛ 0 is the permittivity of free space. The polarization can also be defined as P = N, 1.49) where N is the total number of atoms. Therefore, Eqs and 1.49 can be equated, yielding ɛ 0 χe = N. 1.50) Almost all of the quantities above are macroscopic and measurable, so in order to calculate the susceptibility, we only need to find the expectation value of the electric dipole operator Density Operator For pure quantum states, an expectation value is defined in terms of the wavefunction Ψ, such that for some operator Λ, Λ = Ψ Λ Ψ. 1.51) However, it is often difficult to find the wavefunction of one atom, much less of the entire system of atoms. Additionally, an atom may exist in a mixed state. While a

27 1.3. Absorption in EIT 15 pure state may be determined by a superposition of stationary states, a mixed state is a statistical mixture of pure quantum states that do not interfere within a system. If for example, the states of the individual atoms in the system are expressed as n and follow a distribution P n, then the expectation value becomes Λ = n P n n Λ n. 1.5) For this reason, it is simpler to define expectation values in terms of the density operator ρ = n P n n n, 1.53) where P n is the probability of an atom being in state n. To see how this works, recall the completeness and orthogonality relations in Eqs. 1.3 and 1.4, and compute, Λ = n = mn P n n Λ n P n n Λ m m n = mn m P n n n Λ m = m m ρλ m Λ = TrρΛ). 1.54) This definition of an expectation value makes no reference to the specific state of the atoms. Instead, it makes uses of statistical distributions within the system as defined in the density operator Von Neumann Equation Since Schrödinger s equation is defined in terms of the wavefunction, a different relation exists in terms of the density operator; this relation is called the von Neumann equation. In order to derive the von Neumann equation, recall that Schrödinger s equation and its adjoint are expressed by d dt ψ = i H ψ, d dt ψ = i ψ H. 1.55)

28 16 Chapter 1. Theory Computing the time derivative of the density operator, we find ρ = n P n ṅ n + n ṅ ) = i P n H n n n n H) n = i Hρ ρh) = i [H, ρ]. 1.56) Again, our model has omitted the possibility of spontaneous emission from states and 3. To count for this, we include an additional term ρ = i [H, ρ] 1 {Γ, ρ}, 1.57) where the straight brackets represent a commutator and the curly brackets represent an anti-commutator. Here, Γ is defined in terms of the decay rate γ n from a state n as Γ = 0 γ 0, 1.58) 0 0 γ 3 and γ 1 = 0 because the atom cannot decay out of the ground state. The additional decay term only accounts for atoms leaving states and 3. It does not specify where they go. In reality, the majority of these atoms may decay into state 1, but since there are already a large number of atoms in the ground state, the addition of the decayed atoms can be neglected. Instead, we treat γ and γ 3 as decay rates of atoms leaving the three-level system completely. In coordinate form, von Neumann s equation becomes ρ ij = [ i H ikρ kj ρ ik H kj ) + 1 ] Γ ikρ kj + ρ ik Γ kj ). 1.59) k Von Neumann Equation in EIT Using von Neumann s equation, written in component form, and the three-level EIT Hamiltonian H, from Eq. 1.7 expressed in the corotating basis the density operator ρ can be calculated in the same basis. The diagonal matrix elements of ρ can be written in terms of ρ as ρ 11 = iω p ρ 31 ρ 13 ), 1.60a) ρ = γ ρ + iω c ρ 3 ρ 3 ), 1.60b) ρ 33 = γ 3 ρ 33 iω p ρ 31 ρ 13 ) iω p ρ 3 ρ 3 ). 1.60c)

29 1.3. Absorption in EIT 17 Additionally, the off-diagonal terms are ρ 1 = ρ 1 = γ ) + i p c ) ρ 1 iω c ρ 13 + iω p ρ 3, 1.61a) ρ 13 = ρ 31 = γ ) 3 + i p ρ 13 + iω p ρ 33 ρ 11 ) iω c ρ 1, 1.61b) ρ 3 = ρ 3 = 1 ) γ + γ 3 ) + i c ρ 3 + iω c ρ 33 ρ ) iω p ρ c) We will now assume that the majority of the atoms occupy the ground state. This is a good approximation both because atoms are constantly being pumped by the control beam from into 1, and also because spontaneously emitted atoms continuously repopulate the ground state. Therefore, since the diagonal terms ρ nn can be thought of as the fraction of atoms in the population of n, we see that ρ 11 1 and ρ ρ This yields ρ 1 = ρ 1 = γ ) + i p c ) ρ 13 = ρ 31 = γ ) 3 + i p ρ 3 = ρ 3 = ρ 1 iω c ρ 13 + iω p ρ 3, 1.6a) ρ 13 iω p iω c ρ 1, 1.6b) γ ) + γ 3 + i c ρ 3 iω p ρ c) Now, the weak probe approximation can be used to drop all terms proportional to Ω p. In the steady state solutions for Eq. 1.6c where ρ 3 = 0), ρ 3 is linearly proportional to Ω p, so then the iωp ρ 3 term in Eq. 1.6a can be dropped. Thus, we have ρ 1 = ρ 1 = γ ) + i p c ) ρ 13 = ρ 31 = γ ) 3 + i p ρ 1 iω c ρ 13, 1.63a) ρ 13 iω p iω c ρ b) These are now in the form of two coupled differential equations. To solve, they can be expressed as a single matrix equation, such that ) ρ 1 γ = + i ) ) ) p c ) iωc ρ1 0 ρ 13 iωc γ 3 + i ) p ρ 13 iωp Now define X = M = A = ρ1 ρ 13 ), 1.65a) ), 1.65b) γ + i p c ) iωc 0 iωp iωc γ 3 + i p ), 1.65c) 1 Obviously states and 3 are not empty, but their populations are so much smaller than that of 1 that they can be neglected.

30 18 Chapter 1. Theory and Eq becomes simply, Ẋ = MX + A. 1.66) The steady state solution to this matrix equation assuming Ẋ = 0 is X = M 1 A. 1.67) This yields solutions for the matrix elements of the density operator, ρ 1 = ρ 13 = Ω c Ω p, iγ 3 c p ) + γ γ 3 i p ) + 4 c p 4 p + Ω c 1.68a) i γ + i c p )) Ω p. iγ 3 c p ) + γ γ 3 i p ) + 4 c p 4 p + Ω c 1.68b) These matrix elements are, however, still in the corotating basis. In order to transform back to the Schrödinger picture, we would have to compute ρ = Ũ ρũ ρ 11 ρ 1 e iωp+φp ωc φc)t ρ 13 e iωp+φp)t = ρ 1 e iωp+φp ωc φc)t ρ ρ 3 e iωc+φc)t 1.69) ρ 31 e iωp+φp)t ρ 3 e iωc+φc)t ρ Complex Susceptibility And Its Implications We now have a matrix expression for the density operator in the corotating frame ρ. To ease the amount of computation, we will first show that ρ 13 is the only matrix element of the density operator in the Schrödinger picture needed to calculate the electric susceptibility χ. Recall that polarization can be expressed as P = N = NTr ρ ) ρ 11 ρ 1 ρ 13 = NTr ρ 1 ρ ρ 3 ρ 31 ρ 3 ρ 33 = NTr ρ ρ 13 3 ρ ρ 1 3 ρ 3 31 ρ 3 3 ρ ρ 3 ρ ρ 33 3 ρ ρ 3 3 = N ρ ρ ρ ρ 3 3 ). 1.70) Using Eq. 1.69, the polarization can be expressed in terms of the rotated density elements as P = N ρ e iωp+φp)t + ρ 3 3 e iωc+φc)t + ρ e iωp+φp)t + ρ 3 3 e iωc+φc)t). 1.71)

31 1.3. Absorption in EIT 19 The polarization can also be expressed in terms of the electric field as defined in Eq such that P = ɛ 0 χe = ɛ 0χω p )E p e iω pt + e iωpt) + ɛ 0χω c )E c e iω ct + e iωct). 1.7) Notice that since χ depends on frequency, there appear two different susceptibilities in the equation above. We are particularly interested in χω p ) χ p because the desired effect is transmission of the probe beam. To isolate this parameter, we match exponentials in Eqs and 1.7 specifically, those that contain a factor of e iωpt. This yields ɛ 0 χ p E p = N ρ e iφpt. 1.73) Recalling that 31 = 13 e iφpt and solving for χ p gives χ p = N 13 ɛ 0 E p ρ ) Finally, using Eq. 1.68b we can write a full expression for the complex electric susceptibility as χ p = N 13 ɛ 0 E p Imχ p ) = N 13 ɛ 0 E p i γ + i c p )) Ω p. 1.75) iγ 3 c p ) + γ γ 3 i p ) + 4 c p 4 p + Ω c The real and imaginary parts can be written Reχ p ) = N 13 γ p + c p ) 4 c p ) p + Ω )) c Ωp ɛ 0 E p γ + 4 c p ) ), γ p) + γ γ c p ) c ) Ω c + Ω 4 c 1.76a) ) γ γ 3 + 4γ 3 c p ) + γ Ω c Ω p γ + 4 c p ) ) γ p) + γ γ c p ) c ) Ω c + Ω 4 c. 1.76b) We recall that the real part of the linear susceptibility is proportional to the dispersion of the medium; likewise, the imaginary part is proportional to the absorption. Referring to the plot of Eq in Fig. 1.4b, we see that the absorption falls off drastically when the probe field is on resonance. This window of transparency is precisely what is meant by EIT. With only one or the other applied field, we would measure a typical absorption spectrum, but with both, we see a cancelation, such that the probe beam remains unabsorbed. Additionally, the dispersion becomes much steeper near resonance. Though this experiment will not dwell much on the dispersion, it is worth noting that such a dispersion relation in a zone of low absorption is conducive of complex optical phenomena such as slow light [6].

32 0 Chapter 1. Theory a b Figure 1.4: Linear susceptibility plotted in arbitrary units as a function of normalized probe frequency. The solid line shows the imaginary part of χ p which is proportional to the absorption), and the dashed line shows the real part of χ p which is proportional to the dispersion). Probe frequency is normalized with respect to γ 3, which allows us to plot in units of γ 3. For a) we set c = 0, Ω c = 0, and γ = 10 4 γ 3. For b) we set c = 0, Ω c = γ 3, and γ = 10 4 γ 3.

33 1.3. Absorption in EIT 1 As one last check, if we set Ω c = 0 corresponding to the case when the control beam is switched off we get Reχ p ) = N 13 Ω p p, ɛ 0 E p γ p 1.77a) Imχ p ) = N 13 Ω p γ 3. ɛ 0 E p γ p 1.77b) As shown in the plots of Eq in Fig. 1.4a, the absorption and dispersion revert back to typical trends as when EIT is not present, as expected.

34

35 Chapter Experiment This experiment aims to measure an absorption curve of rubidium atoms in a magnetic field, effectively establishing the existence of EIT. The theory discussed in the previous chapter was derived for standard EIT which makes use of two lasers. The theory also generalizes to Zeeman EIT. The major distinction is that the multiplelevel system is created by applying a magnetic field across atoms with degenerate energy levels. This splits the levels creating a suitable energy scheme for EIT. Due to this distinction, it is possible to use only one laser and scan the magnitude of the magnetic field surrounding the atoms to acquire data. Details on how this is possible will be discussed in this chapter..1 Rubidium and Zeeman Splitting Rubidium gas was chosen to be the atomic medium in this study for its energy structure and the availability of applicable optical equipment. Additionally, since rubidium is ubiquitous in the study of EIT and saturated absorption spectroscopy, it makes for easy comparison to past research. Specifically, the D transition in the 87 Rb isotope is used. This transition is between the ground state 5s S 1/ and the excited state 5s P 3/ and is resonant at a wavelength of nm. Within these states there exist several hyperfine levels denoted by their quantum numbers F for the ground state and F for the excited state. The hyperfine levels used in this experiment are F = 1 and F = 1. These levels were chosen because of the prominence of the peak associated with their atomic transition on a hyperfine absorption spectrum and for the simplicity of their Zeeman splitting. The Zeeman effect occurs when states of the same energy but distinct angular momenta are introduced to a magnetic field. Here, angular momentum is denoted by the quantum numbers m F and m F for the ground and excited state, respectively. Each state F = 1 and F = 1) can have three momenta m F = 0, ±1, so they each split into three distinct levels. The resulting six-level system is shown in Fig..1. In theory, it may seem simpler to use the F = 0 level because this excited state does not split due to the Zeeman effect. However, in practice, its associated absorption line is too small to detect using the equipment in

36 4 Chapter. Experiment F 1 m F ' 1 m F ' 0 F 1 F m F 0 m F 1 Figure.1: The F = 1 F = 1 transitions with the Zeeman split levels. Solid arrows represent transitions induced by σ + and dashed lines represent that of σ. this experiment. In order to construct a Λ-type level structure, two circularly polarized fields are applied. A left-circularly polarized σ + beam will increase an atom s angular momentum by one quantum number. That is, σ + will excite m F = 1 m F = 0. Similarly, a right-circularly polarized σ beam will take m F = 1 m F = 0. At the same time, the m F = 0 state will also excite into m F = ±1 states forming a V-type structure. This combination of Λ and V schemes is the same energy structure shown to have EIT dark states in Section Luckily, EIT is seen for both of these configurations when the Zeeman splitting is minimal, or when the applied magnetic field is close to zero. This means their transparency windows will occur concurrently on the absorption curve. Due to spontaneous decay from the upper levels, atoms will tend to settle in the Λ states, so the contribution of the Λ scheme to the EIT signal will be much greater that that of the V scheme. Therefore, we still expect absorption similar to that of the three-level system. In terms of the states of the three-level Λ system discussed in Chapter 1, the m F = ±1 states act as 1 and exchangeably, depending on the sign of the applied magnetic field 1. The m F = 0 state acts as 3.. Preliminary Setup The setup consists of a diode laser which is parked at the resonant frequency using saturated absorption spectroscopy. The beam is then properly polarized and shone through a cell of rubidium surrounded by magnetic field coils. The transmitted beam is then detected by a power meter, and an absorption curve is obtained. 1 When the magnetic field is positive, the m F = 1 state will have higher energy than m F = 1; when the magnetic field is negative, the energy splitting is reversed.

37 .. Preliminary Setup 5..1 Diode Laser The 780 nm diode laser was a product of a past thesis at Reed College [14]. It has a Littrow configuration in which an external cavity is formed between the rear facet of the diode and a diffraction grating. The grating is angled such that the first-order diffraction is reflected back into the diode to form a feedback loop. Zeroth-order reflection exits the cavity as the lasing beam. The lasing wavelength is determined by the length of the external cavity. The laser is driven by a ILX Lightwave LDX-355 precision current source and cooled by a Peltier cooler configured with a ILX Lightwave LDT-5910B PID precision thermo-electric temperature controller. The frequency of the laser can be fine tuned by a piezoelectric crystal which changes the cavity length by small amounts. A voltage is applied across the piezo crystal by a Thorlabs MDT694A piezo controller, which is driven by an Agilent 3310A arbitrary waveform signal generator. This allows the laser to be scanned across a range of wavelengths according to the output voltage of the signal generator. The piezo crystal can scan the laser frequency over a range of 1- GHz without mode jumping. The exiting laser light enters a Conoptics Model 713A optical isolator and is split by a 95-5 beamsplitter. The transmitted beam is much stronger, and it continues into the primary EIT setup. The weaker reflected beam is directed toward the saturated absorption spectroscopy setup described in the following section... Saturated Absorption Spectroscopy Doppler-free saturated absorption spectroscopy is a standard experimental method used to park the laser onto a particular frequency resonant with a hyperfine transition [15]. We will use this technique to tune the laser to the F = 1 F = 1 transition in 87 Rb. Without saturated absorption spectroscopy, this would prove to be a difficult task; since the atoms in a cell of gas will have some spread of velocities according to a Maxwell-Boltzmann distribution, each velocity group will appear to have an atomic resonance frequency that is Doppler-shifted by some amount. For this reason, the hyperfine peaks will be Doppler-broadened and, in a typical absorption spectrum, smoothed into one conglomerate absorption peak. Saturated absorption spectroscopy is a technique used to resolve the hyperfine peaks by counterpropagating a pump and probe beam and saturating the atoms in their excited state. A schematic of the saturated absorption spectroscopy setup is shown in Fig... The beam from the laser is reflected twice off a glass plate once off the front facet of the plate and again off the back. These are the probe beams which pass through a rubidium cell before being detected by photodiodes. Another much stronger pump beam is transmitted through the glass plate. The pump is redirected to counterpropagate with one of the probe beams as they pass through the medium. Due to the opposing Doppler shifts, the counterpropagating beams cannot be si- It is important to note that the pump and probe beams in the saturated absorption spectroscopy setup are distinct from those in the primary EIT setup and those mentioned in the theory sections.

38 6 Chapter. Experiment Signal Generator Temp. Controller Piezo Controller Current Controller Figure.: Schematic of saturated absorption setup. multaneously resonant except with the zero-velocity group of atoms. In this scenario, the pump beam will excite the vast majority of the atoms into an excited state saturating that transition. This allows the probe beam to pass through with little absorption. The laser is scanned in frequency by sending a positive triangle waveform through the piezo controller. Two spectra result: one from the unsaturated probe beam a typical Doppler-broadened absorption spectrum and another from the saturated beam consisting of narrow peaks at the hyperfine resonance lines. The difference of these two signals yields a resolvable hyperfine spectrum. Examples of such spectra are shown in Fig..3. To park the laser onto the F = 1 F = 1 peak, the frequency scanning range is decreased in amplitude while observing the detector signal. This effectively narrows the spectrum onto the desired peak. Then, the scan is stopped altogether. The piezo voltage can also be manually adjusted to ensure the laser is stabilized at the right frequency. Since we do not actively lock the laser frequency, the signal will drift. However, the laser stays at resonance for about 3-5 minutes, which is long enough to take about one run of measurements. In future experiments, it may be necessary to use feedback control electronics to stabilize the laser for longer amounts of time [17].

39 .3. Primary EIT Setup 7 V spectrum V 6 F' 1 F' V spectrum V F' F' V scan V Figure.3: Hyperfine spectra of 87 Rb. F = 1 F transitions left) and F = F transitions right). These measurements are consistent with the results of MacAdams et al. [16]..3 Primary EIT Setup The majority of the light emitted by the laser now parked on the hyperfine line of interest passes through the primary setup where EIT is observed..3.1 Components of Polarization The novel aspect of this setup is that only one laser is used. To achieve this, the single beam will be composed of two components of orthogonal polarizations. These components will act as independent control and probe beams, though they make up the same beam. For this reason, it is easiest to discuss how optical elements affect each polarization component individually. As shown in Fig..4, the light first passes through a half-wave plate. Based on the angle of the half-wave plate, the resulting light will have some specific combination of linear polarizations, which will later become the control and probe components of the beam. The angle of the half-wave plate is chosen such that the ratio of intensities of control polarization to the probe polarization is 9:1. The beam then passes through a quarter-wave plate, which converts the linearly polarized components to circular polarizations. This produces the σ + and the σ needed to excite the atomic medium. After the beam passes through the rubidium cell, it is converted back to linear polarization by another quarter-wave plate, and the linear components are split by a polarizing beamsplitter. The power of the resulting probe beam is measured by a ThorLabs PM100D optical power meter. To see that this works explicitly, we will use Jones calculus [18] to calculate the relevant polarizations at the lettered points in Fig..4. Assume that the polarization output by the laser is the y-direction, where the y-direction points upwards perpendicular to the optics table. 3 Therefore, the initial Jones vector at point A and the 3 The starting polarization may very well be some admixture of x- and y-polarizations, but this can be corrected for by properly setting the angle of the half-wave plate.

40 8 Chapter. Experiment D Rb Cell Λ 4 E PBS C Λ 4 B Constant Voltage Source Λ Power Meter A SAS Setup Optical Isolator Tunable Diode Laser Figure.4: Schematic of the primary EIT setup. Abbreviations used in this figure are: λ/ - half-wave plate; λ/4 - quarter-wave plate; PBS - polarizing beamsplitter; BD - beam dump. Points A-E are included solely for ease of reference. Jones matrix associated with the half-wave plate are ) 0 cos θ sin θ P A =, J 1 HWP = sin θ cos θ ),.1) respectively, where θ is the angle of the half-wave plate s fast axis with respect to the y-direction. After the half-wave plate, the polarization at point B will be P B = J HWP P A cos θ sin θ = sin θ cos θ sin θ = cos θ ) 0 1 ) )..) Clearly, θ can be chosen to create any control-to-probe ratio desired. In this experiment, only the 9:1 ratio will be studied, but changing this ratio leads to other

41 .3. Primary EIT Setup 9 interesting observable effects, such as coherent population trapping [19] and electromagnetically induced absorption [0]. The Jones matrix for a quarter-wave plate with its fast axis 45 from the y- direction is J QWP = 1 ) 1 i..3) i 1 Therefore, the polarization at point C will be P C = J QWP P B = 1 ) ) 1 i sin θ i 1 cos θ = 1 ) sin θ i cos θ i sin θ cos θ = 1 ) 1 1 [sin θ i cos θ i i = σ + sin θ σ i cos θ,.4) where the left-circular and the right-circular polarizations are defined to be σ + 1 ) 1, σ i 1 ) 1,.5) i respectively. Thus, the polarization at point C has a component of each circular polarization. When a beam reflects off a mirror, left-circularly polarized light becomes rightcircularly polarized and vice versa. Therefore, the polarization at point D is P D = σ sin θ σ + i cos θ.6) = 1 ) sin θ i cos θ.7) i sin θ + cos θ )] Finally, the beam goes through another quarter-wave plate at 45. polarization at point E will be The resulting P E = J QWP P D = 1 1 i i 1 sin θ = cos θ ) 1 cos θ + i sin θ i cos θ sin θ )..8) The beam has now returned to a composition of linearly polarized components such that the polarizing beamsplitter can pick off the probe component to be measured. It is unclear at this point which polarization corresponds to the control beam and which to the probe beam: σ + or σ. In fact, it does not matter all that much. In Chapter 1, we assumed that the probe field was resonant with the transition from )

42 30 Chapter. Experiment the lowest energy state and the control was resonant with that of the higher energy ground state. However, as we saw in Section.1, when the magnetic field around the rubidium cell is swept from positive to negative values, the two ground states trade places. For this reason, either choice will yield similar results. It took a series of steps to set the wave plates to the correct angles. First, the two quarter-wave plates and the rubidium cell were removed from the setup. This allowed the beam to pass through only the half-wave plate and the polarizing beamsplitter. The half-wave plate was adjusted until the transmitted beam was maximized. This corresponds to having only y-polarized light mw of power was transmitted through the beamsplitter, and only 43 µw was reflected. Then, one of the quarter-wave plates was put in place and adjusted until the transmitted and reflected beams were as close to equal as possible. This alignment of the quarter-wave plate converts linear polarization to circular. The transmitted beam had power 5.40 mw and the reflected beam had 5.79 mw. The first quarter-wave plate was removed and replaced by the remaining quarter-wave plate, and this process was repeated. This time, the transmitted power was 5.98 mw, and the reflected power was 5.40 mw. Finally, after all of the wave plates were returned to their spots in the optical setup, the half-wave plate was readjusted such that the ratio of the power of the beams transmitted and reflected through the polarizing beamsplitter was 9:1. Their true powers were 9.9 mw and mw..3. Zeeman Apparatus EIT actually occurs within the rubidium cell. In order to induce this effect, the cell is placed within a Zeeman apparatus. A section and 3D rendering of the apparatus are displayed in Fig..5. The cell is enclosed in an acrylic tube wrapped in a solenoid of copper wire. The total resistance of the coil is 14.6 Ω. For the experiment, a 100 Ω external resistor was added to attain a finer absorption curve. The wire exits the apparatus at its end caps and is driven by a constant voltage source to induce a magnetic field within the solenoid along the apparatus axis. The solenoid is about three times longer than the cell, while their radii are comparable, so near the center of the apparatus, the induced magnetic field can be approximated as constant. This will be shown more explicitly later in the section. Outside of the solenoid are three cylindrical layers of mu metal. Mu metal is a material with a very high magnetic permeability for the purpose of shielding the cell from exterior magnetic fields. The shielding we used had a relative permeability 4 of [1]. The geometry of the shielding three concentric cylinders was chosen because inserting gaps, in fact, makes the shielding much more effective. This configuration was also relatively simple to construct. The effectiveness of the shielding was determined by measuring the magnetic field at various positions within the apparatus without current passing through the 4 Relative permeability is a dimensionless quantity defined as µ R = µ/µ 0 where µ is the absolute permeability and µ 0 is the permeability of free space.

43 .3. Primary EIT Setup x cm z cm Figure.5: Section top) and 3D rendering bottom) of the Zeeman apparatus. Images by Jay Ewing.