Electromagnetically induced transparency in cesium vapor

5.2.1. Brief review of EIT

Since its first demonstration in 1991 by Harris et al. [63], EIT has attracted much attention, and its capability to slow down [147] and even stop light [66] was shown. In 2001, an EIT based setup for light storage in atomic vapor at room temperature was first demonstrated [71, 65]. Storage times of up to 0.5 ms have been reached. In the meantime, even single photons have been stored in such setups [69], but with shorter storage times on the order of a fewµs. EIT can also be utilized for all-optical control of amplitude and phase of a light pulse down to the single photon level. Such a control facilitates loading of single photons into an optical cavity in quantum interfaces [15] or their storage in an atomic ensemble [72]. Cross-phase modulation on the single photon level mediated by EIT [149, 150] is important for establishing optical quantum gates [151]. Both regimes can be described within a common theoretical framework applicable for any Λ-type system [152]. More details will be given in section 5.2.2. Most of the EIT experiments so far have used rubidium atoms [71, 72]. Cesium, however, offers certain advantages, e.g., theF = 3→F = 4 hyperfine ground state clock transition allowing the realization of all optical atomic clocks [73]. The ¹³³Cs D1-line at 894nm (see figure A.1 in the appendix) lies well within the wavelength regime of exciton emission from InAs quantum dots [74], which is relevant for possible coherent interfaces between atomic and solid-state systems [75]. Details of EIT experiments using Cs will be demonstrated in section 5.3.

Very recently, Reim et al. [76] demonstrated coherent storage and retrieval of sub-nanosecond low-intensity (several thousand photons) light pulses with spectral band-widths exceeding 1GHz by utilizing a far off-resonant two-photon Raman transition.

In this experiment cesium provided the advantage of smaller Doppler linewidth, i.e.,

∼380 MHz in¹³³Cscompared to ∼540 MHz in⁸⁷Rb at room temperature, and larger hyperfine splitting of 9.2 GHz and 6.8 GHz, respectively. The latter sets a limit to the maximum storage bandwidth. In spite of the potential advantages there have been no experiments so far on the single photon level with cesium vapor, for the most part be-cause of the problem to filter the strong coupling beam. This problem was overcome in

5.2. Electromagnetically induced transparency in cesium vapor this work, and EIT experiments on the single photon level will be described in the next chapter 6.

5.2.2. Theoretical background

In this thesis the focus is on EIT in cesium vapor in a gas cell at room temperature. The used level schema of the D1-line in ¹³³Cs is Λ-shaped. It is displayed in Figure 5.1 and consists of the two hyperfine levels of the 6²S_1/2 ground state and of the 6²S_1/2(F = 4) level. The two ground state levels |bi and|ci are coupled via dipole-allowed transitions to the common excited level|aiby a coherent coupling field of Rabi frequency Ω and by a weak probe field of Rabi frequencyα, respectively.

In case of homogeneous broadening EIT can be achieved if the intensity of the cou-pling laser is larger than the product of the decay rates of the coherence between the lower level γbc and the homogeneous linewidth γ. It can be shown, that in case of an inhomogeneous broadened system, e.g., in case of Doppler-broadening relevant for¹³³Cs at room temperature, EIT can still be achieved if Ω²γ_bcγ [153].

Coupling laser

Ω

6 P² _1/2

6 S² _1/2

Probe laser

α

F=4 F=3

F=4 a

c b

γ γ

γ_bc

Figure 5.1.: Parts of the ¹³³Cs D1-line form a Λ-shaped three level atomic system. The upper level|aidecays to|biand|ciwith a decay rateγ. The relaxation rate between the lower levels is called γbc.

Optical Bloch equations The theoretical description of the system is based on the atom-field interaction Hamiltonian

H =~ω_a|ai ha|+~ω_b|bi hb|+~ω_c|ci hc| −hαe^2πiνpt|ai hb| −hΩe^2πiνct|ai hc|+ H.c. . (5.1) Here, ~ω_x (x∈ {a, b, c}) is the energy of the corresponding atomic level and ν_c and ν_p are the laser frequencies of coupling and probe field. Using the Liouville equation

ρ˙=−i

~[H, ρ]

time evolution is described. In order to account for incoherent processes like damping or dephasing, one makes use of the master equation formalism including the coupling to

a reservoir in thermal equilibrium [154]. The equations of motion for the density matrix elements in a rotating frame are then given by the optical Bloch equations [155]

ρ˙_ab=−(iω_ab+γ_ab)ρ_ab+iαe^−2πiνpt(ρ_bb−ρ_aa) +iΩe^−2πiνctρ_cb ρ˙_ac=−(iω_ac+γ_ab)ρ_ac+iΩe^−2πiνct(ρ_cc−ρ_aa) +iαe^−2πiνptρ_bc ρ˙_cb=−(iω_cb+γ_bc+γ_deph)ρ_cb+iΩ^∗e^2πiνctρ_ab−iαe^−2πiνptρca

ρ˙aa=−γ_aρaa+iαe^−2πiνptρba−iα^∗e^2πiνptρab+iΩe^−2πiνctρca−iΩ^∗e^2πiνctρac

ρ˙_bb=−γ_bcρ_bb+γ_bcρ_cc+γ_abρ_aa−iαe^−2πiνptρ_ba+iα^∗e^2πiνptρ_ab ρ˙_cc=−γ_bcρ_cc+γ_bcρ_bb+γ_acρ_aa−iΩe^−2πiνctρ_ca+iΩ^∗e^2πiνctρ_ac

withγ_ab=γ_ac =γ+^1γ₂^bc). Here the following assumptions apply: (1) the decay rates of the transitions |bi ↔ |ci are equal (γ_bc =γ_cb), which is generally the case because this decay is determined by the time of flight of the atoms through the interaction region;

(2) all other dephasing mechanisms are summarized in the dephasing termγ_deph that is later handled as a fit parameter; (3) co-propagating coupling and probe lasers as well as a frequency difference between the transitions (|ai → |bi) and (|ai → |ci) that is small enough to ignore the residual Doppler-shift.

Applying the rotating wave approximation [154] a stationary solution of the optical Bloch equations is obtained by assuming ˙ρ≡0 and normalization Tr(ρ) = 1. In case of a weak probe field the calculations can be limited to the first order in the Rabi frequency of the probe fieldα. The absorption of the probe field is then governed by the first-order solution of the off-diagonal density matrix elementρab. In the steady state this can be found to be

ρ⁽¹⁾_ab = −iα

(γ_ab+i∆_ab)(γ_bc+γ_deph+i∆_bc) + Ω²· [γbc+γdeph+i∆bc(ρ⁽⁰⁾_aa −ρ⁽⁰⁾_bb )+

Ω²

γac−i∆ac)(ρ⁽⁰⁾_cc −ρ⁽⁰⁾_aa)]

(5.2)

with the detuning terms ∆_ab = (ωa−ω_b)−νp, ∆_ac = (ωa−ωc)−νc, and ∆_bc =

∆_ab−∆_ac= ∆_cb.

Susceptibility Now, from equation 5.2 the susceptibility can be calculated [156, 157]

χ=η (ρ⁽¹⁾_ab

α )

with η= 3

8π²NCsγaλ³_ba (5.3) with N_Cs being the density of the Cs gas, γ_a the total decay rate of the excited state andλba= _(ω ^c

a−ω_b) the wavelength of the probe laser transition.

As the experiment is performed at room temperature, Doppler-broadening needs to be accounted for. This can be achieved by convoluting the susceptibility with the velocity

5.2. Electromagnetically induced transparency in cesium vapor

The velocity distribution can be approximated by a Lorentzian distribution [153, 158]

with the Doppler-linewidth ∆w_D in order to obtain analytical results for χ. To describe atoms with the velocity −→v, the substitutions ∆_ac → ∆_ac +δ, ∆_ab → ∆_ab +δ and ∆_bc → ∆_bc with δ := −→v−→

k are performed in ρ⁽¹⁾_ab, and ρ⁽⁰⁾_ii . Now equation 5.4 can be solved by contour integration in the complex plane.

Using the definition of the complex index of refraction ˜n = n−i κ with extinction coefficient κ and using χ = ˜n²−1, one finds Im[χ] = Imn²−κ²−2inκ−1. Since n ∼= 1 in dilute vapor, Im[χ] ∼= −2κ follows, which allows the transmission T to be calculated using the absorption coefficient α_EIT = _λ^2π

baIm [χ] [101] and the length of the gas cell LCs

T =e^−αEITLCs. (5.6)

It can be further shown, that the width of the transparency window is scaling with

∆ν_EIT^Dopp ∼Ω²/∆w_D in the case of Doppler broadening instead of ∆ν_EIT ∼Ω²/γ_a for a gas at absolute zero temperature [155].

According to reference [159] dispersive properties like the index of refraction

n∼Re[1 +χ]¹ (5.7)

can be calculated. The first summand inv_gaccounts for frequency dispersion, the second is attributed to spatial dispersion. Experimentally, the EIT caused pulse delay in Cs τEIT cell will be studied,

τ_EIT =L_Cs 1 v_g −1

c

!

. (5.8)

The dependence of absorption and index of refraction on probe laser detuning is simulated in figure 5.2 using equations 5.6 and 5.7. In the figure the solid line gives the situation for a coupling laser of Rabi frequency Ω = 0.5 MHz while the dashed line shows the situation without a coupling laser for reference. Figure 5.2(a) shows the absorption coefficient, the transparency window is visible only in the presence of the

1Due to the definition of the Rabi frequencies no additional factor 2πis required.

coupling laser. Figure 5.2(b) exhibits the corresponding index of refraction with a steep slope if the coupling laser is applied.

-10 -5 0 5 10

AbsorptionCoefficient

Probe Laser Detuning (MHz)

-10 -5 0 5 10

1

IndexofRefraction

Probe Laser Detuning (MHz)

(a) (b)

Figure 5.2.: (a) Absorption and (b) index of refraction around the probe laser transition.

The absorption shows the typical feature of high transmission on resonance while the index of refraction exhibits a steep slope as required for low group velocities. The dashed lines show the situation without coupling laser for reference.

Im Dokument Narrow-band single photons as carriers of quantum information (Seite 68-72)