El último carnaval: una historia verdadera (1998): el carnaval como un escenario donde confluyen mitos

In the final section of this chapter, we will consider some of the Rydberg properties in the context of a recently demonstrated application [97]. The potential of Rydberg atoms for detection of microwave (MW) and terahertz10_{fields, has been recognised}

in early experiments [48, 50, 52]. For example, authors in Ref. [48] excited atoms in a atomic beam to a high-lying state |r1〉, where big dipole matrix elements ∝

n2

∗ between nearby states would make the medium optically dense for incoming

black-body radiation in the narrow spectral range corresponding to the transition |r1〉 → |r2〉 where |r2〉 is Rydberg state with higher energy. Atomic population in

different Rydberg states can be detected with state-selective ionization (Sec. 2.3.5) providing readout of incoming radiation intensity. The ultimate detection sensitivity limit of this approach is given by the noise due to the collisional and black-body radiation induced population redistribution of |r1〉. Authors of Ref. [50] managed to

control the black-body radiation by enclosing the interaction region of the atomic beam in a cooled (∼ 14 K) box which included metallic meshes in the holes for beam input that were dense enough to prevent penetration of the MW radiation from the hot atom source. This background shielded detection region was then used for directly measuring microwave radiation from black-body sources at different temperatures, using the same approach as in Ref. [48].

More recently, the intensity of a coherent source was measured with all-optical methods in a Rydberg vapour. Coherent MW driving between the two Rydberg states induced dressing of the state, and the corresponding Autler-Townes splitting can be detected if the transition to either of the Rydberg states is probed with a laser scan [160, 199]. However, the glass cell reflects microwaves, producing a

10_{Note that in the early papers the terahertz part of the spectrum was referred to as far-infrared}

complicated interference pattern in the measurement volume [200] that, due to the integral nature of all-optical detection via transmission measurements, cannot be directly detected in a single measurement. Also the interference pattern has to be known in order to correctly model the Autler-Townes splitting spectra.

In recent experiments by C. Wade et.al. [97] that interference pattern was converted to optical domain and measured along the laser beam in a single shot, by spatially imaging atomic fluorescence. Thermal caesium vapour enclosed in a quartz glass

Material n α (cm−1) Silica 1.96 0.62 - 7.8 Pyrex 2.1 7 - >90 BK7 2.5 14 - >90 SF7 3.5 43 - >90 PTFE 1.44 1.4 - 2.8 HDPE 1.53 1.4 - 2.8

Table 2.1: Refractive indexes n and ab- sorption coefficients α for common glasses and polymers for EM radiation in range 0.5- 2 THz. All materials exhibit stronger absorp- tion at higher frequencies. Data for glasses is from [201] and for polymers from [202].

cell was used as active medium (for details of the experimental setup, see Ref. [97]). The choice of glass is crucial for efficient sampling of terahertz radiation, since other common glasses with higher ionic fraction have order of magnitude higher absorption coefficients (Table 2.1). Common polymers used for THz lenses (PTFE, HDPE) have similar absorption coefficients as Silica glass. PTFE with maximal usable temperature up to ∼ 250◦_{C and low thermal conductivity, can also be used as an}

insulator for vapour-cell ovens if minimum absorption is needed. Alkali metals have a number of transitions in the THz regime featuring strong coupling [Fig. 2.26(a)], that forms a dense frequency comb in the terahertz window (0.3-3.0 THz) of the EM spectra, allowing measurement of radiation electric field amplitude via Autler- Townes splitting, as described in the previous paragraph. Measurements are relative to the fixed atomic standard that, in principle, can be absolutely calibrated. If both laser and THz driving are detuned from resonance ∼ 200 MHz a new regime emerges. Due to reflections of the THz wave typically a standing wave will form (Fig. 2.25). In the nodes of the THz field, the laser driving is off-resonant with Rydberg transitions and therefore no atoms will be excited to this highly excited state. This is different compared to the previously discussed techniques in Ref. [48, 50], where states could be always populated through collisional processes, introducing a background signal11_{. In the points in space where there is a non-zero THz field present, the}

combination of optical and THz fields with equal detunings (Fig. 2.25, inset) will drive two-photon stimulated Raman transition to the Rydberg state |r〉. The natural lifetime of that state is 4 µs, but the population from that state will be redistributed through BBR and collision induced processes to other states, most of them capable of decaying in the visible spectrum. On these time scales hot atoms (∼ 60◦_{C) can travel}

distances of ∼ 1 mm before they emit fluorescence in the visible spectrum. Crucially, since the experiment uses a three-step ladder excitation scheme, the only atoms that can be excited to the Rydberg state will be those selected by the resonance condition of the lasers. If the laser beams are not too strong so as to induce additional dressings, THz

799 nm 21 P3/2

21 S1/2

Figure 2.25: Two-photon optical-terahertz excitation of the Rydberg atomic states maps intensity of the terahertz standing wave into the fluorescence pattern of the Rydberg atoms. Atoms (blue dots) in the off-resonant laser beam can be excited (green spherical clouds) to Rydberg state if they are in the areas where field of terahertz standing wave (grey oscillating strip) is non-zero, as the two-photon resonance condition is fulfilled (in- sets). Image below illustration is experimental data from C. Wade [97].

and if they are set on resonance with the zero-velocity class, the atoms’ velocity in the direction of the laser beam will be significantly reduced (∼ 5 m/s) compared to the average velocity in the transverse direction (∼ 200m/s). This means that after excitation the atom motion will not smear the fluorescence pattern in the direction

of laser beam propagation, thus maintaining high-resolution record of THz driving

field in the fluorescence pattern. At the same time lower-lying Rydberg states that have strong THz transitions, have lifetimes of the order of ∼ 1 − 10 µs. That sets an ultimate theoretical limit on maximum frequency of THz field amplitude modulation at ∼100 kHz, if intensity changes are to be resolved in time. Realistic rates are also limited by the finite exposure time required to capture an image with good signal to noise ratio. This depends on the fluorescence intensity, the detection acceptance

11_{Note that for very dense vapours collisional redistributions and light-assisted collisions will ultimately}

Detection frequncy (THz) Detection frequency (THz) |¯µT Hz | (a0 e) ¯µ 2 Rydberg ¯µ 2 THz (a 4 0e 4) (b) (a) _Cs Rb K

Figure 2.26: Relative sensitivity of alkali metals to terahertz radiation. (a) For reson- ant detection of coherent radiation measured via Autler-Townes splitting of the Rydberg-state ex- citation resonances, reduced dipole matrix ele- ments |¯µTHz| = |〈r1||er||r2〉| for terahertz trans-

itions between the Rydberg states |r1〉 ↔ |r2〉

gives relative sensitivities in the terahertz range. For off-resonant imaging schemes, the fluores- cence rate will depend on both coupling between the Rydberg states ¯µTHzand coupling from the

lower excited state to the Rydberg state ¯µRydberg

as ¯µ2_Rydbergµ¯2_THz(for the fixed Rydberg laser intens- ity), calculated on (b) for example of caesium excitation from the lower excited state 7 S1/2, as

in Ref. [97] (THz transition used there is high- lighted with a red dot).

angle and efficiency, and dark-noise. Rates of ∼ 12 Hz have been demonstrated with the standard consumer photo-camera [97], which is already good for real-time applications.

The excitation rate of Rydberg states in the off-resonant laser beam will be pro- portional to the square of the two-photon Rabi frequency (ΩoΩTHz/∆)2where Ωo

and Ω_THzare the Rabi frequencies of the optical and terahertz transitions respect- ively. The initial Rydberg population will then be proportional to the square of the corresponding dipole matrix elements. For all possible caesium transitions from Fig. 2.26(a), that quantifies relative sensitivities for resonant probing of the fields, Fig. 2.26(b) quantifies this relative Rydberg excitation rates for the off-resonant fluorescence imaging. The ultimate intensity of the fluorescence depends also on the lifetimes and decay channels of the state. The resolution of this method in the axial direction is limited by the residual Doppler-velocity of the excited atoms in that direction to ∼ 20 µm (theoretical estimate of lower bound). In the other two dimensions (in the radial direction) resolution is limited by the size of the probing laser beam, similarly to Ref. [200].

As can be seen on Fig. 2.26(b), this method as it stands offers narrowband detection, which has limited discrete tunability, achieved by changing the principal quantum number of the state. Detuning the excitation laser offers limited continuous tunability, since for bigger detunings ∆ the two-photon transition rate is quickly diminished, while for smaller detunings the direct single-photon excitation of the states will cause a background signal. In principle, fluorescence from directly excited P states will have different frequencies than decays from S states populated in a Raman transition, but quick population redistribution processes will quickly contaminate all the fluorescence channels. A viable alternative for expanding detection ranges is by using Stark shifts of the state (Sec. 2.4.1), as in Ref. [48]. For example, scalar polarizability of the 21 P3/2 mj = 1/2 and 21 S1/2 mj = 1/2 caesium states are

α(P)₀ =1.36 MHz cm2/V2_{and α}(S)

0 =0.08 MHz cm2/V2respectively, providing tuning

of αP− αS= 1.28 MHz cm2/V2with an applied electric field, i.e. S to P resonance shift of 6.4 GHz through application of an electric field of 100 V/cm. In stronger fields, one has to take care of the strong state admixing (Sec. 2.4.1) that would not only change coupling constants, but also possibly allow driving of two-photon transitions to states that are normally forbidden by the ` selection rules for the dipole operator of the unperturbed atomic states.