Ultra-high Resolution Spectroscopy with atomic or molecular Dark Resonances:Exact steady-state lineshapes and asymptotic profiles in the adiabatic pulsed regime

Ultra-high Resolution Spectroscopy with atomic or molecular Dark Resonances:
Exact steady-state lineshapes and asymptotic profiles in the adiabatic pulsed regime

Thomas Zanon-Willette111E-mail address: thomas.zanon@upmc.fr
UPMC Univ. Paris 06, UMR 7092, LPMAA, 4 place Jussieu, case 76, 75005 Paris, France CNRS, UMR 7092, LPMAA, 4 place Jussieu, case 76, 75005 Paris, France
   Emeric de Clercq LNE-SYRTE, Observatoire de Paris, CNRS, UPMC, 61 avenue de l’Observatoire, 75014 Paris, France    Ennio Arimondo Dipartimento di Fisica ”E. Fermi”, Università di Pisa, Lgo. B. Pontecorvo 3, 56122 Pisa, Italy
July 5, 2019
Abstract

Exact and asymptotic lineshape expressions are derived from the semi-classical density matrix representation describing a set of closed three-level atomic or molecular states including decoherences, relaxation rates and light-shifts. An accurate analysis of the exact steady-state Dark Resonance profile describing the Autler-Townes doublet, the Electromagnetically Induced Transparency or Coherent Population Trapping resonance and the Fano-Feshbach lineshape, leads to the linewidth expression of the two-photon Raman transition and frequency-shifts associated to the clock transition. From an adiabatic analysis of the dynamical Optical Bloch Equations in the weak field limit, a pumping time required to efficiently trap a large number of atoms into a coherent superposition of long-lived states is established. For a highly asymmetrical configuration with different decay channels, a strong two-photon resonance based on a lower states population inversion is established when the driving continuous-wave laser fields are greatly unbalanced. When time separated resonant two-photon pulses are applied in the adiabatic pulsed regime for atomic or molecular clock engineering, where the first pulse is long enough to reach a coherent steady-state preparation and the second pulse is very short to avoid repumping into a new dark state, Dark Resonance fringes mixing continuous-wave lineshape properties and coherent Ramsey oscillations are created. Those fringes allow interrogation schemes bypassing the power broadening effect. Frequency-shifts affecting the central clock fringe computed from asymptotic profiles and related to Raman decoherence process, exhibit non-linear shapes with the three-level observable used for quantum measurement. We point out that different observables experience different shifts on the lower-state clock transition.

pacs:
32.70.Jz, 32.80.Qk, 37.10.Jk, 06.20.Jr
preprint: APS/123-QED

I Introduction

In the 1930s, molecular-beam magnetic resonance techniques achieved very high precision allowing the observation of atomic/molecular systems essentially in total isolation Kellogg:1946 (). The Rabi method revealed coupling interactions between internal energy states and provided plenty of information not only on atomic and molecular structure, but also on nuclear properties. In the 1950s, N.F. Ramsey realized a scheme with much higher resolution by increasing the interaction time between the atom or molecule and the oscillating field Ramsey:1956 (). Still today, this technique provides the highest resolution in order to follow a dynamical evolution of wave functions and probe its phase accumulations. Control and elimination of systematic frequency shifts dephasing a wave function oscillation at a natural Bohr frequency are fundamental tasks to achieve precision measurement Cronin:2009 ().
An alternative tool to probe by the Rabi or Ramsey sequences, for a dipole-forbidden transition, is to radiatively mix the atomic or molecular states. As an example, without natural state mixing from spin-orbit interaction, a long-lived Raman coherence between a ground state and a long-lived (as in alkalis) or metastable (as in two-electron atoms) level, to be referred as clock states, can be established by a two-photon process via an upper excited level, thus forming a three-level system. The properties of such a system are strongly determined by an optical pumping mechanism leading to a formation of a Dark Resonance associated with a trapping of the atomic population in a coherent superposition of states Hansch:1970 (); Brewer:1975 (); Whitley:1976 (); Arimondo:1976 (); Alzetta:1976 (); Gray:1978 (); Orriols:1979 (). Since such quantum superposition states are radiatively stable, the associated Raman coherence production leads to extremely narrow Dark Resonances allowing high-resolution frequency measurements Shah:2010 (). Such coherences were explored for single trapped ions Siemers:1992 (); Janik:1985 (), microwave clocks Vanier:2005 (), microwave chips Treutlein:2004 (); Farkas:2010 (), optical lattice clocks Santra:2005 (), multi-photon excitations Hong:2005 (); Champenois:2007 (), or nuclear clocks Peik:2002 (). Similar coherent superpositions are used in solid-state physics for quantum information Greilich:2006 (), in super-conducting circuits Dutton:2006 (); Kelly:2010 (), in a single impurity ion inserted into a crystal Santori:2006 (), in quantum dots Xu:2008 () with protection against random nuclear spin interactions Issler:2010 (), and in opto-mechanical systems Weis:2010 (); Safavi:2011 (). They are also actively considered within the future challenge of realising nuclear systems for quantum optics in the X-ray region Coussement:2002 (); Burvenich:2006 (). Dark Resonances, largely exploited in Quantum Optics, have been extended to the preparation of molecules in ro-vibrational ground states Mark:2009 (), and to coherent superposition of atomic-molecular states in order to measure atomic scattering lengths and lifetimes of exotic molecular states Winkler:2005 (); Dumke:2005 (); Moal:2006 (); Moal:2007 ().
Three-level narrow resonances are associated to quantum interferences produced by amplitude scattering into different channels and are strongly dependent on the configuration intensities and detunings Lounis:1992 (); Stalgies:1998 (). Such different lineshapes were associated to quantum interferences as the Autler-Townes (AT) splitting of the resonance Autler:1955 (), the Fano-Feshbach (FF) canonical form Fano:1961 (), the Dark Resonance (DR) lineshape Alzetta:1976 () known as Coherent Population Trapping (CPT) Whitley:1976 (); Arimondo:1996 () or Electromagnetically Induced Transparency (EIT) Harris:1990 (); Fleischhauer:2005 ().
Accurate calculation of the lineshape for a quantum superposition resonance requires numerical integration of the Bloch’s equations. The literature reports on efforts to establish approximate analytic equations applicable to each particular case and, in a few cases, exact but rather complex Brewer:1975 (); Orriols:1979 (); Swain:1980 (); Swain:1982 (); Kelley:1994 (); Wynands:1999 (); Lee:2003 (); MacDonnell:2004 (). The application of schemes to high accuracy atomic clocks, in microwave or optical domains, requires to determine precisely the physical processes affecting the resonance lineshape and the shifts of the clock frequency. For that purpose this work provides a careful analysis of the lineshape dependence on different parameters characterizing the atomic or molecular system under investigation.
The standard clock interrogation of a three-level system involves continuous excitation of the two lower states while sweeping through the Raman resonance. For that regime, starting from the steady-state analytical solution of three-level Optical Bloch Equations we derive the exact expression of the resonance lineshape where the role of the relaxations and dephasing rates determining the absorption profile is expressed with physical meaning. Our detailed discussion of key lineshape parameters expands previous analysis Whitley:1976 (); Orriols:1979 (); Brewer:1975 (); Janik:1985 (); Lounis:1992 (); Stalgies:1998 (). We show that Autler-Townes, CPT, EIT and FF lineshapes are associated to a universal two-photon resonance lineshape depending on system parameters Zanon-Willette:2005 (). The analytical expressions for the frequency shift associated either to the FF extrema or to the EIT resonance point out dependencies not obvious in a perturbation treatment.

An alternative clock operation scheme is based on a Raman-Ramsey scheme with the application of time separated but resonant two-photon pulses, experimentally introduced in the microwave domain Zanon:2005-PRL (); Zanon:2005-IEEE () and extended to rubidium cold atoms Chen:2010 (). This operation was inspired by the Ezekiel’s group work at MIT on a thermal beam of sodium atoms Thomas:1982 (). While in the standard Ramsey approach, a coherent superposition of clock states in the bare atomic/molecular basis is dynamically produced by a pulse depending on pulse duration and laser power, the coherent superposition, in the three-level two-photon approach, is created by an optical pumping process long enough to reach a steady-state. This scheme overcomes the power broadening mechanism of the continuous wave resonance allowing to obtain high contrasted signals in a saturation regime. This idea was extended in refs. Zanon-Willette:2006 (); Yoon:2007 (); Yudin:2010 () to the realization of EIT-Raman (and hyper-Raman) optical clocks with alkaline-earth-metal atoms. The time-separated and individually tailored laser pulses may be designed to create an atomic coherent superposition while eliminating off-resonant ac Stark contributions from external levels modifying the optical clock resonance Zanon-Willette:2006 (). For the regime of the first laser pulse long enough to produce an efficient coherent superposition, we present here a detailed analysis describing the dependence of the DR lineshape on the system parameters.
Within the quantum clock framework, the determination of lineshapes and resonance shifts in different experimental configurations remains an important issue, to be carefully investigated within the present work. An important result of the present analytical and numerical analysis for the resonance frequency-shift of a three-level quantum clock, is that different lineshapes versus the optical detunings are obtained depending on the experimentally detected population or coherence observable.
The system and the Bloch’s equations for an homogeneous medium are introduced in Sec. II, where an adiabatic analysis of the time dependent equations determines the approximated time scale required to produce an efficient atomic/molecular coherent superposition. Sec. III establishes an exact treatment of the excited state steady-state regime and derives the key informations on the Dark Resonance lineshape. In Sect. IV, we derive the steady-state profile of a two-photon resonance between clock states. In Sec. V, we focus our attention to the Raman coherence lineshape observed between clock states. Finally Sec. VI analyzes DR fringes produced with resonant two-photon pulses separated in time mixing steady-state properties and Ramsey oscillations. A detailed analysis of the fringe properties is derived in the adiabatic regime where the first pulse establishes a steady-state solution and the probe pulse duration vanishes. Instead only dynamical properties of these phase-shifts were demonstrated in refs Hemmer:1989 (); Shahriar:1997 (); Zanon-Willette:2006 (); Yoon:2007 (). In the Appendix A, we rewrite lineshape population solutions in terms of generalized multi-photon transition rates enhancing one and two photon transition rates in the three-level system. We finally derive in Appendix B a first order analytical expression of the central fringe Raman frequency-shift associated to the pulsed Dark Resonance lineshape.

Ii Three-level Optical Bloch equations

Figure 1: (Color online) Closed three-level configuration in the density matrix representation including relaxation rates , and decoherences . Optical detunings are , . The parameter defines the Raman resonance condition. and define the couplings with the applied laser fields. and are the clock states while is the excited state.

The Doppler-free three-level system presented in Fig. 1 is described by the density matrix () obeying the Liouville equation

(1)

The coupling of the three atomic or molecular states to two coherent radiation fields, see Fig. 1, is described within the rotating wave approximation (RWA) by the following Hamiltonian:

(2)

where are the detunings of the two fields. Depending on the transition the Rabi frequencies and driving the system are determined by the product either between electric dipole and electric field amplitude or between magnetic dipole and magnetic field amplitude. It is worth noticing that Rabi frequencies defined here are half of the definition of ref. Cohen-Tannoudji:1998 (). The matrice taking into account relaxation and decoherence phenomena, is written

(3)

The total spontaneous emission rate is composed by the rates (with ) describing either alkaline () or akaline-earth () three-level decay configuration. Optical coherences are relaxed with terms where . In a pure radiative process Cohen-Tannoudji:1998 (), optical decoherences are related to spontaneous emission rates by the relation . The decoherence is described by the dephasing term. The optical Bloch equations describe the temporal evolution of the density matrix elements in the RWA as Zanon-Willette:2005 ():

(4)

with . The population conservation of the closed system is given by . The optical detunings will be related to the Raman detuning by and . represents the common optical detuning for a configuration where one laser is fixed while the other is frequency scanned. Notice that within the approach of deriving the laser modes from a single source by modulation at frequency , with and , the light-shift derivation should be modified.
Eqs. (4) describe the transient dynamics and the steady state of populations and quantum coherences. A complete state mixing is reached when all atoms or molecules have been pumped efficiently into the dark state, a coherent superposition of the lower states. Thus, a pumping time is required to achieve an optimal atomic fraction trapped into the coherent state superposition. We derive such time scale evolution from Eq. (4) by an adiabatic elimination of the time derivative, for pulse durations greater than , as the population and optical coherences and evolve more quickly than the populations , and the Raman coherence .
We investigate the dynamics of the three-level systems using various combination of long and short two-photon pulses separated in time. A straightforward temporal analysis of the resulting adiabatic set, similar to NMR equations Jaynes:1955 (); Torrey:1949 (), exhibits two damping times: , determining the phase memory of the Raman coherence precession (equivalent to a transversal or spin-spin relaxation rate), and which determines the typical population transfer into the dark state superposition (similar to a longitudinal or spin-lattice relaxation rate) Abragam-Schoemaker (). At low optical saturation , we have

(5)

where we have introduced the following generalized relaxation rates:

(6)

The generalized branching ratio difference is

(7)

with the normalized branching ratio given by . The and timescales play a key role on the population transfer between atomic or molecular states Zanon-Willette:2006 (). Indeed, optical coherences are efficiently generated only when the and Rabi frequencies are applied for a time exceeding . A short pulse duration having will be Fourier limited and will lead to a weak contrast resonance profile, whereas a long pulse with will eliminate all time dependencies in lineshape and frequency shifts. This regime is latter examined within the following section.

Iii The steady-state lineshape of

iii.1 The Dark Resonance

Figure 2: (Color online) Three-level spectra vs Raman detuning observed on the population. For all spectra , . (a) AT spectrum at , , and . (b) Dark/EIT resonance at , , and . (c) FF resonance for , , . Solid lines from the analytic solution Eq. (8) and dots from the numerical integration of Eq. (4) at .

In examining the steady-state situation with all time derivatives in Eq. (4) set to zero, we find the exact expression for the population of the upper state

(8)

where is the Raman detuning. The expression of , the frequency shift affecting the Raman detuning, and , halfwidth of the two-photon resonance, are reported in the following subsection. The coherence decay rate is with the saturation rate of the Raman coherence given by

(9)

Notice that coincides with for the resonant laser case.
of Eq. (8) represents the signal amplitude. It contains the broad features of the dependence on the optical detunings . The fraction on Eq.(8), whose values lie in the [0,1] interval, determines the narrow variation with the Raman detuning . The three-level signal amplitude is given by

(10)

where we introduced the normalized dimensionless Rabi frequencies note0 ()

(11)

The saturation parameter driving the population exchange between energy levels is determined from Einstein’s rate equations as:

(12)

The imaginary parts of optical coherences are related to the excited state lineshape expression by the relation:

(13)

Therefore their lineshape is equivalent to that of .
In Appendix A, we recast all population lineshapes in terms of multi-photon transitions rates, pointing out the light-shift contributions to the optical detuning terms. We verified that the numerical results to be presented in the following can be derived also from that solution.
Depending on the detuning and intensity of the lasers or microwaves driving the three-level system, the lineshapes associated to Eqs. (8) and (13) present very different features, known as AT spectra, Dark/EIT resonance, and FF profile, associated to different degrees of interference between two-photon transition amplitudes.
The AT profile appears when one Rabi frequency is much larger than the natural linewidth of the excited state ( and in addition , or viceversa. Two splitted resonances, a doublet structure, appears in the frequency spectrum, as shown in Fig. 2(a).
At Rabi frequencies smaller than the excited state width, we reach the DR or EIT configuration where a narrow two-photon resonance is established from the quantum destructive interferences between the transition probability amplitudes Arimondo:1996 (); Fleischhauer:2005 () as seen in Fig. 2(b). The system is placed in the dark state uncoupled from the driving fields. Note that at exact resonance, , when . These regimes are characterized , with for the Dark Resonance, and for the EIT resonance.
The FF lineshape plotted in Fig. 2(c) is originated when one Rabi frequency is much larger than the second one, and in presence of an optical detuning from the excited state(, ). Two resonances appear in the FF spectrum, one broad corresponding to the saturated one-photon resonance. The second sharp feature exhibits a characteristics asymmetric response, highly sensitive to changes in the system parameters and centered around the Raman detuning. Its minimum is associated to the DR, or EIT dip, while the maximum is the Raman peak, or bright resonance, associated to the preparation of the coherent superposition of and states coupled to the driving electromagnetic fields. Both the EIT dip and the Raman peak are manifestations of the interference between the one-photon and two-photon amplitudes Lounis:1992 (). The asymmetry of the FF profile is reversed by changing the relative ratio between the Rabi frequencies.

Figure 3: (Color online) Frequency shifts of the Dark/EIT dip and Raman peak observed on the excited state population versus the optical detuning from Eq. (19)(solid blue and red lines, respectively), and from the numerical integration of Eq. (4) (solid dots and open dots , respectively). Parameters are , , and . In (a) ; in (b) leads to a vanishing shift for the EIT resonance.

iii.2 Raman linewidth

The sub-natural EIT resonance of Fig. 2(b) experiences a linewidth which is power broadened by the optical saturation rate of Eq. (9). Let’s note that such power broadening is important even for a laser intensity where the saturation is negligible on the optical transitions. In fact ref. Kocharovskaya:1990 () introduced a coherence saturation intensity, defined by , smaller than the optical saturation intensity. The exact expression of the Raman halfwidth is

(14)

where the factor is

(15)

is very small () in a quasi-resonant laser interaction (according to the condition ) and a pure radiative process. As long as , determines the half-linewidth of the sub-natural resonance. In that regime and for the pure radiative case, is well approximated by .

iii.3 Raman frequency-shift

The frequency shift correcting the Raman detuning condition is given by

(16)

with the light-shift (LS) expression including the saturation effect given by

(17)

Notice that the second term of the above expression vanishes for the symmetric scheme, i.e., with (pure radiative process). In that case, the first term of could be associated to the light-shift expression, as pointed out in Brewer:1975 (). The decoherence shift (DS) depends on the rate as

(18)

Let us emphasize that is always null at resonance note ().

iii.4 Approximated frequency-shifts of EIT and FF resonances

Instead of the previous Section exact expression correcting the Raman detuning condition in the denominator, it is useful to derive the effective shift of the Dark/EIT resonance minimum, which is very relevant for precision spectroscopy or clock resonance. The calculation of the DR/EIT and FF frequency shifts requires to examine Eq. (8) with the Raman detuning as a free parameter. A valid approximation for the EIT and Raman-peak shifts in various excitation configurations can be found when optical detuning are tuned around the Raman condition . A differentiation of Eq. (8) versus the parameter leads to roots of a quadratic equation defining the following extrema of the EIT/FF lineshapes:

(19)

The solutions refer to the extrema of the FF lineshape. The minus (plus) sign holds for EIT dip (Raman Peak) when , and the opposite when .

Figure 4: (Color online) Three-level spectra versus the Raman detuning observed on the population using Eq. (21), solid blue line, and the numerical integration of Eq. (4), solid dots (). In (a) AT resonance, in (b) Lamb dip lineshape, and in (c) FF resonance. System parameters as in Fig. 2. In (a) and (b) Rabi frequencies as in Fig .2. In (c) , , and .
Figure 5: (Color online) Population inversion resonance between the and states, monitored on and , for unbalanced decay rates, , . Solid dots () are the result of the numerical integration of Eq. (4). Other parameters are , , and .
Figure 6: (Color online) Frequency shift of the population inversion resonance observed on (or ) as derived from Eq. (25) (solid blue line) and frequency-shift of the EIT resonance from Eq. (19) (solid red line) versus the optical detuning , for unbalanced decay rates. Solid dots () are from the numerical integration of Eq. (4) with parameters , and . In (a) , . Note that the shift is indistinguishable from the shift. In (b) , .

Fig. 3 shows the continuous-wave frequency shift versus the common optical detuning using Eq. (19). Results for both and are presented. The EIT decoherence shift, proportional to exhibits a linear dependence on the optical detuning when radiative decay terms are symmetrical. As pointed in Orriols:1979 (), when the decoherence term vanishes (), there is no shift of the EIT minimum note2 () except if we take into account external small off-resonant level contributions Declercq:1983 (). The Raman peak shift produced by light and decoherence varies with the inverse of the optical detuning as discussed by Stalgies:1998 () and observed experimentally in Siemers:1992 (). For a quasi-resonant interaction we can further simplify the above expression for the Raman Peak and the EIT dip when and . Near the two-photon resonance, the shifts of Eq. (19) can be accurately approximated as

(20)

The EIT dip frequency-shift is thus roughly given by the product of the decoherence rate and the Raman frequency shift divided by the linewidth of the sub-natural resonance. Such a dependence was pointed out by Santra:2005 () and by Wynands:1999 () based on a theoretical analysis of Fleischhauer:1994 (), and as mentioned in MacDonnell:2004 (), was earlier derived in Kofman:1997 ().

Iv Steady-state lineshapes of clock state populations

iv.1 The two-photon resonance

We focus now on clock-state resonances observed on the , populations and linked to the Raman coherence between those states. The clock state populations may be expressed in an exact form similar to that of Eq. (8) as

(21)

where

(22)

and

(23)

with

(24)

Fig. 4 reports the population resonance under various saturation conditions, matching the AT, EIT and FF lineshapes of Fig. 2. Notice that the EIT regime of Fig. 4(b) corresponds to the case of Rabi frequencies smaller than the natural decay rate of the excited state. The ”Lamb-dip” like lineshape for the resulting quasi saturated transition can conveniently be observed in the three-level configuration. This method of spectroscopy without Doppler broadening was proposed and experimentally accomplished by Javan and Schlossberg in Schlossberg:1966 (); Schlossberg-bis:1966 (). In such situation, the dip can be narrower than the homogeneous linewidth of the population resonance as in Fig. 4(b).
The clock state populations depend strongly on the normalized branching ratio difference and on the Rabi frequencies driving atomic or molecular transitions. A numerical analysis of Jyotsna:1995 () demonstrated the occurrence of a strong population transfer for unequal decay rates. Fig. 5 shows the steady-state complete population transfer for and using the asymmetric decay rates associated to an alkaline-earth three-levels system as strontium atoms Santra:2005 (); Zanon-Willette:2006 (). A large coherent population transfer is achieved when for or for .

iv.2 Approximated frequency-shift of the two-photon resonance

In an alkaline-earth frequency clock probing scheme, the large population transfer regime of Fig. 5 may be used to detect one lower state population, or the population difference between clock states. Thus, it is important to derive the two-photon shift also in this scheme. An analysis equivalent to the derivation of Eq. (19) when , we obtain, the frequency shift of . For , we make a similar derivation also using the population conservation condition. We obtain the following expression for and similarly for

(25)

For the specific radiative configuration of alkaline-earth species shown in Fig. 5, only the solution with the minus sign is needed but solutions generally refer to the extrema of a dispersive lineshape. Our standard choice of the laser detunings and introduces a very small difference in expressions of the frequency-shifts affecting each clock state population. For a quasi-resonant interaction when with and , the expression may be simplified to yield:

(26)

The frequency shift versus the common mode optical detuning affecting the (equivalently ) resonance is plotted in Fig. 6(a) and (b) for two particular ratios of the relaxation rates by spontaneous emission. In both cases, the shift of the two-photon resonance measured on the or observables has a dispersive lineshape versus the optical detuning . The slope is completely reversed owing to a nonlinear behavior when the ratio as in Fig. 6(b). A comparison with the frequency-shift of the excited state is also included in the figure. Notice the difference in the / shifts for the case of a large asymmetry in the spontaneous decay rates.

V Steady-state lineshape of Raman coherence

Figure 7: (Color online) (a) Steady-state lineshapes of the square modulus versus using Eq. (27) for , , , and . In (a) ; in (b) ; in (c) . Solid dots () are from the numerical integration of Eq. (4).
Figure 8: (Color online) (a) Frequency shift of the Raman coherence resonances observed on the as derived from Eq. (29) (solid blue line) versus the optical detuning . In (a) we have a symmetric radiative configuration with where , and . In (b) we have an asymmetric radiative configuration with , where , and . Solid dots () are from the numerical integration of Eq. (4).

v.1 The Raman coherence resonance

We are now focusing on the steady state Raman coherence resonance given by

(27)

with

(28)

When the dipole transition is allowed, the Raman coherence resonance can be detected in several manners. If we deal with alkaline atoms such as Cs or Rb, the hyperfine Raman coherence might be detected as a microwave emission proportional to , inserting the atomic medium into a micro-wave cavity Vanier:1998 (). Fig. 7 shows versus the Raman detuning condition when at different values of . The dispersive behavior of leads to a second resonant peak, appearing when as seen in Fig. 7(b) and Fig. 7(c).

v.2 Approximated frequency-shifts of the Raman coherence resonance

We derive here an accurate expression for the frequency-shifted resonance of when . A cubic equation is derived from the analytical differentiation of of Eq. (27) with respect to the Raman detuning . Using Cardan’s solutions, the three roots are written as:

(29)

with , the and given by

(30)

and

(31)

Eq. (29) allows us to obtain the frequency shift as a function of the common mode optical detuning plotted in Fig. 8. However an estimate of that shift is obtained looking only at the real part of the coherence solution which mainly describes the lineshape emission. From the square modulus of the real part , simple cubic solutions for the Raman coherence frequency-shift can be derived as:

(32)

or

(33)

When and , the Raman coherence frequency-shift expression corresponding to the maximum emission becomes:

(34)

For that case, we recover the usual dispersive shape related to the light-shift affecting clock states. The frequency shift versus the common mode optical detuning derived from Eq. (29) is shown in Fig. 8. for the case of a symmetric radiative configuration with while in (b) for . The central dispersive feature, related to the Raman shift expression , corresponds to the maximum of the coherent emission. Other branches of the shift correspond to the extrema of the second resonance appearing for as from the lineshape simulation of Fig. 7(b). A direct comparison of Fig. 8(b) with frequency shifts reported in Fig. 6 with similar conditions, yield to Raman coherence shifts larger by more than an order of magnitude than population frequency shifts.

Vi Dark Resonance fringes

vi.1 Pulsed regime lineshape

The clock operation may be based on a pulsed Raman-Ramsey scheme, illustrated in Fig. 9, with beating oscillations observed whichever variable is monitored. This detection approach originally introduced in Thomas:1982 (), was refined in Zanon:2005-PRL (); Zanon:2005-IEEE () and discussed in refs Zanon-Willette:2006 (); Yoon:2007 (); Yudin:2010 (). It allows to reach a higher precision in the clock frequency measurement, as typical of the Ramsey fringes. The present work focuses on the laser pulse scheme where the first pulse is long enough to allow the atomic or molecular preparation into the dark state superposition and the second delayed short pulse probing that superposition. Because the pumping time of Eq. (5) is required to reach the steady-state atomic or molecular preparation into the dark state, the length of the first pulse should satisfy . From a mathematical point of view the steady state solution of the three-level optical Bloch equations may thus be used as initial condition for determining the evolution at later times. At time the laser fields are switched off in order to allow for a free evolution over the time T. Finally a readout operation is performed by applying a short pulse whose time duration is limited by . In this adiabatic regime where , the short probe pulse operation is well described using the limit. For a readout pulse as long as the preparation pulse, all atoms or molecules are repumped into a new dark state erasing interference fringes. Interferences fringes are detectable on all observables as a function of the time delay, with very short readout pulses required for optical coherences and the excited state population fraction. Instead longer probing times are required for monitoring fringes on lower state populations due to slow time evolution of the clock states and the Raman coherence.

Figure 9: (Color online) Pulsed Dark Resonance detection scheme to perform high resolution spectroscopy of three-level systems. is the Ramsey time when both laser light fields are switched off. The first pulse is long enough to reach the stationary regime. During the second pulse, the probing time can be short to observe Dark Resonance fringes or long to recover a cw Dark Resonance as a new preparation stage for the next pulse.
Figure 10: (Color online) (a) Dark Resonance fringes in the weak field limit. (b) Frequency span of the lineshape. Lines from Eq. (35) and dots from Bloch’s Eqs. (4). Common parameters: , , and . In (a) and (b) Rabi frequencies , free evolution time , probe time . Very good agreement between Eq. (4) and Eq. (35) results.
Figure 11: (Color online) DR lineshapes in strong laser fields computed from Eq. (35) versus Raman detuning for different Ramsey time . Parameters , ,