Frequency shift by optical coherent control

PHYSICAL REVIEW A 81, 023405 (2010)

Frequency shift by optical coherent control Emilio Ignesti,1 Roberto Buffa,1,2 Lorenzo Fini,3,4 Emiliano Sali,3,4 Marco V. Tognetti,2 and Stefano Cavalieri3,4 1

Dipartimento di Fisica, Universit`a di Siena, Via Roma 56, I-53100 Siena, Italy 2 CNISM, Unit`a di Siena, Via Roma 56, I-53100 Siena, Italy 3 Dipartimento di Fisica, Universit`a di Firenze and CNISM, Via G. Sansone 1, I-50019 Sesto Fiorentino, Firenze, Italy 4 European Laboratory for Non-linear Spectroscopy (LENS), Universit`a di Firenze, Via N. Carrara 1, I-50019 Sesto Fiorentino, Firenze, Italy (Received 15 July 2009; published 8 February 2010) We report the experimental observation of an optically controllable shift of the central frequency of a laser pulse, using a scheme based on dynamical electromagnetically induced transparency. This is evidence of frequency shift controllable by a coherent process. Original theoretical results are in agreement with the experimental data. DOI: 10.1103/PhysRevA.81.023405

PACS number(s): 32.80.Qk, 42.50.Gy

In recent years a great effort has been devoted to all-optical control of several characteristics of the propagation of a light beam in a medium, such as its velocity, absorption, storage, and retrieval. The possibility of temporal shaping has also been investigated. The technique of electromagnetically induced transparency (EIT), proposed in 1990 and based on quantum interference [1], appears as a powerful technique in order to obtain all these degrees of control [2–9] due to its intrinsic capability to modify the optical characteristics of a prepared medium. Other techniques have been proposed and tested for optical control, such as coherent population oscillations [10,11], stimulated Brillouin and Raman scattering [12–17], spectral hole burning [18], and double absorbing resonances [19]. Very recently, another issue arose in the field of optical control: the possibility to shift the central frequency of a laser pulse. It was first shown theoretically that in a photonic crystal resonator the central frequency of a pulse can be changed dynamically by optically varying the refractive index of the medium while the pulse propagates inside it [20]. Two experimental demonstrations of this process have also been obtained using silicon resonators [21,22]. In this article we report an experimental observation of optically controllable shift of the central frequency of a laser pulse using a coherent process. Our starting idea was to explore the possibility of achieving a controllable frequency shift of a light pulse by inducing a dynamical change in the optical characteristics of an EIT-modified medium, in particular its dispersive properties. In this work we achieved this result by inducing a dynamical change of the EIT medium by using a time-dependent control field. When a second, probe, field propagates through the medium slightly detuned from resonance, we obtain a controllable shift of its central frequency ranging from −12 to +13 GHz. Figure 1 shows our experimental apparatus and, in the inset, the atomic levels and the transitions involved. The system under consideration is a three-level ladder scheme in sodium, involving the atomic states |1
= |2p 6 3s J = 1/2
, |2
= |2p6 3p J = 1/2
, and |3
= |2p6 3d J = 3/2
. The probe field, whose wavelength λp = 2π c/ωp can be tuned across the resonance of the transition |1
–|2
in vacuum at λ12 = 589.756 nm, is provided by a frequency-tunable multimode dye laser pumped by a frequency-doubled Q-switched Nd:YAG laser at a repetition rate of 10 Hz. The dye laser pulses have a measured spectral bandwidth δω/2π = 1.8 GHz and a multipeaked temporal structure of few nanoseconds of 1050-2947/2010/81(2)/023405(4)

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duration. The control field at λc = 2π c/ωc = 818.550 nm, resonant with the transition |2
–|3
, is provided by a frequencytuneable, single-longitudinal-mode titanium-sapphire (Ti:S) laser [23] delivering pulses with temporal full width at half maximum equal to 40 ns. The wavelength of the pulsed emission from the Ti:S laser is monitored using a wavelength meter with a resolution of 1 pm. The same instrument is also used to measure the central wavelength of the probe pulse, before and after the propagation in the cell, during the experiment of frequency shift. The two laser beams are linearly polarized along the same direction and they overlap, both temporally and spatially, inside the cell. A counterpropagating configuration is arranged for the purpose of reducing the effect of Doppler broadening. The temporal synchronization of the laser pulses is obtained by mutually adjusting the triggers of the two Nd:YAG pump lasers. The sodium sample is contained in a cylindrical cell heated up to a maximum temperature of 250 ◦ C for a length L = 1 m, corresponding to an estimated density-length product N L ≈ 1015 –1016 cm−2 . The measurements were done with a probe-pulse peak intensity inside the cell of approximately Ip  0.8 kW/cm2 . The control-pulse peak intensity was varied from Ic  25 kW/cm2 to Ic  150 kW/cm2 .

Ti:Sapphire ring cavity

Wavelength meter

FIG. 1. (Color online) Experimental setup. ECDL, extendedcavity diode laser; ND, neutral-density filter; DM, dichroic mirror; GP, glass plate. (Inset) Scheme of sodium levels involved. ©2010 The American Physical Society

EMILIO IGNESTI et al.

PHYSICAL REVIEW A 81, 023405 (2010)

16

12 Frequency shift (GHz)

Frequency shift (GHz)

12

16 (a)

8 4 0 -4 -8 -12

8 4 0 -4 -8 -12

12

12 (b)

(b)

8

8 Frequency shift (GHz)

Frequency shift (GHz)

(a)

4 0 -4 -8 -12

4 0 -4 -8 -12

-16 -50 -40 -30 -20 -10 p

0

10

20

30

40

-16

50

-50 -40 -30 -20 -10

(pm) p

0

10

20

30

40

50

(pm)

FIG. 2. (Color online) Measured values of the central frequency shift as a function of the detuning of the probe laser from resonance at a temperature of T = 250 ◦ C for several control field peak intensities. Black squares, Ic = 150 kW/cm2 ; solid red circles; Ic = 50 kW/cm2 ; open blue circles, Ic = 25 kW/cm2 . The results of graph (a) were obtained with the probe pulse overlapped with the rising edge of the control pulse, whereas in (b) the probe pulse was overlapped with the falling edge of the control pulse. The insets show the temporal overlapping of the two pulses at z = 0 (cell entrance) in the two cases. The dashed line is the control pulse, while the solid line is the probe pulse. The two beams are arranged in a counterpropagating configuration.

FIG. 3. (Color online) Measured values of the central frequency shift as a function of the detuning of the probe laser from resonance for a control peak intensity Ic = 150 kW/cm2 and for several temperatures of the sample. Black squares, T = 250 ◦ C; solid red triangles, T = 225 ◦ C; open blue squares, T = 200 ◦ C. The results of graph (a) were obtained with the probe pulse overlapped with the rising edge of the control pulse, whereas in (b) the probe pulse was overlapped with the falling edge of the control pulse. The insets show the temporal overlapping of the two pulses at z = 0 (cell entrance) in the two cases. The dashed line is the control pulse, while the solid line is the probe pulse. The two beams are arranged in a counterpropagating configuration.

Figure 2 shows the results that we obtained at T = 250 ◦ C for different values of the control peak intensity, while Fig. 3 shows the results obtained with Ic = 150 kW/cm2 and different sodium temperatures. In both figures, panel (a) shows the results obtained with the probe pulse overlapping the rising edge of the control pulse, whereas panel (b) shows the results obtained with the probe pulse overlapping the falling edge of the control pulse: the two sets of data are approximately symmetrical with respect to the vertical axis passing through the resonance wavelength, as expected. As the probe field propagates and the control field intensity increases [see insets of Figs. 2(a) and 3(a)], the central frequency of the probe field shifts in one direction, while when the control field intensity decreases [see insets of Figs. 2(b) and 3(b)], the central frequency of the probe field shifts in the opposite direction. The sign of the effect also depends upon the sign of the detuning: This provides a further parameter for the control. We observe that the maximum frequency shift scales with both Ic and the sodium density, showing the possibility of controlling optically

the central frequency of our pulse. We obtain a maximum frequency shift of 13 GHz, with an output energy of 1/3 the value measured at resonance. We stress that this maximum value of 13 GHz is several times (7) the laser bandwidth, showing that the shifting process is not due to a filtering effect of the EIT line shape. This effect of frequency shift is a completely new process compared to all others connected to EIT. In particular, it is completely different from EIT-enhanced frequency conversion, as in that case one has the generation of discrete new frequency components, whereas in our case we demonstrate the possibility of continuously shifting the central frequency of the laser pulse in a controllable fashion. Ignoring hyperfine structure, our experimental results admit a theoretical interpretation based on the solution of the probepulse propagation in the presence of dynamical EIT. If we consider a counterpropagating configuration, the probe field envelope Ep obeys the following propagation equation:

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∂Ep ωp d12 1 ∂Ep + = −iN ρ12 , ∂z c ∂t 0 c

(1)

FREQUENCY SHIFT BY OPTICAL COHERENT CONTROL

PHYSICAL REVIEW A 81, 023405 (2010)

while the strong control field envelope Ec propagates in the opposite direction unaffected by the atomic sample, that is, Ec (z, t) = Ec (t + z/c). Here N is the atomic density, and d12 is the electric dipole moment of the 1–2 transition. In the hypothesis of weak probe field p  c , which implies ρ11 = 1, ρ22 = 0, ρ33 = 0, and ρ23 = 0 during the interaction process, the coherence ρ12 which appears in Eq. (1) satisfies the following atomic Liouville equation: ρ˙12 = −ip − ic ρ13 − (i p + γ12 )ρ12 ρ˙13 = −ic ρ12 − (i p + γ13 )ρ13 ,

(2)

(3)

2 where, if (c − | p |)2  γnm throughout the probe-pulse propagation,
 2 2p Nωp d12 p − iγ13 ∂c 1 + χ1 = i 2   2¯h0 c p − 2c c 2 − 2 2 ∂t p c

2 ∂Ec , c Ec ∂t
 2 2p + 2c N ωp d12 2 Ec . χ2 = −   + 2¯h0 c 2 − 2 2 c

+

Im(χ10 − χ1L ) χ2L

 2   cL − 2c0 2p − 2cL = p    2 , 4¯h0 2p − 2c0 2p + 2cL + Nω 2p − 2cL d2 p 12

(4)

where χ1L and χ2L are obtained by substituting Ec = EcL = Ec (tL + L/c) into Eq. (4). Equation (7) shows that no frequency shift is present at resonance ( p = 0) and for constant control field (cL = c0 ). Moreover, it has to be remarked that, while p and c0 can be viewed as independent variables in Eq. (7), cL depends in general on p , c0 , and the temporal shape of c through Eq. (6). However, Eq. (7) provides explicit expressions in the two limits of small and large detunings. For small detunings ( p  c ), tL is independent of p and δωL shows a linear dependence on p . For large detunings ( p  c ), where tL  L/c, δωL shows a linear dependence on both the control intensity and the atomic density and it decreases as −3 p . This is exactly the qualitative behavior shown in Figs. 2 and 3. These results, hence, predict a frequency shift controllable by a coherent process. Figure 4 shows a theoretical fit of the experimental data reported in Fig. 2(a) obtained at T = 250 ◦ C and Ic = 150 kW/cm2 . The fit was performed using Eqs. (6) and (7) and treating the density N and the control laser peak intensity as free parameters. The best fit is obtained with N L = 1016 cm−2 and Ic = 190 kW/cm2 , which are values compatible with the estimated experimental data. The theoretical curve is not reported for those values of p for which we verified by a numerical integration of Eq. (2) that the first-order approximation, at the basis of Eq. (3), is not fulfilled for the entire propagation time of the probe pulse through the cell. In this range of values

c

In the hypothesis that at z = 0 the probe field is overlapped by a constant region of the control field of value Ec0 the solution of Eq. (3) is given by     t+z/c χ20 χ1 χ20 Ep (z, t) = Ep 0, ξ exp χ10 ξ − Ec dt , χ2 Ec0 χ2 0 (5)

t+z/c Ec where ξ = z + 0 dt and χ10 and χ20 are obtained by χ2 substituting Ec = Ec0 into Eq. (4). The shift of the central frequency of the probe pulse is given by the imaginary part of the coefficient of the term inside the exponential in Eq. (5) which depends linearly upon the time t. We can think of this process as due to a dynamical EIT inducing a time-dependent phase on the probe field. Therefore, by developing the argument of the exponential around the exit time tL of the probe pulse from the cell, defined implicitly trough  tL +L/c dt , (6) L= 2 2p + 2c Nωp d12 2 0 + 2 2¯h0 c ( 2 − 2 ) c p c

16 12

Frequency shift (GHz)

p

δωL =

(7)

where p = d12 Ep /(2¯h) and c = d23 Ec /(2¯h) are the Rabi frequencies, d23 is the electric dipole moment of the 2–3 transition, p is the the static part of the probe detuning from resonance (i.e., the probe detuning at the cell input), and γnm ’s represent all kinds of dephasing rates. It is possible to find an analytical solution for Eq. (2) to the first order in δωp / c and δωc /c , where δωp,c are the spectral bandwidths of the probe and the control field, respectively [24]. By numerical integration of Eq. (2) we have verified that this first-order approximation provides correct results for small ( p  c ) and large ( p  c ) detunings. In this case Eq. (1) can be cast in the form ∂Ep 1 ∂Ep 1 ∂(χ2 Ep ) + = χ1 Ep + , ∂z c ∂t Ec ∂t

we find that at the cell output the central frequency of the probe pulse is shifted by an amount

8 4 0 -4 -8 -12 -16 -50

-40

-30

-20

-10 p

0

10

20

30

40

50

(pm)

FIG. 4. (Color online) Black squares, experimental data obtained at T = 250 ◦ C and Ic = 150 kW/cm2 as a function of the detuning of the probe laser from resonance. Red curves, theoretical fit performed using Eqs. (6) and (7) and treating the density N and the control laser peak intensity as free parameters. The curve is reported only in the range of validity of Eqs. (6) and (7).

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EMILIO IGNESTI et al.

PHYSICAL REVIEW A 81, 023405 (2010)

of p , Eqs. (6) and (7) lose their physical validity and any attempt of comparison between experimental and theoretical results is meaningless. The comparison shown by Fig. 4 reveals a very good agreement between experimental and theoretical results in the range of validity of Eqs. (6) and (7). Another issue of interest is whether and how the spectrum of the probe pulse changes upon propagation in the EIT medium. Indeed, what happens to our pulse spectrum upon propagation in the cell is under study in our group, and we plan to present the results in a forthcoming article. Preliminary experimental results, obtained for small detunings ( p  c ), show that the probe-pulse bandwidth increases in the conditions of Figs. 2(a) and 3(a). The Fourier transforms of Eq. (5) confirm this observation.

In conclusion, we reported evidence of an all optically controlled shift of the central frequency of a laser pulse obtained using a coherent process. The experimental data show that the optical control of the shift can be obtained by varying the intensity of a control laser in an EIT scheme. The maximum shift that we obtain is several times the laser bandwidth. Our experimental data are in agreement with original theoretical predictions and are presented for several values of density and detuning from exact two-photon-resonance conditions. These experimental results show the possibility of controlling optically, through an EIT scheme, not only velocity, absorption, storage, and retrieval of a light pulse, but also the shift of its central frequency.

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