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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 8 — Apr. 14, 2008
  • pp: 5350–5361
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Investigation of quantum coherence excitation and coherence transfer in an inhomogeneously broadened rare-earth doped solid

Byoung S. Ham  »View Author Affiliations


Optics Express, Vol. 16, Issue 8, pp. 5350-5361 (2008)
http://dx.doi.org/10.1364/OE.16.005350


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Abstract

Quantum coherence excitation onto spin ensembles by resonant Raman optical fields and coherence transfer back to an optical emission are discussed in a three-level optical system composed of inhomogeneously broadened spins, where the spin decay time is much slower than the optical decay time. Dynamic quantum coherent control of the spin excitations and coherence conversion are also discussed at a strong coupling field limit for practical applications of optical information processing.

© 2008 Optical Society of America

1. Introduction

Coherence excitation in an optical medium by resonant optical fields occurs whenever an atomic population transfer is involved: absorption. The medium’s optical parameters such as optical susceptibility are modified during the action of optical excitations, so that a dynamic behavior of the refractive index of the medium can be utilized. When two orthogonal photons interact with an optically dense medium, the absorption spectrum can be modified to be transparent. This nonabsorption resonance phenomenon is called coherent population trapping [1

1. H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr., “Coherent trapping of atomic populations,” Opt. Lett. 3, 218–220 (1978). [CrossRef] [PubMed]

] or electromagnetically induced transparency (EIT) [2

2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

]. Thus, two-photon excitations have been studied for the refractive index modification toward a group velocity control of traveling light [3–6

3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

]. The group velocity modification gives a great benefit to optical information processing such as an optical buffer memory [7

7. F. Xia, L. Sekaric, and L. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006). [CrossRef]

] and quantum information [8

8. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005). [CrossRef] [PubMed]

] owing to lengthened interaction time based on enhanced nonlinear optics.

In an optical medium experiencing refractive index modification, photon density per unit volume is inversely proportional to the group velocity of traveling light. With a dramatic decrease of the group velocity of a traveling light pulse, highly efficient nonlinearity (giant Kerr effect) has also been studied [9–15

9. S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611 (1999). [CrossRef]

] and presented for potential applications of quantum information [13–18

13. D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split ublevels,” Phys. Rev. A 65, 033833 (2002). [CrossRef]

]. In this case, the resulting strong modification of the dispersion spectrum can be applied for π-phase shift on a weak signal field, while keeping nearly zero absorption under EIT. Thus, the EIT-based giant Kerr effect can be a good candidate for Schrödinger’s cat [13

13. D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split ublevels,” Phys. Rev. A 65, 033833 (2002). [CrossRef]

], quantum entanglement [14

14. M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev. A 67, 023811 (2003). [CrossRef]

], quantum switching [16

16. S. E. Harris and Y. Yamamoto, “Quantum switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998). [CrossRef]

, 17

17. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84, 4080–4083 (2000). [CrossRef] [PubMed]

], and quantum wavelength conversion [10–12

10. Y. Zhang, A. W. Brown, and M. Xiao, “Matched ultraslow propagation of highly efficient four-wave mixing in a closely cycled double-ladder system,” Phys. Rev. A 74, 053813 (2006). [CrossRef]

,18

18. S. A. Moiseev and B. S. Ham, “Quantum manipulation of two-color stationary light: Quantum wavelength conversion,” Phys. Rev. A 73, 033812 (2006). [CrossRef]

].

Researches on quantum coherent control in an optical system composed of spin inhomogeneous broadening have been carried out in a rare-earth doped solid medium [4

4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

,17

17. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84, 4080–4083 (2000). [CrossRef] [PubMed]

,19–22

19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

]. The spin inhomogeneous broadening is a common phenomenon in solid media especially rare-earth doped crystals, whose spin decay rate is much slower than the optical counterpart. The slow spin decay rate is an origin of the (optical) spectral hole-burning phenomenon [24

24. K. Holiday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SIO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993). [CrossRef]

]. The hole-burning phenomenon gives a great benefit using a rare-earth doped crystal to optical information processing owing to spectral modifications [4

4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

, 17

17. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84, 4080–4083 (2000). [CrossRef] [PubMed]

, 19–22

19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

]. Thus, the rare-earth doped crystals have been intensively studied for quantum optical memories utilizing photon echoes [20–23

20. L. Rippe, M. Nilsson, S. Kroll, R. Klieber, and D. Suter, “Experimental demonstration of efficient and selective population transfer and qubit distillation in a rare-earth-metal-ion-doped crystal,” Phys. Rev. A 71, 062328 (2006). [CrossRef]

] and EIT [4

4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

, 19

19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

]. Theoretical studies of quantum optical memories using EIT in rare-earth doped solids, however, have not been performed well yet. Here, we discuss the role of the spin inhomogeneous broadening to the quantum coherent control such as the light-matter coherence conversion process [4

4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

, 17–19

17. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84, 4080–4083 (2000). [CrossRef] [PubMed]

,22

22. G. He′tet, J. J. Longdell, A. L. Alexander, P. K. Lam, and M. J. Sellars, “Electro-optic quantum memory for light using two-level atoms,” Phys. Rev. Lett. 100, 023601 (2008). [CrossRef]

]. Understanding the coherent transfer mechanism between photons and spin ensembles (inhomogeneously broadened spins) is important for the intensive studies of slow-light based optical routing [25

25. B. S. Ham, “Observations of delayed all-optical routing in a slow light regime,” Phys. Rev. Lett. (To be published); ibid. arXiv:0801.3501 (2008). [PubMed]

] as well as quantum optical data storage time versus spin decay rate. Coherence excitations by optical Raman fields onto the spin ensemble and the spin coherence transfer back into optical emission will be discussed below.

2. Theory

The density matrix approach is an excellent tool to denote ensemble behavior of light-matter interactions in an optical system. In a lambda-type three-level optical system interacting with two optical Raman fields ΩP and ΩC (see Fig. 1(a)), the time-dependent density matrix equation is denoted by [26

26. M. Sargent III, M. O. Scully, W. E. Lamb, Jr., and Laser Physics, (Addison-Wesley, 1974), Chap. 7.

]:

dρdt=iħ[H,ρ]12{Γ,ρ},
(1)

where {Γ,ρ}=Γρ+ρΓ. Interaction Hamiltonian H is

H=ħ{δ111δ222δ33312(Ω113+Ω223)+H.c.},
(2)

where δ131P, δ232C, ωijij, and ωP and ωC are the frequencies of the probe (ΩP) and coupling (ΩC) fields, respectively. In Fig. 1(a), the thick line on level |2> stands for spin inhomogeneous broadening between |1> and |2>.

Fig. 1. (a). Energy level diagram and (b). Pulse sequence: ΩPC=20kHz; γ3132=50kHz; Γ3132=0.5kHz; γ21=0.2kHz; Γ21=0; Δspin inh=50kHz.

3. Results and discussions

Figure 1(b) shows the pulse sequence of the applied Raman optical fields for Fig. 1(a). The coupling field ΩC and the probe field ΩP temporally overlap each other. Initially equal population distribution between two low lying levels |1> and |2> are assumed: ρ11=1/2; ρ2233=1/2. Even if optical transitions ΩP for (|1>-|3>) and ΩC for (|2>-|3>) satisfy the resonant two-photon condition, the spin inhomogeneous broadening between |1>-|2> makes an intrinsic detuning. We choose only δ2 as an intrinsic detuning for the inhomogeneously broadened spins to the two-photon optical fields (in this case only to the coupling field ΩC) for the following numerical calculations.

Fig. 2. Spin inhomogeneous broadening.

In Fig. 2, a Gaussian shaped spin inhomogeneous broadening is assumed. The optical system, however, is homogeneous. For the calculations using the density matrix equations given in Eq. (1), the spin inhomogeneous broadening, whose full width at half maximum is 50 kHz, is divided into 180 groups from δ2=-90 kHz to δ2=+90 kHz at a step of 2 kHz.

For the numerical calculations, experimental parameters in Pr3+ doped Y2SiO5 are used [19

19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

, 24

24. K. Holiday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SIO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993). [CrossRef]

, 27

27. R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995). [CrossRef]

]. Each applied laser Rabi frequency of ΩP and ΩC is 20 kHz for a weak field limit, where the optical phase decay rate is γ3132=50 kHz. The atom-field interaction time of 60 µs) is identified as much longer than the optical phase decay time T2 (T2=1/(πγ32=6.4 µs).

In Fig. 3, numerical calculations for optical (i13)) and spin coherence (r12)) are presented. For the calculations all groups of atoms (spins) with different detuning δ2 are considered. As seen in Figs. 3(a) and 3(b), the probe field absorption (i13)) and spin coherence (r12)) are affected by both population differences between levels |3> and |1> and between |2> and |1>. According to the theory of coherent population trapping (in a weak field limit) the maximum spin coherence (r12)) is -0.5, where the current coherence magnitude in Fig. 3 should be reduced by a higher optical decay rate γ and the spin inhomogeneous broadening.

Fig. 3. Coherence vs. (a) optical population difference, and (b) spin population difference.

In Fig. 4, however, we demonstrate population-difference-independent spin coherence excitation for a detuning zero atom group (center in Fig. 1) satisfying two-photon resonance for ΩP and ΩC. Unlike Fig. 3 discussed for population-dependent coherence excitations, here the spin coherence is independent of the population difference between levels |2> and |1>. This is because the optical system in Fig. 1(a) is population shelved, in which the optical population decay rate is much weaker than the applied optical Rabi frequencies. Moreover, both optical Rabi frequencies are exactly the same in magnitude and phase. Thus, the two optical fields incur population difference only between levels either |1> and |3>, or |2> and |3>, under perfect resonance condition (δ2=0).

According to the density matrix equations, the time derivative of the spin coherence is composed of three parts (spin coherence ρ12, optical coherence ρ13, and optical coherence ρ23), where the population difference should not be a factor. However for the optical coherence the population difference must be considered as shown in Fig. 3. This means that the population difference between two low-lying ground levels is not a necessary condition to induce the spin coherence excitation by resonant two-optical fields.

dρ12dt=i2ΩCρ13+i2ΩPρ32i(δ1δ2)ρ12γ12ρ12,
3(a)
dρ13dt=i2ΩP(ρ11ρ33)i2ΩCρ12iδ1ρ13γ13ρ13,
3(b)
dρ23dt=i2ΩPρ21i2ΩC(ρ22ρ33)iδ2ρ23γ23ρ23.
3(c)
Fig. 4. Coherence vs. interaction time for two-photon resonance (δ2=0). Dotted line: ρ2211; pink (red) curve: ρ1122=0.5:0.5 (1:0); blue curve: r12).

Figure 5 shows all groups of atoms for spin coherence excitation versus ground levels population difference. As seen in Fig. 5, at line center of the spin inhomogeneous broadening, the spin coherence is excited even without population difference (see the red-dotted lines). This fact is important for coherence retrieval processes such as photon echoes using reversible optical inhomogeneous broadening for the population reverse in a two-level system, because no population difference is created before and after the retrieval action (This is beyond the scope of this paper and will be discussed elsewhere). For all other atoms which two-photon resonance is not satisfied, the spin coherence excitation depends on the population difference (ρ2211) via those between optical transitions as discussed in Fig. 4.

Fig. 5. (a). Spin population difference, and (b) Spin coherence
Fig. 6. Spin coherence (r(ρ12)) evolution vs. two-photon detuning δ2.

For detuned atoms (spins) in Fig. 5, it is clear that the excited coherence oscillates with δ2 as a function of interaction time (see Fig. 6): The more detuning it has, the faster oscillation it experiences. Of course its amplitude becomes weaker as the detuning gets wider. As is well known in the area of optical transient phenomena such as free induction decay as shown in Fig. 3, the overall coherence decay time becomes shorter than the spin phase decay time T2 Spin as the detuning becomes wider: Here T2 Spin is 1/(πγ21)=1,590 µs. Thus, Figs. 3 and 6 prove a spin coherence transient effect induced by two optical fields under the spin broadened system: nutation. This optical field-induced spin coherence transient effect is important in analyzing the EIT-based nonlinear Kerr effect such as nondegenerate four-wave mixing processes as experimentally demonstrated already [4

4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

,17

17. B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84, 4080–4083 (2000). [CrossRef] [PubMed]

,19

19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

,28

28. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Efficient phase conjugate via two-photon coherence in an optically dense crystal,” Phys. Rev. A 59, R2583– R2586 (1999). [CrossRef]

]. Unlike spin homogeneously broadened atomic systems [3

3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

,5

5. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003). [CrossRef] [PubMed]

,6

6. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493, (2001). [CrossRef] [PubMed]

,10–12

10. Y. Zhang, A. W. Brown, and M. Xiao, “Matched ultraslow propagation of highly efficient four-wave mixing in a closely cycled double-ladder system,” Phys. Rev. A 74, 053813 (2006). [CrossRef]

], the read-out pulse must come in a shorter time compared with the spin T2 [28

28. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Efficient phase conjugate via two-photon coherence in an optically dense crystal,” Phys. Rev. A 59, R2583– R2586 (1999). [CrossRef]

].

Fig. 7. (a). Spin coherence vs. interaction time for initial condition of ρ11=1 and ρ22=0. (b) Spin coherence vs (c) spin population difference. ΩPP=20 kHz.

In general, however, the population-independent spin-coherence excitation is not satisfied if either balanced optical fields (ΩPC) or equal population distributions on the low lying two ground levels (ρ1122) are not provided. Figure 7 shows a general case of coherence excitation induced by two optical fields whose Rabi frequency is the same, where initial population distribution is unbalanced: ρ11=1; ρ2233=0. As seen in Fig. 7, the spin coherence excitation strongly depends on the population difference (see also Eqs. 3(b) and 3(c)).

Fig. 8. Coherence excitation vs. interaction time (a) for all atoms, and (b) only for δ2=0: ΩPP=20 kHz; ρ1122=0.5; γ3132=50kHz; Γ3132=0.5kHz; γ21=0.2kHz; Γ21=0.

Now we increase each optical Rabi frequency up to 50 kHz which is equal to the optical phase decay rate. In this case, one can expect a damped oscillation phenomenon in coherence excitations. Figure 8 shows coherence excitations versus population differences for the initial condition of equal population (ρ1122). All other parameters are the same as above. As seen in Figs. 8, spin coherence excitation (r12)) results from optical absorption (i13)) with a phase delay. The phase delay is a common characteristic of the response function in a physical system. For example, the absorption and dispersion relation is characterized as the Kramers-Kronig relation, where the absorption and dispersion are coupled to each other resulting in a response function. Here the relation between linear absorption (i13)) and two-photon dispersion (r12)) mimics the Kramers-Kronig relation as a response function. Surprisingly time evolution of the spin coherence (r12)) matches that of the population difference between the excited and ground levels (ρ3311). This is because the atom population on both levels must be Rabi flopping, and then the coherence excitation directly results from the population difference: See Eqs. (3). Thus, spin coherence excitation can be controlled only by adjusting one of the optical fields’ Rabi frequencies. Figure 8(a) depicts overall atoms, and 8b only two-photon resonant atoms (δ2=0). As seen in Fig. 8(b), the ground levels population difference is negligible for the spin coherence excitation.

Fig. 9. Coherence vs. interaction time: ΩP=10 kHz; ΩC=200 kHz; ρ1122=0.5; γ3132=50kHz; Γ3132=0.5kHz; γ21=0.2kHz; Γ21=0.

So far we have discussed resonant Raman field induced spin coherence excitations at a moderate or strong probe limit. However, optical signals in information processing, in general, are expected to be weak. Owing to the discussion in Fig. 8 that one optical laser field (coupling ΩC) can be used to control the spin coherence excitation, the following is considered: ΩP≪γ; ΩP≪ΩC; γ≪ΩC.

Figure 9 shows a quantum coherent control for spin coherence excitation by using a strong coupling field with a weak probe field. As we have discussed above, due to a very weak optical decay rate (population shelved model) and weak probe field ΩP, each atomic population on levels |2> and |3> strongly depends on the coupling field ΩC: Rabi flopping. The Rabi flopping-induced coherence oscillation at a frequency of 200 kHz is the same as the coupling Rabi frequency in Fig. 9. So is the population difference between levels |3> and |1> as wells as between |2> and |1>: See the blue dotted and red curves, respectively. Unlike the weak coupling limit (ΩC≪γ) discussed Figs. 1–8, spin coherence excitation now strongly depends on the population on level |2> via the coupling field interaction. Region A in Fig. 9 shows spin free-induction decay, where the decay time is 1/Δspin: Δspin=50 kHz.

Figure 10 represents coherence damping, whose decay time relies on the optical phase decay rate γ32 (=50 kHz), where the optical phase decay time is T2=1/(πγ32)=6.4 µs. In Fig. 10(b) and 10(c) the dotted exponential curve is for the best fit to the coherence envelope: Fig. 10(b) shows overall inhomogeneous spins, while Fig. 10(c) shows only two-photon resonance spins at line center in Fig. 2. As seen in Fig. 10, the spin coherence (r12)) evolution nearly perfectly follows the population change on level |2> (ρ22). For the calculations in Fig. 10(c), both optical and spin population decay rates are set to zero for analysis purposes. From these results, we conclude that the spin coherence excitation depends on both optical phase decay rate and Rabi flopping by the coupling field ΩC.

Fig. 10. (a). Pulse sequence, (b) overall spin coherence, and (c) two-photon resonant spin coherence.

So far we have discussed resonant Raman optical field excited spin coherence behavior for both a weak field limit and a strong field limit of the coupling field. In the strong field limit, we have demonstrated that the magnitude of the spin coherence excitation can be controlled by adjusting only the coupling field interaction time. Thus, one may seek relationship between the spin coherence and the Kerr effect in nonlinear optics. To answer to this question, the coherence transfer phenomenon will be discussed below.

Fig. 11. Coherence excitation in a lambda-type three-level system of Fig. 1(a).

Figure 11 repeats the resonant Raman field excited spin coherence shown in Figs. 1–3, but for a reference of coherence transfer from the spin coherence to the optical coherence. Here, we discuss the relation between the spin coherence and the optical coherence (emission) caused by a third optical pulse of ΩA immediately following the excitation pulse composed of ΩP and ΩC, in which ΩA has the same transition as ΩC.

In Fig. 12, three different cases of coherence transfer are demonstrated. From Fig. 12(a) to 12(c) the spin coherence magnitude created by the Raman excitation pulse is chosen in the order of degradation. The Raman excitation pulse length for Figs. 12(a), 12(b), and 12(c) is 25, 30, 40 µs, respectively, as shown in Fig. 12(d), 12(e), and 12(f). The retrieval pulse length of ΩA is 10 µs for all cases. From Figs. 12, we conclude that the spin coherence is transferred to optical coherence resulting in optical emission by the action of the retrieval pulse ΩA, and the strength of the optical coherence (magnitude of emission) is proportional to that of the spin coherence (denoted by dotted arrows). This phenomenon is actually EIT-based nondegenerate four-wave mixing experimentally demonstrated in Pr3+ doped Y2SiO5 [28

28. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Efficient phase conjugate via two-photon coherence in an optically dense crystal,” Phys. Rev. A 59, R2583– R2586 (1999). [CrossRef]

].

Fig. 12. Spin coherence retrieval to optical coherence.
Fig. 13. Coherence transfer from spin coherence to optical coherence.

For a detailed investigation of the coherence transfer process, we apply a time derivative to the spin coherence and plot the results in Fig. 13. The Eq. (4) is obtained intuitively from the delay relation in Eq. (3). For the numerical calculations in Fig. 13, Fig. 12(b) is chosen as a model. The result of Eq. (14) for the region (dash-dot, 30<t<40 µs) of retrieval pulse ΩA is denoted by blue-colored “o,” which is a perfect match to the optical coherence (red curve) obtained in Fig. 12(b) for the zero detuned spins (δ2=0, green curve). For overall spins spanned over 50kHz, the coherence transfer induced optical emission lags a little bit behind (see the black curve).

E(t)ddt[r(ρ12(t))]
(4)
Fig. 14. 3D simulations for optical absorption and spin coherence.

Figure 14 shows three-dimensional simulations of the one-photon absorption (i13)) and the spin coherence (r12)) for all inhomogeneous spins. As seen in Fig. 14, the overall one-photon optical coherence (emission; red area in Fig. 14(a)) is from the retrieval of the overall two-photon spin coherence (blue area in Fig. 14(b)) as discussed in Fig. 13. This phenomenon has already been studied experimentally in Pr3+ doped Y2SiO5 [28

28. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Efficient phase conjugate via two-photon coherence in an optically dense crystal,” Phys. Rev. A 59, R2583– R2586 (1999). [CrossRef]

].

Fig. 15. Coherent control at strong field limit: ρ1122.

So far we have discussed resonant Raman optical field excited spin coherence and its coherence transfer back to an optical emission at a weak control limit. To comply with dynamic quantum coherent control at higher speed, we need to discuss it at a strong coupling limit. As we mentioned above in Figs. 9 and 10 at a strong coupling field limit with weak probe field, both optical and spin coherence excitations depend on the control field directly affecting the population on level |2>, while remaining nearly unchanged on level |1>.

Figures 15 and 16 demonstrate dynamic coherence retrieval processes with a retrieval pulse ΩA. Figure 15 shows coherence oscillation at a strong coupling limit. Surprisingly, without population inversion between the ground level |1> and the excited level |3>, Figs. 15(b) and 15(e) show photon emission: ι(ρ13)>0; (ρ3311)<0. This is due to the coupling induced Rabi flopping on level |3>, so that the slopes of population differences for both transitions are negative (see Fig. 15(c)), which means photon emission. As discussed in Fig. 12, the retrieval pulse whose Rabi frequency is 100 kHz follows the two-photon excitation pulse composed of ΩP and ΩC with no time delay (see Fig. 16). To demonstrate a dynamic coherence transfer of the excited spin coherence to optical one-photon coherence, we choose three different moments at t=12, 15, and 20 µs (see the green dashed line in Fig. 15(b)). For the coherence transfer at t=15 µs and t=20 µs (see respectively Figs. 16(b)/(e)/(h) and (c)/(f)/(i)), the photon emission at t=20 µs is bigger because of a stronger spin coherence excited. Moreover, there are extra photon emission peaks as a result of Fig. 15(c): see pink shaded area in Fig. 15(c). This kind of dynamic coherence control is important in a population shelved system to avoid unwanted signals.

Fig. 16. Dynamic coherence transfer to optical emission.

4. Conclusion

We have presented a quantum coherent control and discussed resonant Raman optical field excited spin coherence and the spin coherence transfer back to the optical field by using a third retrieval pulse for both a weak and a strong coupling limit. In the weak coupling limit, we numerically demonstrated that the spin coherence excitation does not have to rely on a population difference between the ground levels, which is important for photon echo like quantum memories [4

4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

,19

19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

]. In a strong coupling limit, however, the spin coherence excitation nearly perfectly depends on population change induced by the coupling field only. This one-field quantum coherent control for spin excitation is important in practical applications using an optically shelved medium such as a rare-earth doped crystal. For the spin coherence transfer into optical coherence (emission), we numerically demonstrated that the magnitude of the one-photon coherence nearly perfectly depends on the magnitude of the spin coherence. For the strong coupling limit in an optically shelved model, we also theoretically proved that an emitted photon pulse train is possible without population inversion.

Acknowledgment

This work was supported by Creative Research Initiative Program (Center for Photon Information Processing) of MOST and KOSEF, S. Korea.

References and links

1.

H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr., “Coherent trapping of atomic populations,” Opt. Lett. 3, 218–220 (1978). [CrossRef] [PubMed]

2.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]

3.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594–598 (1999). [CrossRef]

4.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of Ultraslow and Stored Light Pulses in a Solid,” Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]

5.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003). [CrossRef] [PubMed]

6.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490–493, (2001). [CrossRef] [PubMed]

7.

F. Xia, L. Sekaric, and L. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006). [CrossRef]

8.

A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Nonlinear optics with stationary pulses of light,” Phys. Rev. Lett. 94, 063902 (2005). [CrossRef] [PubMed]

9.

S. E. Harris and L. V. Hau, “Nonlinear Optics at Low Light Levels,” Phys. Rev. Lett. 82, 4611 (1999). [CrossRef]

10.

Y. Zhang, A. W. Brown, and M. Xiao, “Matched ultraslow propagation of highly efficient four-wave mixing in a closely cycled double-ladder system,” Phys. Rev. A 74, 053813 (2006). [CrossRef]

11.

D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Phys. Rev. Lett. 93, 183601 (2004). [CrossRef] [PubMed]

12.

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves,” Phys. Rev. Lett. 88, 143902 (2002). [CrossRef] [PubMed]

13.

D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms with Zeeman-split ublevels,” Phys. Rev. A 65, 033833 (2002). [CrossRef]

14.

M. Paternostro, M. S. Kim, and B. S. Ham, “Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime,” Phys. Rev. A 67, 023811 (2003). [CrossRef]

15.

M. G. Payne and L. Deng, “Quantum entanglement of Fock states with perfectly efficient ultraslow single-probe photon four-wave mixing,” Phys. Rev. Lett.91, 123602 (2003); S. A. Moiseev and B. S. Ham, Phys. Rev. A 71, 053802 (2006). [CrossRef] [PubMed]

16.

S. E. Harris and Y. Yamamoto, “Quantum switching by quantum interference,” Phys. Rev. Lett. 81, 3611–3614 (1998). [CrossRef]

17.

B. S. Ham and P. R. Hemmer, “Coherence switching in a four-level system: Quantum switching,” Phys. Rev. Lett. 84, 4080–4083 (2000). [CrossRef] [PubMed]

18.

S. A. Moiseev and B. S. Ham, “Quantum manipulation of two-color stationary light: Quantum wavelength conversion,” Phys. Rev. A 73, 033812 (2006). [CrossRef]

19.

B. S. Ham, M. S. Shshriar, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997). [CrossRef]

20.

L. Rippe, M. Nilsson, S. Kroll, R. Klieber, and D. Suter, “Experimental demonstration of efficient and selective population transfer and qubit distillation in a rare-earth-metal-ion-doped crystal,” Phys. Rev. A 71, 062328 (2006). [CrossRef]

21.

A. L. Alexander, J. J. Longdell, M. J. Sellars, and N. B. Manson, “Photon echoes produced by switching electric fields,” Phys. Rev. Lett. 96043602 (2006). [CrossRef] [PubMed]

22.

G. He′tet, J. J. Longdell, A. L. Alexander, P. K. Lam, and M. J. Sellars, “Electro-optic quantum memory for light using two-level atoms,” Phys. Rev. Lett. 100, 023601 (2008). [CrossRef]

23.

S. A. Moiseev and S. Kroll, “Complete reconstruction of the quantum state of a single-photon wave packet absorbed by a Doppler-broadened transition,” Phys. Rev. Lett. 87, 173601 (2001). [CrossRef] [PubMed]

24.

K. Holiday, M. Croci, E. Vauthey, and U. P. Wild, “Spectral hole burning and holography in an Y2SIO5:Pr3+ crystal,” Phys. Rev. B 47, 14741–14752 (1993). [CrossRef]

25.

B. S. Ham, “Observations of delayed all-optical routing in a slow light regime,” Phys. Rev. Lett. (To be published); ibid. arXiv:0801.3501 (2008). [PubMed]

26.

M. Sargent III, M. O. Scully, W. E. Lamb, Jr., and Laser Physics, (Addison-Wesley, 1974), Chap. 7.

27.

R. W. Equall, R. L. Cone, and R. M. Macfarlane, “Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5,” Phys. Rev. B 52, 3963–3969 (1995). [CrossRef]

28.

B. S. Ham, M. S. Shahriar, and P. R. Hemmer, “Efficient phase conjugate via two-photon coherence in an optically dense crystal,” Phys. Rev. A 59, R2583– R2586 (1999). [CrossRef]

OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(270.1670) Quantum optics : Coherent optical effects

ToC Category:
Quantum Optics

History
Original Manuscript: February 15, 2008
Revised Manuscript: March 30, 2008
Manuscript Accepted: March 30, 2008
Published: April 2, 2008

Citation
Byoung S. Ham, "Investigation of quantum coherence excitation and coherence transfer in an inhomogeneously broadened rare-earth doped solid," Opt. Express 16, 5350-5361 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5350


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References

  1. H. R. Gray, R. M. Whitley, and C. R. StroudJr., "Coherent trapping of atomic populations," Opt. Lett. 3, 218-220 (1978). [CrossRef] [PubMed]
  2. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633-673 (2005), and references are there in. [CrossRef]
  3. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature 397, 594-598 (1999). [CrossRef]
  4. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, "Observation of Ultraslow and Stored Light Pulses in a Solid," Phys. Rev. Lett. 88, 023602 (2002). [CrossRef] [PubMed]
  5. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Superluminal and slow light propagation in a room-temperature solid," Science 301, 200-202 (2003). [CrossRef] [PubMed]
  6. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature 409, 490-493 (2001). [CrossRef] [PubMed]
  7. F. Xia, L. Sekaric, and L. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2006). [CrossRef]
  8. A. Andre, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Nonlinear optics with stationary pulses of light," Phys. Rev. Lett. 94, 063902 (2005). [CrossRef] [PubMed]
  9. S. E. Harris and L. V. Hau, "Nonlinear Optics at Low Light Levels," Phys. Rev. Lett. 82, 4611 (1999). [CrossRef]
  10. Y. Zhang, A. W. Brown, and M. Xiao, "Matched ultraslow propagation of highly efficient four-wave mixing in a closely cycled double-ladder system," Phys. Rev. A 74, 053813 (2006). [CrossRef]
  11. D. A. Braje, V. Balic, S. Goda, G. Y. Yin, and S. E. Harris, "Frequency mixing using electromagnetically induced transparency in cold atoms," Phys. Rev. Lett. 93, 183601 (2004). [CrossRef] [PubMed]
  12. L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, "Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves," Phys. Rev. Lett. 88, 143902 (2002). [CrossRef] [PubMed]
  13. D. Petrosyan and G. Kurizki, "Symmetric photon-photon coupling by atoms with Zeeman-split ublevels," Phys. Rev. A 65, 033833 (2002). [CrossRef]
  14. M. Paternostro, M. S. Kim, and B. S. Ham, "Generation of entangled coherent states via cross-phase-modulation in a double electromagnetically induced transparency regime," Phys. Rev. A 67, 023811 (2003). [CrossRef]
  15. M. G. Payne and L. Deng, "Quantum entanglement of Fock states with perfectly efficient ultraslow single-probe photon four-wave mixing," Phys. Rev. Lett. 91, 123602 (2003);S. A. Moiseev and B. S. Ham, Phys. Rev. A 71, 053802 (2006). [CrossRef] [PubMed]
  16. S. E. Harris and Y. Yamamoto, "Quantum switching by quantum interference," Phys. Rev. Lett. 81, 3611-3614 (1998). [CrossRef]
  17. B. S. Ham and P. R. Hemmer, "Coherence switching in a four-level system: Quantum switching," Phys. Rev. Lett. 84, 4080-4083 (2000). [CrossRef] [PubMed]
  18. S. A. Moiseev and B. S. Ham, "Quantum manipulation of two-color stationary light: Quantum wavelength conversion," Phys. Rev. A 73, 033812 (2006). [CrossRef]
  19. B. S. Ham, M. S. Shshriar, and P. R. Hemmer, "Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid," Opt. Lett. 22, 1849-1851 (1997). [CrossRef]
  20. L. Rippe, M. Nilsson, S. Kroll, R. Klieber, and D. Suter, "Experimental demonstration of efficient and selective population transfer and qubit distillation in a rare-earth-metal-ion-doped crystal," Phys. Rev. A 71, 062328 (2006). [CrossRef]
  21. A. L. Alexander, J. J. Longdell, M. J. Sellars, and N. B. Manson, "Photon echoes produced by switching electric fields," Phys. Rev. Lett. 96043602 (2006). [CrossRef] [PubMed]
  22. G. He’tet, J. J. Longdell, A. L. Alexander, P. K. Lam, and M. J. Sellars, "Electro-optic quantum memory for light using two-level atoms," Phys. Rev. Lett. 100, 023601 (2008). [CrossRef]
  23. S. A. Moiseev and S. Kroll, "Complete reconstruction of the quantum state of a single-photon wave packet absorbed by a Doppler-broadened transition," Phys. Rev. Lett. 87, 173601 (2001). [CrossRef] [PubMed]
  24. K. Holiday, M. Croci, E. Vauthey, U. P. Wild, "Spectral hole burning and holography in an Y2SIO5:Pr3+ crystal," Phys. Rev. B 47, 14741-14752 (1993). [CrossRef]
  25. B. S. Ham, "Observations of delayed all-optical routing in a slow light regime," Phys. Rev. Lett. (To be published); ibid. arXiv:0801.3501 (2008). [PubMed]
  26. M. Sargent III, M. O. Scully, and W. E. Lamb Jr., Laser Physics (Addison-Wesley, 1974), Chap. 7.
  27. R. W. Equall, R. L. Cone, R. M. Macfarlane, "Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5," Phys. Rev. B 52, 3963-3969 (1995). [CrossRef]
  28. B. S. Ham, M. S. Shahriar, and P. R. Hemmer, "Efficient phase conjugate via two-photon coherence in an optically dense crystal," Phys. Rev. A 59, R2583-2586 (1999). [CrossRef]

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