Stokes photon

The principle of photoacoustic Raman spectroscopy (107) is similar to that of CARS. When two laser beams, vp (pump beam) and vs (Stokes beam), impinge on a gaseous sample contained in a cell (Fig. 3-43), these two beams interact when the resonance condition, vp — vs = vM, is met, where vM is a frequency of a Raman-active mode. This results in the amplification of the Stokes beam and the attenuation of the pump beam. Each Stokes photon thus... 

Figure 3-43 Schematic representation of the photoacoustic Raman scattering (PARS) process, (a) A simple energy level diagram illustrating the Raman interaction that occurs in the PARS process, (b) Basic elements of the PARS experimental arrangement. The pump beam is attenuated and the Stokes beam is amplified by the stimulated Raman process that takes place where the beams overlap in the gas sample cell. For each Stokes photon created by the Raman process, one molecule is transferred from the lower state to the upper state of the transition. Collisional relaxation of these excited molecules produces a pressure change that is detected by a microphone. (Reproduced with permission from Ref. 107.)...

The number of detected anti-Stokes photons per shot induced by the pump pulse is given by... 

Fig. 5.4 (a) The energy diagram of the three-color CARS process, where the pump and Stokes photons (tUpujjjp and Wsfokes) are provided from the single broadband pulse and the probe photon (cOprobe) from the delayed narrowband pulse, (b) Experimental setup of a time-frequency 2D-CARS microscope BPF band-pass filter, PCF photonic crystal fiber, LPF longpass filter, SPF shortpass filter... 

Figure 21c shows the respective probabilities of one and two ( >i pump photon absorption tPp-1, Pp. 2. and of one (02 Stokes photon emission and absorption /Js i. yjs. 1. defined with the respective formulas of the probabilities of coj photons emissions and of (02 photons emissions... 

To that effect, we choose the parameters 8 = 2flo and Qmax = 4.4 f>o, corresponding to the path (c) on the surfaces in Fig. 19. As shown in Fig. 22, the solution of the semiclassical Schrodinger equation (321) leads to nearly complete population transfer from state 11) to state 3). The analysis of the surfaces shows that the state 1 0,0) connects 3 —3,3). Thus the complete population transfer from the bare state 1) to the bare state 3) must be accompanied with absorption of three pump photons and emission of three Stokes photons at the end of the process. This is confirmed by the numerical solution of the dressed Schrodinger equation (308) with the initial state as a number state for the photon field 11 0,0), shown in Fig. 23a the dressed state vector /(f) approximately projects on the transfer eigenvectors during the process. It shows... 

The direction, delay time Td, and rate of retrieval are determined by the direction, timing, and intensity of the retrieve laser, allowing control over the spatio-temporal properties of the retrieved pulse (referred to as the anti-Stokes pulse). Since the storage and retrieval processes ideally result in identical photon numbers in the Stokes and anti-Stokes pulses [Lukin 1999], this technique should allow preparation of an n-photon Fock state in the anti-Stokes pulse conditioned on detection of n Stokes photons. 

Fig. 2 A shows the average number of detected Stokes photons per unit time (photon flux) in the write channel as a function of time during the 1.6 /is-long write pulse. The magnitude of the photon flux (and hence the total number of photons in the pulse) is controlled by varying the excitation intensity. After a time delay Td, we apply the retrieve beam to convert the stored spin wave into anti-Stokes photons, as shown in Fig. 2 B. The duration and peak flux of the anti-Stokes pulse can be controlled by the intensity of the retrieve laser.

Figure 2. A - Experimentally measured and theoretically calculated values of dns/dt, the number of Stokes photons per unit time emitted from the atomic vapor cell. For each plot, ns = f dt dns/dt represents the total number of photons emitted from the cell. The write laser power is varied from 25 mW to 100 mW. B - Experimentally measured and theoretically calculated values of dnAs/dt, the number of anti-Stokes photons per unit time emitted from the atomic vapor cell. The experimental pulse shapes correspond to a Stokes pulse with ns 3 photons, and the theoretical curves assume an initial spin wave with nspin = 3 excitations and an optical depth of 20. Each curve is labeled with the power of the retrieve laser. Inset theoretical calculation of the number of flipped spins per unit length dnspin/dt as a function of position in the atomic cell, for nspin = 3. C - Measured anti-Stokes pulse width (full-width at half-maximum) and total photon number as a function of the retrieve laser intensity. Lines are intended only to guide the eye.

The shape of this spin excitation is directly mirrored in the shape of the retrieved anti-Stokes pulse. The resonant classical read laser beats with the ground-state coherence S(z, t) and converts the atomic spin wave into an anti-Stokes field as(s, t) the number of anti-Stokes photons emitted per unit time from the end of the cell is driAs(t)/dt = c S L, t) as(L, t))/L. Note that the read laser establishes an EIT configuration for the generated anti-Stokes field, so that the anti-Stokes light propagates unabsorbed through the... 

In practice, decay of the spin coherence during the delay time t,i and finite optical depth flatten and broaden the anti-Stokes pulse, reducing the total number of anti-Stokes photons which can be retrieved within the coherence time of the atomic memory. For weak retrieve laser intensities, the total photon number per pulse increases with increasing laser power because the time required to read out the spin wave is longer than the characteristic decoherence time of our atomic memory ( 3gs, see Fig. 3 b). After accounting for dead-time effects, we find that once the retrieve laser power increases to 25 mW, all of the spin wave is retrieved in a time shorter than the decoherence time, resulting in a constant anti-Stokes number versus retrieve power. 

These correlations between Stokes and anti-Stokes pulses allow for the conditional preparation of the anti-Stokes pulse with intensity fluctuations that are suppressed compared with classical light. In order to quantify the performance of this technique, we measured the second-order intensity correlation function giis MS ) and mean number of photons fi for the anti-Stokes pulse conditioned on the detection of ns photons in the Stokes channel (see Fig. 4). (For classical states of light, (f1] > 1, whereas an ideal Fock state with n photons has anti-Stokes photons grows linearly with ns, while (AS1) drops below unity, indicating the nonclassical character of the anti-Stokes photon states. In the presence of back-ground counts, gks (AN) does not increase monotonically with ns, but instead exhibits a minimum at ns = 2. The Mandel Q parameter [Mandel 1995] can be calculated using = n f((jns (AS) — 1) from these measurements we... 

Figure 4. Conditional nonclassical state generation. Conditional (f f (.4,9) as a function of the number of detected Stokes photons. Diamonds show experimentally measured values, which are calculated from the two arms of the anti-Stokes beam-splitter via g (A.S ) = (AS AS-2)/ AS ) AS-2) (see Fig. 1 C). The measured mean photons number on the Stokes and anti-Stokes channels were fis = 1.06 and has = 0.36 respectively. The solid line shows the result of a theoretical model including background and loss on both the Stokes and anti-Stokes channels. The overall detection efficiency (a) and number of background photons (hbg ) used in the model were as = 0.35, n G = 0.27 (qas = 0.1, rdfs = 0.12) on the Stokes (anti-Stokes) channel, and were estimated from experimental measurements. For these measurements an optically-pumped 87Rb cell was used to filter the Stokes photons from the write laser. The dotted line represents < ns (AS) corrected for loss and background on the anti-Stokes channel, obtained by setting the anti-Stokes channel loss and background to zero in this model. Inset measured mean anti-Stokes number n s conditioned on the Stokes photon number ns- The solid line represents n s as predicted by the model.

We now turn to a quantitative examination of the feasibility of conditional Fock state generation using our preparation and retrieval technique. For applications in long-distance quantum communication, the quality of the atomic state preparation is the most important quantity. Assuming perfect atom-photon correlations in the write Raman processes, we can find the density matrix p for the number of atomic spin-wave excitations conditioned on the detection of ns Stokes photons. Here we consider only the spin-wave modes correlated with our detection mode. For example, in the absence of losses and background, the conditional atomic density matrix is simply p(ns) = ns)(ns. Loss on the Stokes channel (characterized by transmission coefficient a.s) leads to a statistical mixture of spin-wave excitations,... 

To determine the unconditional probability distribution for the spin-wave excitations Psw(n), we must find the effective number of transverse modes which contribute to the Raman processes. We identify two extreme regimes which permit analytic treatment a single mode regime where the number of excitations in the 87Rb cell follows Bose-Einstein (thermal) statistics and a multimode regime where it follows Poisson statistics. We find in both cases that the quantities F and Q depend on two experimental parameters 0 ( number of lost Stokes photons) and v ( noise to signal ratio), which are defined in Tab. 1. 

Table 1. Scaling for the anti-Stokes pulse Q-parameter and Fock state fidelity F. n refers to the mean number of excitations in the rubidium cell, nfG is the mean photon number of background counts in the write channel ( we assume they are mainly due to leak of the write drive and so follow Poisson statistics), as is the Stokes detection efficiency and ns is the number of Stokes photons on which we condition. The mean number of atomic excitations is calculated via (nsw) = Tr (pas nsw), similarly (n2sw = Tr (pas h2sw). The subscript T (P) refers to thermal (Poisson) photon statistics of the unconditional Stokes light.

A second laser is required to produce fhe Stokes photons for the CARS interaction. This can be achieved with a separate Stokes laser with its own pulse control system or, as is shown in Figure 13.8, some of the laser energy from the pump laser can be used to optically pump a laser suited to the CARS system. The symchro-nization of the timing between the pump and Stokes laser systems must be accurate to within 1 ms to ensure that the two laser beams arrive at the same place at the same time. The advantage of fhe latter laser system is that only one timing circuit is needed to keep the pump... 

The linewidths of spontaneous and stimulated Raman lines depend on the linewidth of the pump laser. For narrow linewidths, however, the width of the stimulated Raman lines becomes smaller than that of the spontaneous lines, which are Doppler-broadened by the thermal motion of the scattering molecules. A Stokes photon hco, which is scattered into an angle 0 against the incident laser beam by a molecule moving with the velocity v, has a Doppler-shifted frequency... 

When the energy in the Stokes wave becomes comparable to that in the incident laser pump, depletion occurs and the gain is reduced. In principle, every photon in the incident pump wave can be converted to a Stokes photon, giving a maximum theoretical conversion efficiency of... 

The ordinary Raman effect can be described as an inelastic scattering of pump photons ha>p by molecules in the energy level E(. The energy loss h(cop — cOs) of the scattered Stokes photons hco is converted into excitation... 

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