Enhanced-contrast optical readout in ultrafast broadband Raman quantum memories

The signal-to-noise contrast of the optical readout in broadband Raman quantum memories is analyzed as a function of the pulse widths and phase properties of tailored optical field waveforms used to write in and read out broadband photon wave packets. Based on this analysis, we quantify the tradeoff between the readout contrast and the speed of such memories. Off-resonance coherent four-wave mixing is shown to provide a source of noise photons, lowering the readout contrast in broadband Raman quantum memories. This noise cannot be suppressed by phase matching, but can be radically reduced with a suitable polarization arrangement and proper field-waveform tailoring.


Heisenberg-picture anti-Stokes readout analysis
We consider optical readout signal generation in a Raman quantum memory scheme as a result of two-step coherent anti-Stokes Raman scattering (CARS) process involving write-in and read-out stages (Fig. 1a). CARS has been shown to provide a versatile spectroscopic protocol for the optical interrogation of long-lived coherent phonon modes in solids [24][25][26][27] , including the phonon modes in diamond, considered promising candidates for sold-state broadband memory [7][8][9][10][11][12][13] . At the first, write-in step, two fields with frequencies ω p and ω s , referred to, in accordance with the standard CARS nomenclature 28,29 , as the pump and Stokes fields, drive a molecular or phonon mode with a central frequency Ω R through a Raman-type resonance, ω p − ω s ≈ Ω R . At the second step, a probe field with a frequency ω r reads out the Raman coherence, giving rise to an anti-Stokes signal at the central frequency ω a = ω p − ω s + ω r , which serves to report the state of the phonon memory.
Field quantization is performed by replacing the slowly varying field envelopes A(t, z) by operators where ω 0 is the central frequency of the wave packet, n 0 is the refractive index at the frequency ω 0 , β ω is the propagation constant at the frequency ω, and ω a( ) is the field annihilation operator such that  ( ) . In the second-quantization picture, the evolution of a quantum memory based on a Raman-active phonon driven by a classical pump field with a spectrum A p (ω) is described by an effective Hamiltonian 30 Figure 1. (a) Coherent anti-Stokes readout off the Raman quantum memory: (left) a frequency-tunable Stokes field maps the broadband pump wave packet into a memory based on coherently driven optical phonon, (right) the probe field applied with a variable delay time reads out the coherence driven by the pump and Stokes fields, giving rise to an anti-Stokes signal, which serves to report the state of the phonon memory. (b) Four polarization arrangements for the Raman quantum memory with anti-Stokes readout. Field polarization vectors are shown (arrows) against crystallographic directions. (c) The spectra of the anti-Stokes readout for different polarization arrangements as specified in panel (b) in the regime when the bandwidths of the pump, Stokes, and probe fields, Δ p , Δ s , and Δ r , are much smaller than the linewidth Γ of the Raman mode.
( ) ] p 2 1/2 , and ϑ is the effective coupling constant. The properties of such a Hamiltonians are discussed in an extensive literature on Raman quantum memory (see, e.g., refs 17,18 ), as well as Raman protocols of quantum communication and quantum information processing 31 . The main focus of this work is on the limitations on the contrast of the optical readout in broadband Raman quantum memories due to the unwanted coherent FWM originating from off-resonance-driven electronic and two-photon excitations. The nonlinear signal produced through such FWM processes interferes with the anti-Stokes readout from the Raman coherence, dramatically lowering its contrast. Central to the analysis of this interference in classical-field models and experiments is the dispersion of the Raman-resonance and nonresonant parts of the pertinent nonlinear susceptibility 28,29,32 , that is, the frequency dependence of the Raman-resonant and nonresonant terms in the effective coupling constant. The goal of our analysis here is to extend these classical models to the Raman schemes operating with quantum states of the Stokes and anti-Stokes fields, such as Raman quantum memories. To this end, we continue treating the pump and readout (probe) fields classically and write the Hamiltonian as [33][34][35]  , P p and P r are the peak powers of the pump and probe fields, Δβ = β a + β s − β p − β r , β j = β(ω j ), j = p, s, r, a, κ εγ = A A p r , ε is the polarization-sensitive numerical factor, P p and P r are the complex amplitudes of the pump and probe fields, respectively, and γ is the nonlinear coefficient related to the pertinent third-order optical susceptibility.
For the full quantum analysis of the phonon modes behind the quantum memory, the evolution equations as dictated by the Hamiltonian (3) need to be included in the model. The Hamiltonian (4) is not intended for such an analysis. Instead, it provides an adequate framework for the description of the spectral interference of the anti-Stokes readout from the Raman coherence with off-resonance FWM, thus enabling a quantitative analysis of the readout contrast in Raman quantum memories.
With the Hamiltonian taken in the form of Eq. (4), the solution to the Heisenberg-picture evolution equations a s a s , , can be written in the input-output form as [33][34][35]  , with u s and u a being the group velocities of the Stokes and anti-Stokes pulses. Here, however, we seek to isolate effects related to the pulse shape, phase, and polarization of optical fields. We therefore choose to work in the approximation where group walk-off effects are neglected -approximation that is broadly accepted even in ultrafast-CARS literature 36 .
We now use the Heisenberg picture to calculate the expectation value for the anti-Stokes field photon-number operator, . With an input Stokes-anti-Stokes field state with no anti-Stokes photons at Notably, with <n s0 > = 0, corresponding to a vacuum input state, ψ = = z ( 0) 0, 0 , Eq. (8) recovers the two-mode squeezed-state result of the input-output Heisenberg-picture analysis of FWM [33][34][35] It is, however, the first, ∝<n s0 > term of Eq. (8) that we are mainly concerned with here in our analysis of broadband Raman quantum memories. For |κ| ≪ |δ|, this part of the anti-Stokes readout can be written as In the case of a classical Stokes field at the input, s  a s  a s a s  ,  ,  1 , , , where P a,s are the peak powers of the anti-Stokes and Stokes pulses and τ a,s are their respective pulse widths, the standard classical-field result is recovered: Off-resonance coherent four-wave mixing as a source of anisotropic photon noise Apart from the complex, frequency-dependent Raman-resonant part, γ R (ω), the FWM nonlinear coefficient γ generally includes a purely real nonresonant term, γ nr , related to electron transitions 28,29,32,37 , and may include a distinct two-photon-resonant term 38,39 , γ tp , which is of special significance in solid semiconductors, For cubic and isotropic materials, the spectrum of the Raman resonance can be approximated with a Lorentzian profile, leading to the following three-term approximation for |γ| 2 near the Raman resonance where N is the number density of Raman-resonant species, Γ is the linewidth of the Raman resonance, α R is the local-field-corrected polarizability, χ 1 and χ 2 are the real and imaginary parts of the effective cubic susceptibility χ eff = χ 1 − iχ 2 , which can be expressed through the components of the electronic third-order susceptibility χ ijkl (3) . The coefficients d 1 , d 2 , and d 3 depend on the polarization arrangement of optical fields, reflecting tensor properties of the electronic susceptibility χ ijkl E (3) and polarizability α R . For four representative arrangements of the pump, Stokes, probe, and anti-Stokes polarization vectors relative to the [100] direction of the diamond lattice sketched in Fig. 1, we find no Raman resonance for polarization geometry 1 (curve 1 in Fig. 1c), d 1 = 0, d 2 = 6, and d 3 = 0 for geometry 2 (curve 2 in Fig. 1c), d 1 = 0, d 2 = 0, and d 3 = 12 for geometry 3 (curve 3 in Fig. 1c), and d 1 = 6, d 2 = 6, and d 3 = 0 for geometry 4 (curve 4 in Fig. 1c).
The anti-Stokes signal is thus a result of spectral interference of the photon fields generated through the CARS-type scattering of the probe field off the Raman coherence, encoding the information written by the pump and Stokes fields, and the nonresonant FWM background related to the γ nr and γ tr terms in Eq. (11). Both the useful anti-Stokes Raman memory readout signal and the FWM background build up as sin 2 (Δβz/2) as functions of the propagation path. As a consequence, the nonresonant background in anti-Stokes photon counts cannot be suppressed by phase-matching adjustments.

Polarization-sensitive anti-Stokes readout contrast enhancement
As can be seen from Eq. (12), the ratio 1 [the factor that appears in front of the frequency denominator in the third term in Eq. (12)] provides a meaningful quantitative measure for the contrast of the anti-Stokes memory readout relative to the FWM nonresonant background. This ratio is highly sensitive to the polarizations of the optical fields used to write information into the Raman memory and to read out the state of this memory.
Analysis of the tensor properties of χ ijkl E (3) and polarizability α R of diamond shows that this material is, in many respects, a highly promising medium for quantum memory. The cubic symmetry of the crystal lattice of diamond provides an ample parameter space for η contrast optimization. At the same time, the 1332-cm −1 zone-center Γ (25+) (F 2g ) symmetry optical phonon in diamond, offers an advantageous combination of a broad, terahertz bandwidth and a sufficiently long coherence time in the range of a few picoseconds, allowing ultrafast Raman memory schemes to be implemented at room temperatures using femtosecond laser pulses [7][8][9][10][11][12][13] .
However, a great care needs to be exercised in choosing polarizations of the laser beam to provide the highest possible anti-Stokes-readout-to-FWM-background ratios. In particular, for the polarization arrangement where the pump field is polarized along the [100] direction of the diamond lattice, while the Stokes and probe fields are polarized along the [010] direction (polarization geometry 2 in Fig. 1b), we have η ≈ 110 cm −1 and |χ 2 /χ 1 | ≪ 1 38,39 , providing a high-contrast anti-Stokes readout against a virtually negligible FWM background (curve 2 in Fig. 1c).
Storing polarization qubits, however, requires two different polarization arrangements of optical fields. Building upon the high η ratios provided by the polarization arrangement 2 in Fig. 1b, it is tempting to use a scheme with all the laser fields polarized along the [010] direction (polarization geometry 1 in Fig. 1b) as the second polarization arrangement for polarization-qubit storage. However, the triply degenerate F 2g -symmetry optical phonon in diamond (as well as in CaF 2 and other homologous fluorides) has zero Raman-resonant cubic susceptibility χ R

1111
(3) 38,39 . The useful anti-Stokes readout vanishes in this polarization arrangement (curve 1 in Fig. 1c). With this exception, diamond provides a vast parameter space for ultrafast broadband quantum memory with a high-contrast anti-Stokes readout (Fig. 1c).

Broadband photon wave packets
In the case of ultrashort pump, Stokes, and probe field waveforms, The anti-Stokes readout of the Raman quantum memory is thus given by r r s p s p s s s 0 As the first important limiting regime, we consider the case when the bandwidths of the laser pulses Δ p and Δ s are much smaller than the linewidth Γ of the Raman resonance, This result recovers Eq. (9) with |γ| as defined by Eq. (12), |Δβz| ≪ 1, and ω ω − ≈ Ω p s R . The highest contrast of the anti-Stokes signal relative to the FWM background achieved in this regime is σ 0 ≈ |η| 2 /Γ 2 . For the 1332-cm −1 zone-center Γ (25+) (F 2g ) symmetry optical phonon in diamond driven and probed in polarization arrangements 2, as shown in Fig. 1b, η 2 ~ 110 cm −1 and Γ ~ 1 cm −1 38,39 . The highest contrast of the anti-Stokes signal in this scheme is σ 0 ~ 10 4 (Fig. 2a). For the polarization arrangement 4 (Fig. 1b), the contrast of the anti-Stokes readout is seen to be about 13 times lower (cf. Fig. 2a,b). This correlates well with the polarization  Fig. 1c is ~13, leading to a 13-fold difference in the highest contrast of the anti-Stokes readout for the polarization arrangements 2 and 4 in Fig. 2a,b.
As the pump-Stokes frequency difference ω p − w s is scanned near and through the Raman resonance ω ω − = Ω p s R , the maximum of the anti-Stokes readout in the Δ p , Δ s ≪ Γ regime closely follows the spectral profile of the Raman mode (pink curve 12 in Fig. 2a,b). In the context of ultrafast broadband quantum memory, however, such a high contrast is achieved at the expense of the operation speed. Indeed, in the case of a diamond-based quantum memory, the condition Δ p , Δ s ≪ Γ is satisfied for laser pulse widths longer than or on the order of 10 ps, which limits the memory speed to the 1-10-GHz rate level.
Similar to their classical-field analogues, Eqs (12, 13, 15 and 17) suggest that the contrast of the anti-Stokes readout can be enhanced through a proper field-waveform tailoring. Indeed, the maximum level of the nonresonant FWM signal is achieved, as can be seen from these expressions, when the pump, Stokes, and probe pulses are all transform-limited. Resonant excitation of Raman-active phonon, on the other hand, does not necessarily require transform-limited pump and Stokes fields. Phase modulated pump and Stokes pulses can still provide a resonant excitation of a phonon mode with a central frequency Ω R as long as their spectral phase functions Φ p (ω) and Φ s (ω) satisfy the condition 36,40 Φ p (ω) = Φ s (ω − Ω R ). In CARS microscopy, this condition, as elegantly shown in refs 36,40 , can be satisfied by applying frequency-shifted step functions to the spectral phases of the pump and Stokes fields. The Φ p (ω) = Φ s (ω − Ω R ) recipe can be extended to Raman quantum information processing and storage protocols operating with quantum states of light with a well-defined phase. In the Raman scheme considered in this work, this implies a well-defined phase of the Stokes field, Φ s (ω). As is also readily seen from Eqs (13, 15 and 17), uncertainty in the phases of optical fields translate into fluctuations of the anti-Stokes readout. The quantum-field treatment of coherent anti-Stokes Raman scattering provided in the previous sections thus remains meaningful beyond Raman memory schemes, providing a closed formalism for a quantitative analysis of the anti-Stokes readout in coherent Raman scattering of quantum optical fields.

Anti-Stokes readout with a time-delayed probe
As a powerful resource for a practical implementation of quantum memory, the probe (reading) pulse in broadband Raman memory can be applied with a variable delay time τ d relative to the pump-Stokes write-in pulse dyad (Fig. 1a). To appreciate the advantages of this modality, it is instructive to transform the nonlinear susceptibility χ ω ω ω ω − ( ; , , ) a p s r (3) , controlling, via Eq. (12), the nonlinear coefficient in the ω a = ω p − ω s + ω r coherent Raman memory, to the time domain: We isolate the Raman-resonant and FWM-nonresonant parts of the time-domain nonlinear susceptibility by representing χ t t t ( , , ) 1 2 3 as R R (3)  1 2 3  1  2  1  3 and nr nr When transformed to the time domain and expressed in terms of χ t t t ( , , ) 1 2 3 using Eqs (19)(20)(21)(22), the expectation value for the anti-Stokes photon-number operator becomes As can be seen from Eqs (23)(24)(25), the anti-Stokes signal is generally a mixture of the Raman memory readout, described by the q R (θ) term in Eq. (23), and FWM photons not related to the Raman memory and represented by the q nr (θ) term in Eq. (23). However, when the pulse widths of the pump, Stokes, and probe laser pulses τ p , τ s , and τ r , are short enough, such that τ p , τ s , τ r ≪ 1/Γ, and the delay time τ d is chosen such that τ d > τ p , τ s , the FWM background in anti-Stokes counts is completely suppressed. showing that, similar to time-resolved CARS with classical laser fields 29,41,42 , the expectation value for the anti-Stokes photon-number operator <n a > measured as a function of the delay time τ d in the quantum version of coherent Raman scattering provides a background-free map of the |q R (τ d )| 2 trace of Raman coherence.
Remarkably, even in this mode of quantum memory, the ratio Δ + Δ Γ ( ) / p s 2 2 2 is meaningful as a fundamental limit on the efficiency of Raman coherence excitation as a part of memory write-in process. Indeed, the integral ∫ ∫ ∫ ( )