Multimode optical parametric amplification in the phase-sensitive regime
Gaetano Frascella, Roman V. Zakharov, Olga V. Tikhonova, Maria V. Chekhova
MMultimode optical parametric amplification in the phase-sensitiveregime
Gaetano Frascella , , Roman V. Zakharov , , Olga V. Tikhonova , ,and Maria V. Chekhova , Max Planck Institute for the Science of Light, Staudtstr. 2, 91058 Erlangen, Germany. University of Erlangen-Nuremberg, Staudtstr. 7/B2, 91058 Erlangen, Germany. Physics Department, Moscow State University, Leninskiye Gory 1-2, Moscow 119991, Russia. Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119234, Russia.February 24, 2021
Abstract
Phase-sensitive optical parametric amplification of squeezed states helps to overcome detection lossand noise and thus increase the robustness of sub-shot-noise sensing. Because such techniques, e.g.,imaging and spectroscopy, operate with multimode light, multimode amplification is required. Here wefind the optimal methods for multimode phase-sensitive amplification and verify them in an experimentwhere a pumped second-order nonlinear crystal is seeded with a Gaussian coherent beam. Phase-sensitiveamplification is obtained by tightly focusing the seed into the crystal, rather than seeding with close-to-plane waves. This suggests that phase-sensitive amplification of sub-shot-noise images should beperformed in the near field. Similar recipe can be formulated for the time and frequency, which makesthis work relevant for quantum-enhanced spectroscopy.
Quantum imaging promises to bring an advantage over classical methods in terms of signal-to-noise ratio(SNR) [4, 16, 19, 29]. Nonclassical light and notably squeezed states of light push the sensitivity to overcomethe shot-noise limit in imaging experiments [3, 32], but their main drawback is fragility. Indeed, squeezing isdeteriorated by optical loss and detection inefficiency; therefore, sub-shot-noise experiments require tailoredhigh-transmission equipment and highly efficient detectors.Noiseless amplification of squeezed fields prior to optical or detection losses can solve both issues atthe same time. This option is realized through phase-sensitive optical parametric amplifiers (OPAs), whichconsist of the same optical components that generate the squeezed states in the first place. The ‘noiseless’regime leaves the SNR unchanged from input to output and it is most notoriously achieved with degenerateOPAs. Meanwhile, non-degenerate OPAs amplify without added noise only if both conjugated modes are fedwith the input signal. When only one mode is fed with the signal and the other is left with vacuum fluctu-ations at the input, the added noise translates into a 3 dB penalty in SNR [21, 30]. Linear amplifiers, basedon stimulated emission in a gain medium, suffer from the same problem due to the amplified spontaneousemission.Amplification without added noise is a longstanding objective in many other fields, such as optical [2] andspecifically fiber [1] communications or optomechanics in the microwave range [28]. From seminal papers inthe eighties [5, 36], even quantum metrology has seen rising interest in phase-sensitive OPAs for detection-losstolerance [17, 6, 26, 13, 14].The most natural application of multimode OPAs remains imaging; early theoretical [20] and experi-mental [10, 7, 11] works demonstrated the possibility of noiseless amplification of images with bulk crystals.Later, the idea proved efficient also for OPAs based on four-wave mixing [9] and for optical parametricoscillators [23]. In general, phase-sensitive and -insensitive amplifiers have been intensely studied in termsof their transfer function and angular bandwidth [15, 7, 9] and their suppression of twin-beams intensity a r X i v : . [ qu a n t - ph ] F e b ifference noise [27, 22]. To avoid amplification of vacuum noise, the overlap of the OPA eigenmodes withthe input radiation is critical [24], but the requirements to phase-sensitive amplification of images are stillmissing, just like an understanding of the process in terms of eigenmodes.In this Letter, we show the conditions for multimode optical parametric amplification in the phase-sensitive regime and confirm them with an experiment. We identify the best strategy to obtain phase-sensitive amplification as projecting a near-field image on the amplifier. Our experiment uses a single modeof an image (a Gaussian coherent seed) fed into a spatially-multimode OPA based on high-gain parametricdown-conversion (PDC). The output intensity modulation quantifies the sensitivity to the input phase andwe study its dependence on the initial beam divergence and tilt with respect to the pump direction. We alsodiscuss the time-domain analogy of this experiment.A spatially multimode OPAs is described by the Hamiltonianˆ H = i ¯ h Γ (cid:90) (cid:90) d x s d x i F ( x s , x i ) a † ( x s ) a † ( x i ) + h . c . , (1)where Γ is a coupling parameter proportional to the pump field amplitude, the length of the medium andthe nonlinearity, while x can be either a transverse spatial coordinate ρ (near-field) or the Fourier-relatedtransverse wave-vector q (far-field). To quantify the amount of amplification, one defines the parametricgain of the OPA as G = (cid:82) Γ dt .The two-photon amplitude (TPA) F ( x s , x i ) describes the joint probability amplitude of emitting photonsinto the signal mode with coordinate x s and the idler mode with coordinate x i . The near-field TPA F near ( ρ s , ρ i ) and the far-field one F far ( q s , q i ) are related by a Fourier transform. Eq.( 1) is valid alsofor multiple modes in time or frequency domain, if x is time or frequency.A simple case to understand our viewpoint on phase-sensitive amplification is the one of very shortmedium and very large pump beam waist. Then, the far-field TPA scales as a delta function F far ∼ δ ( q s + q i ), and the Fourier transform yields the near-field TPA in the form F near ∼ δ ( ρ s − ρ i ). TheHamiltonian in Eq.( 1) is simplified toˆ H far = i ¯ h Γ (cid:90) d q s a † ( q s ) a † ( − q s ) + h . c . , (2)ˆ H near = i ¯ h Γ (cid:90) d ρ s (cid:2) a † ( ρ s ) (cid:3) + h . c . . (3)In the far field (Eq.( 2)), an analogy can be made with the two-mode-OPA Hamiltonian, while in the nearfield (Eq. (3)) with the single-mode-OPA one.For a two-mode-OPA, if one input is fed with a coherent beam and the other with vacuum, the coherentbeam is amplified and a conjugated beam appears regardless of the input phase (phase-insensitive or phase-preserving [8]). If both signal and idler inputs are fed with coherent states with two independent phases,the output beams will show amplification or de-amplification depending of the sum of the phases of theinput beams (phase-sensitive) and the original phases are not preserved at the output. This means that, inthe far field, the OPA described with Hamiltonian (2) is phase-sensitive if plane wave modes at conjugatedtransverse wavevectors at q s and q i = − q s are fed with coherent beams simultaneously. In the single-modeFigure 1: Experimental setup for optical parametric amplification of a large-divergence (with the lens inthe dashed box) or a small-divergence (without it) seed. PA, piezoelectric actuator; L1,L2 lenses; BBO, β -barium borate crystal; DM, dichroic mirror; BF, bandpass filter; CCD, charge-coupled device camera.2ase, the amplification is shown to be always dependent on the phase of the input coherent state. Therefore,seeding in the near field, i.e. at one point of the OPA, results in phase-sensitive amplification.We realize experimentally far-field and near-field seeding of a multimode OPA, respectively, with a small-and large-divergence coherent beam and we test the modulation of the signal output intensity with the phase.We also vary the seed central angle to show the transition between seeding conjugated and non-conjugatedinput transverse wavevectors. In our experiment, the OPA is a χ (2) -nonlinear crystal working in the collineardegenerate regime. Signal and idler beams are discerned by considering the emission in one half of the spatialemission spectrum as the signal and the other half as the idler, given the anti-correlation of the two beamsin transverse wavevector.Fig. 1 shows the experimental setup. The coherent beam comes from a Ti:Sapphire laser at 800 nmproducing 1 . ± µ m with lens L1 and 2 . ± . . ± . . ± .
01 mrad, respectively.The pump is the second harmonic of the same laser and its waist position is the same as the one of thecoherent seed. Here, we place the 2-mm β -barium borate crystal BBO. The pump has intensity FWHM240 ± µ m and an average power of 65 mW, and it is rejected with the dichroic mirror DM. We obtaina value of G = 3 . ± . π/ ∼ ∼ F ( x s , x i ) = (cid:88) n,p (cid:112) λ n,p U n,p ( x s ) V n,p ( x i ) , (4)where n and p are, respectively, azimuthal and radial indices, λ n,p are the Schmidt weights and U , V are,respectively, Schmidt signal and idler modes. This is a simplified model, in which the shapes of the modesdo not depend on the gain; a rigorous theory, proved by experiment, shows that the mode shapes changevery little [34].Given this decomposition, the Hamiltonian can be rewritten as a weighted sum of the products of col-lective signal and idler creation operators. Each term of the sum corresponds to a two-mode OPA withthe parametric gain G n,p ≡ G (cid:112) λ n,p . If the parametric gain is increased, the effective number of modesis reduced [33], and so will be the imaging resolution. The choice of these two parameters is always acompromise.This multimode formalism is powerful because it allows to calculate the moments of the observables ofinterest, and, in particular, the output signal photon number ˆ N s averaged for an input state of the OPA in3igure 2: Two-dimensional angular spectrum at the OPA output with amplified (a) and de-amplified coherentbeam (b) with large divergence and central angle 5 mrad. The spectra are integrated over the dashed redregion to obtain the signal intensity. Visibility of the signal intensity phase-sensitive modulation (the insetshowing the time dependence for two marked points) versus the seed central angle for large (c) and small(d) divergence.a specific mode [12].For a strong coherent input beam with complex amplitude α in the spatial mode f ( x ), we calculate theaverage signal photon number: (cid:104) α, f | ˆ N s | α, f (cid:105) ≈ (cid:80) n,p | α | (cid:104) | β n,p | cosh G n,p + (cid:12)(cid:12) β (cid:48) n,p (cid:12)(cid:12) sinh G n,p + sinh 2 G n,p | β n,p | (cid:12)(cid:12) β (cid:48) n,p (cid:12)(cid:12) × cos (cid:0) α − arg β n,p − arg β (cid:48) n,p (cid:1)(cid:3) , (5)where β n,p = (cid:82) d x f ∗ ( x ) U n,p ( x ) and β (cid:48) n,p = (cid:82) d x f ∗ ( x ) V n,p ( x ) are the overlap integrals of the seed withsignal/idler modes. The non-seeded PDC emission is neglected, since its number of photons does not dependon α .The phase-insensitive part of Eq. (5) is described by the first two terms in the square brackets. If thecoherent input mode overlaps with the signal (idler) mode ( n, p ), | α | sinh G n,p photons will be added tothe mode, previously populated by | α | (zero) photons. Meanwhile, the last and phase-sensitive term inEq.( 5) depends on the phase arg α of the coherent beam only if, for at least one value of n and p , theoverlap integrals β n,p and β (cid:48) n,p have simultaneously non-zero modulus. In other words, the condition ofphase-sensitive amplification is valid only if the coherent beam mode overlaps with signal and idler modessimultaneously.This overlap condition is easy to satisfy in the near field, where the signal and idler modes are non-zero atthe same points; it is more difficult in the far field, where these modes peak at opposite wavevectors [22, 7].This suggests that one should be imaging the seed beam waist in the photon creation region of the OPAto obtain noiseless amplification, or, ultimately, one should seed by a spherical wave rather than by a planewave. 4igure 3: a) Simulated phase-sensitive modulation visibility as a function of the seed divergence and centralangle with respect to the pump direction; along the white dashed line the central angle is half the divergence.b) The seed (solid line) along the pump direction (arrow) always undergoes phase-sensitive amplification,with a divergence limited by the OPA non-seeded emission (dashed line). c) When the seed is tilted, theamplification is partly phase-sensitive (stronger shading) and partly phase-insensitive (lighter shading).Two other important aspects of Eq.( 5) are: i) for a fixed n and p , each term in the squared brackets isproportional to an exponential function of G n,p , which means that modes with higher weights are strongeramplified [18]; ii) for a degenerate OPA, this formula can be simplified because the signal and idler modescoincide in modulus | U n,p | = | V n,p | and the relation between the overlap integrals β (cid:48) n,p = β − n,p can bederived.Fig. 3 a shows the calculated visibility of phase-sensitive intensity modulation for a Gaussian seed asa function of divergence and central angle for type-I degenerate PDC with the same parameters as in ourexperiment. The seed spatial spectrum has the form f ( q ) ∝ exp (cid:104) − λ | q − q | / (cid:0) π θ (cid:1)(cid:105) with divergence θ and transverse wavevector q .For a seed beam propagating along the pump direction, the amplification depends strongly on the coherentseed phase for any value of the seed divergence, but a maximum is obtained when the seed matches the fullangular range of the non-seeded emission of the OPA. Fig. 3 b shows a sketch of seed (solid line) propagatingalong the pump direction (arrow); the seed is amplified in the phase-sensitive regime even if it does notmatch the angular bandwidth of the non-seeded PDC emission (dashed line) from the nonlinear crystal.For a seed tilted with respect to the pump direction, the sensitivity to the seed phase is high when thecentral angle is smaller than half the divergence (below the white dashed line in Fig. 3 a). Fig. 3 c showsa sketch of this case: if the seed can populate modes with opposite wavevectors, there will be a part ofphase-sensitive amplification (stronger shading) with good intensity modulation but also a phase-insensitivepart (lighter shading).The results described here for OPAs with multiple modes in space have a time-domain analogy. Indeed,photons of a pair are correlated both in position (near-field coordinate) and in time; meanwhile, they are anti-correlated in momentum (far-field coordinate) and frequency. Therefore, to have phase-sensitive (noiseless)amplification one has to amplify a time pattern, rather than a spectrum. A straightforward conclusion isthat short pulses will be always amplified noiselessly, but with the limitation imposed by the frequencybandwidth of the OPA. This conclusion is very useful for enhanced sub-shot-noise spectroscopy, becausecomplex temporal profiles can be studied through the analysis of the phase-sensitive intensity modulation.Our work is readily applicable in the field of quantum imaging, where phase-sensitive amplification beforedetection can provide loss tolerance [19]. For a single mode of an image, both experimental and theoreticalevidence point towards near-field or high-divergence seeding to obtain phase-sensitive amplification. Incontrast to far-field amplification, projecting the near-field image on the OPA gives phase-sensitive intensitymodulation tolerant to the input central angle. Our calculations can be adapted to real-world case scenariosand help to model theoretically experiments conducted with multimode OPAs.Our experiment is based on a bulk-crystal OPA, but these results on multimode phase-sensitive amplifica-tion are valid also for systems based on four-wave mixing. The main difficulty for the latter is that amplified5ignal is hard to be distinguished from the pump; therefore, two far-field replicas of the image are fed intoconjugate modes with wavevectors slightly off from the pump direction. Meanwhile, nonlinear crystals canbe used in the collinear regime and dichroic mirrors easily reject the pump radiation.The above analysis of the contribution of different modes to the phase-sensitive modulation can provevery useful in the context where multimode OPAs are used to achieve sub-shot-noise precision to phaseshifts. This is the case of SU(1,1) interferometers in the seeded regime [31, 25] and the squeezing-assistedinterferometer with output optical parametric amplification [14].The advantage of squeezing, especially in quantum imaging and sensing, is mostly limited by detectionefficiency. Our work makes a step further in the analysis of multimode phase-sensitive parametric opticalamplification, which, if employed before detection, can compensate for inefficiencies. Acknowledgments
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