Quantum hologram of macroscopically entangled light via the mechanism of diffuse light storage
L. V. Gerasimov, I. M. Sokolov, D. V. Kupriyanov, M. D. Havey
aa r X i v : . [ qu a n t - ph ] N ov Quantum hologram of macroscopically entangledlight via the mechanism of diffuse light storage
L.V. Gerasimov , I.M. Sokolov , D.V. Kupriyanov ,M.D. Havey Department of Theoretical Physics, St-Petersburg State Polytechnic University,195251, St.-Petersburg, Russia Department of Physics, Old Dominion University, Norfolk, VA 23529E-mail: [email protected]
Abstract.
In the present paper we consider a quantum memory scheme for lightdiffusely propagating through a spatially disordered atomic gas. The diffuse trappingof the signal light pulse can be naturally integrated with the mechanism of stimulatedRaman conversion into a long-lived spin coherence. Then the quantum state of thelight can be mapped onto the disordered atomic spin subsystem and can be storedin it for a relatively long time. The proposed memory scheme can be applicable forstorage of the macroscopic analog of the Ψ ( − ) Bell state and the prepared entangledatomic state performs its quantum hologram, which suggests the possibility of furtherquantum information processing.PACS numbers: 34.50.Rk, 34.80.Qb, 42.50.Ct, 03.67.Mn
Keywords : Cold atoms, Light storage and quantum memory, Entangled states uantum hologram of macroscopically entangled light
1. Introduction
At present cold atomic systems have shown themselves as promising candidates foreffective light storage, see recent reviews of the problem in Refs.[1, 2]. However furtherimprovement of atomic memory efficiencies in either cold or warm atomic vapors isa rather challenging, and not so straightforward, experimental task. In the case ofwarm atomic vapors any increase of the sample optical depth meets a serious barrierfor the electromagnetically induced transparency (EIT) effect because of the rathercomplicated, and mostly negative, influence of atomic motion and Doppler broadening,which manifest themselves in destructive interference among the different hyperfinetransitions of alkali-metal atoms [3]. In the case of cold and dilute atomic gases, preparedfor instance in a magneto-optical trap (MOT), for some unique experimental designs anoptical depth around several hundred is attainable [4], but there is are certain challengesin accumulation of such a large number of atoms in a MOT and in making such a systemcontrollable.One possible solution implies special arrangements of effective light storage in acold atomic sample in the diffusive regime, see [5]. In this case, in an optically denseatomic sample with a given number of atoms, the actual random optical path of lighttransport becomes much longer than for the single passage of the same sample in theforward direction either under conditions of the EIT effect or in the near resonancetransparency spectral window. As a rough estimate, if for an atomic medium formed ina MOT the optical depth on a closed resonance transition is b , then the actual diffusivepath can be b Σ ∼ b i. e. b times longer. For typical parameters b ∼ −
50 wehave a very promising enhancement resource for the light storage via stimulated Ramanconversion of the signal pulse as it interacts with the atomic sample in the diffusiveregime.In the present report we discuss such a diffusive quantum memory mechanismin the context of its application to storage of macroscopically entangled light.One experimental technique for generation of polarization entangled light by nearsubthreshold SPDC type-II light source is well established now, see Ref.[6]. Analternative approach under development utilizes the polarization self rotation effect togenerate significantly polarization squeezed light Refs.[7]-[10]. This approach has thesubstantial advantage of generating quantum states of light with narrow bandwidthand tunability in the vicinity of atomic resonances, important characteristics for anatomic physics based quantum memory. In either case, the generated light possessesthe quantum information encoded by some mechanism and, in the case of Ref. [6], inthe strongly correlated photon numbers related to different polarization modes. Eachmode can be stored in a particular memory unit via transforming the unknown numberof photons to the equally unknown number of atoms repopulating the signal level.Importantly, the specifics of the considered memory scheme and of the stored quantumstate is that there is no need for further recovering of the signal light in its original modeor modes. We show that all the stored quantum correlations can be observed in the uantum hologram of macroscopically entangled light Figure 1. (Color online) The mechanism of diffuse storage of light for the exampleof light trapping on the F = 3 → F = 4 closed transition in Rb. The diffusepropagation of a signal mode of frequency ω is indicated by pink thin lines and arrows.The diffusion process is affected by a strong control mode of frequency ω c indicatedby red and thick arrows. This converts a signal pulse into a long-lived spin coherencein the atomic subsystem. standard interferometric technique via relevant operations with the prepared quantumhologram.
2. The mechanism of diffuse light storage
The considered light storage mechanism is based on stimulated Raman conversion ofa signal pulse into a long-lived spin coherence. The main difference with traditionalapproaches, see Refs.[1, 2], requires that the pulse diffusely propagates through anatomic sample. The hyperfine energy structure of heavy alkali-metal atoms, such asrubidium or cesium, allows convenient integration of these processes Ref.[5]. In Figure1 we illustrate this through the example of the F = 3 → F = 4 closed transition in Rb. The crucial point for the protocol is the presence and strong action of the controllaser mode repopulating the atoms from the background F = 3 hyperfine sublevel tothe signal F = 2 sublevel via Raman interaction with the upper F = 3 , ω pulse ≪ Γ AT b Σ ∼ O (1) Ω c ∆ γ b Σ (1) uantum hologram of macroscopically entangled light AT ∼ O (1) Ω c ∆ γ , where γ is the natural radiative decay rate, is an estimate forthe bandwidth of the Autler-Townes (AT) resonance, created by the control field. Thebandwidth is expressed by the averaged Rabi frequency for the control field Ω c and byits averaged detuning ∆ from those upper state hyperfine sublevels, which are involvedin the Raman process. In our case ∆ can be estimated by the hyperfine splitting ∆ hpf between the F = 4 and F = 3 sublevels in the upper state. The dimensionless opticaldepth b Σ can be estimated as b Σ ∼ n ( λ/ π ) L Σ , where n is a typical density of atomsin the sample, λ is the radiation wavelength and L Σ is the length of a diffusive path ofthe signal pulse in the sample, see Figure 1. Physically the condition (1) constrains thespectral domain where dispersion effects are manifestable. However for the spectrallynarrow pulses there can be a strong influence of the spontaneous Raman losses initiatedby the scattering on the absorption part of the AT resonance. To guarantee that thespontaneous scattering is a negligible effect, the following inequality, as an alternativeto (1), should be fulfilled∆ ω pulse ≫ Γ AT p b Σ ∼ O (1) Ω c ∆ γ p b Σ (2)Considered together both the inequalities (1) and (2) leads to an analog of the wellknown requirement for atomic memory units b Σ ≫
1, see Ref.[1], i. e. in our case thesignal pulse should pass a long transport path in the medium. As we pointed out above,in the diffusive regime this path can be made very long.The optimal pulse spectrum seems ∆ ω pulse . γ which justifies the substantialtrapping of light via the multiple scattering mechanism. In this case the variationof the Rabi frequency of the control mode is bounded by the above inequalities, whichtogether comprise the losses and dispersion effects. In reality even a standard onedimensional realization of the quantum memory protocol requires serious optimizationefforts, see Ref.[1]. Apparently in the discussed three dimensional configuration, whichis principally based on the D -line multilevel energy structure of alkali-metal atom, theoptimization scheme is expected to be much more complicated. Inequalities (1) and(2) give us a physically clear but only rough approximation, which we can consideras only the simplest qualitative recommendation. In Figure 2 a typical spectraldependence for the dielectric susceptibility of the atomic sample, modified by thepresence of the control mode, is shown. The spectra are reproduced in the vicinityof the F = 3 → F = 4 resonance line of Rb and ∆ is the relevant frequency detuning.The sample susceptibility is scaled by the dimensionless density of atoms n ( λ/ π ) andwe address the reader to Ref.[5] to see the calculation details. The upper curve indicatesthe overall ”absorption” profile, which is actually responsible not for absorption but forincoherent light scattering, and the lower curves select the contribution of the absorptionand dispersion parts of the AT-resonance only. The initial polarization direction for thesignal mode e and the polarization of the control mode e c are related to the referenceframe and the excitation geometry shown in Figure 1. The spectrum of the signalpulse, shown by the pink filled area, is narrower than the original Lorentz profile ofthe non-disturbed atomic resonance F = 3 → F = 4 but essentially broader than the uantum hologram of macroscopically entangled light Figure 2. (Color online) Dielectric susceptibility of the sample versus the pulsespectrum. The spectra are shown in the vicinity of the F = 3 → F = 4 resonanceline of Rb and ∆ is the relevant frequency detuning. The susceptibility is scaled bythe dimensionless density of atoms n ( λ/ π ) . The upper curve indicates the overallabsorption profile modified by the presence of the control mode and the lower curvesselect the contribution of the absorption and dispersion parts of the Autler-Townesresonance only. The initial polarization and propagation directions for the signal mode e , k as well as the polarization and propagation directions of the control mode e c , k c are related to the reference frame and the excitation geometry shown in Figure 1. Thepulse spectrum is shown by the pink filled area. AT resonance. We consider the flat spectral profile to follow how the photons havinga frequency uncertainty randomly distributed in the selected spectral area could bepotentially delayed via the diffuse memory protocol.In Figure 3 we demonstrate a portion of our Monte-Carlo simulations of the process.The performed calculations have been done for a spherical atomic cloud consisting of Rb atoms with a Gaussian-type radial distribution characterized by a squared variance r . The optical depth for a light ray propagating through the central point of the cloudis given by b = √ π n σ r , where n is the peak density of atoms and σ is theresonance cross-section for the F = 3 → F = 4 transition. In our calculations we used b = 20, which is an example of attainable depth in cold atom experiments with alkali-metal atom samples prepared in a MOT. The graphs of Figure 3 subsequently show howthe delay effect associated with the control field is accumulated as the scattering orderis increased. In our numerical simulations we assumed the simplest atomic distributionwith equal population of all the Zeeman sublevels in the background state ( F = 3). Thiscreates the AT resonance with rather small amplitude, see Figure 2. The resonance couldbe essentially enhanced for the atomic ensemble consisted of the spin oriented atoms,which would select the Λ-type optical transitions with the highest coupling strength.However even in the case of weak AT resonance for high scattering orders the delayeffect becomes quite visible such that a significant part of the light can be stored in the uantum hologram of macroscopically entangled light Figure 3. (Color online) These graphs subsequently show the delay induced by thecontrol field to those pulse fragments, which are freely passed forward (upper panel),and scattered in the forward direction in the 50-th (central panel) and the 100-th (lowerpanel) orders of multiple scattering. The gray dashed curve indicates the original profile(with arbitrary scaled amplitude) of the pulse incident on the sample spin subsystem. As was pointed out in the introduction, for diffusive propagation thelight can experience several hundred scattering events before it leaves the sample.The Monte Carlo simulations normally give a rather realistic approximation oflight diffusion to experimental situation but it cannot demonstrate the real potentialfor the scheme of diffuse light storage in its optimal regime. As known from manydiscussions of more simple and traditional one dimensional realizations of either EITor Raman memory protocols, see review [1] and reference therein, the ”write-in” and”readout” stages of the protocol are not completely symmetric parts of the entire process. uantum hologram of macroscopically entangled light
3. Quantum hologram
The proposed memory scheme is applicable and adjusted to the situation when thequantum information is originally encoded in the total number of photons in thesignal light beam(s), this number considered as a quantum variable. Such variablesare insensitive to either spatial or temporal mode structure of the signal light pulse.Physically this means that an unknown number of informative photons can be mappedonto the atomic subsystem via Raman-induced repopulation of the equivalent unknownnumber of the atoms to the signal level while light diffusely propagates through thesample. Such a situation takes place with storage of the macroscopic analog of theBell state, this consisting of pairs of photons with either (orthogonal) horizontal (H) orvertical (V) polarizations having unknown but strongly correlated photon numbers, seeRef. [6]. In the present report we shall consider, as an example, the following entangledquantum state of light | Ψ ( − ) i = X m,n Λ ( − ) mn | m i H | n i V | m i V | n i H (3)where Λ ( − ) mn = ( − ) n ¯ n m + n [1 + ¯ n ] m + n +1 (4) uantum hologram of macroscopically entangled light Figure 4. (Color online) The quantum hologram of the macroscopically entangledstate of light | Ψ ( − ) i , which can be prepared by a SPDC process, see Ref.[6]. Theunknown photon numbers in each polarization are subsequently stored in four memoryunits. which possesses completely anti-correlated polarizations in the light beams 1 and 2,these beams propagating in different directions. The unique property of this state isthat detection of a certain number of photons in any polarization (not only in H or V but also in any elliptical polarization state) in beam 1 guarantees the detection of thesame number of photons in beam 2 but always in an orthogonal polarization state. InRef.[6] such a state was called a macroscopic analog of a singlet-type two-particle Bellstate. Its Schmidt decomposition, given by Eqs. (3) and (4), can be found via basicexpansion for the two mode squeezed state, see Ref. [12]. The quantum state (3) canbe parameterized by the average number of photons in each light beam ¯ n .In Figure 4 we illustrate how the quantum hologram of this light can be realizedwith the memory units, as described above. Each memory unit stores the photons ofa particular polarization and frequency, which diffusely propagate through an atomiccloud and are substantially trapped on the closed transition and the stimulated Ramanprocess is initiated by interaction with the control mode ω c on other hyperfine sublevels( F = 3 ,
2) as is shown in the transition diagram of Figure 1. The more subtle point ofthis process is that it creates a pure quantum state in the atomic subsystem, which isentangled among the four clouds and has an unknown but strongly correlated numberof atoms repopulated onto the signal level F = 2 in each cloud, such that their furthermeasurement would demonstrate the presence of quantum non-locality in the mattersubsystem.A standard strategy for a quantum memory normally aims towards a goal ofrecovery of the signal pulse in its original mode. In our situation there is no need todo that since all the quantum information is encoded into the numbers of repopulatedatoms. The observation or detection of these numbers can be organized with the Mach-Zehnder interferometer, as is shown in Figure 4. The interferometer can be adjusted forbalanced detection of the signal expressed by the difference of the photocurrents from uantum hologram of macroscopically entangled light Figure 5. (Color online) Schematic diagram of the Mach-Zehnder interferometerfor detecting a small number of atoms stored in a particular cloud. The weakprobe coherent mode | α i of frequency ω p is applied near the resonance of the closed F = 2 → F = 1 transition to avoid effects of Raman scattering. The sensitivity of theinterferometer can be enhanced via sending a portion of the squeezed light | s i to thesecond input port of the interferometer, see Ref. [13]. the output ports. Then the measured signal associated with the small informative phaseshift δφ , induced in one arm of interferometer, is given by i − = i − i ∝ ¯ i δφ, δφ = ξ n (5)where the phase shift is proportional to the number of detected atoms n and to a smallfactor ξ ≪
1, which depends on geometry (sample size, aperture of the light beametc.) and reflects the weakness of the signal. The D -line energy structure of an alkali-metal atom allows one to tune the probe near the resonance associated with the closedtransition and to avoid any negative presence of the Raman scattering channel. In thecase of Rb, that is the F = 2 → F = 1 transition. Then the standard sensitivity ofthe measurement is limited by the shot-noise, Poissonian level, but it can be essentiallyimproved via sending a portion of squeezed light to the second port of interferometer,see Ref.[13].Let us make the following remark concerning the above detection scheme, which inan ideal situation would perform a certain type of quantum non demolition measurement(QND), see Ref.[14]. At first sight the scheme seems it is specifically related to thepolarization basis | H i and | V i , which was used in expansion (3) and in the storageprotocol shown in Figure 4. However, an identical expansion could be rewritten in anyother basis of arbitrary orthogonal elliptical polarizations and this would be describedby the same expansion coefficients. In other words the state | Ψ ( − ) i is insensitive tothe type of polarization beamsplitters used for the hologram creation. Then the aboveQND operation with the hologram can be interpreted as postponed detection of thephotons transmitted by the particular beamsplitters. With variation of the beamsplittertypes the measurement statistics could demonstrate violation of the classical probabilityprinciples. In a particular case of a rare flux consisting of the photons’ pairs preparedin a ”singlet state” the measurements would show violation of the Bell inequalities. Wecan also point out that the hologram yields various of interferometric operations and uantum hologram of macroscopically entangled light Acknowledgements
We thank Maria Chekhova and Timur Iskhakov for fruitful discussions, which initiatedthis work. The work was supported by RFBR (grant 10-02-00103) and NSF (grant NSF-PHY-1068159) and by Federal Program ”Scientific and scientific-pedagogical personnelof innovative Russia on 2009-2013” (contract
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