Temporal imaging for atomic single-photon systems

Adam Leszczyński,1, 2 Mateusz Mazelanik,1, 2, ∗ Michał Lipka,1, 2 Michał Parniak,1, 3 and Wojciech Wasilewski1 Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

The temporal degree of freedom of both classical and quantum states of light enables or enhances a plethora of quantum information processing tasks [1][2][3][4]. In the development of quantum network architectures and novel quantum computing and metrology solutions, a significant effort has been devoted to quantum memories based on atomic ensembles, offering multi-mode storage and processing [5][6][7][8], high efficiency [9] or long storage-times [10]. Feasible implementations of protocols merging the flexibility of atomic systems and temporal processing capabilities inherently require an ability to manipulate and detect temporal photonic modes with spectral and temporal resolution matched to the narrowband atomic emission. A versatile approach leveraging spectro-temporal duality, is to perform a frequency to time mapping -Fourier transform -in an analogy with far-field imaging in position-momentum space. To preserve quantum structure of non-classical states of light, systems relying on the concept of a time lens, are employed [20][21][22]; however, presently existing physical implementations are well suited for high-bandwidth systems and involve either electro-optical phase modulation [23][24][25], sum-frequency generation [17,[26][27][28][29] or four-wave mixing [11,15,30,31] in solid-state media. Fig. 1 localizes the existing schemes in the bandwidthresolution space. Methods relying on the time-lensing concept enable spectral shaping [32][33][34], temporal ghost imaging [35][36][37][38] and bandwidth matching [39] for photons generated in dissimilar nodes of a quantum network. While those solutions offer spectral resolution suitable for high-bandwidth photons generated in spontaneous parametric down conversion (SPDC) or quantum-dot singlephoton sources, their performance is severely limited in the case of spectrally-narrow atomic emission, interfaces with color-centres [40] or optomechanical systems [41].
field temporal imaging which is inherently bandwidth compatible with atomic systems, a regime previously unexplored as seen in Fig. 1, and works at the singlephoton-level. This allows preservation of quantum correlations and characterization of time-frequency entanglement of photons from atomic emission. Our method is analogous to far-field imaging in position-momentum space which employs a single lens with an object and image in the focal planes. We realize an analog of the lens by imposing a quadratic phase in the time domain (chirping the signal), and an analog of the free-space propagation by imprinting a quadratic spectral phase. Our approach employs the time-space mapping in a magnetic gradient echo memory (GEM) scheme [42] (see Fig. 2. An auxiliary field couples the signal to the cold-atomic GEM, which maps different frequencies onto different positions along the atomic ensemble. The coupling field is positively-chirped, effectively imposing a quadratic tem- Figure 2. (a) Ligth-atom interface. Chirped control field simultaneously allows mapping of the signal optical field onto the atomic coherence ρ hg and realizes the temporal lens. (b) Projection of signal spectral components onto atomic coherence spatial components in GEM with magnetic field gradient β. (c) After the writing process, the spatial phase of the atomic coherence is modulated with a parabolic Fresnel profile which realizes a temporal equivalent of free-space propagation. The coherence is converted back to light which is further registered with SPCM. (d) Evolution of the spectrotemporal Wigner function on subsequent stages of far-field temporal imaging which effectively rotates the initial Wigner function of two pulses (equivalent to a cat state) by π/2 as given by Eq. 1.
poral phase onto the signal light during storage, which realizes the temporal lens. Free-space propagation component involves imposing a spatial (equivalently -spectral) quadratic phase profile onto the signal stored in the memory using the spatially-resolved ac-Stark shift which we call spatial spin-wave modulation (SSM) [7,[43][44][45]. Finally, the first stage is reversed via a negatively-chirped coupling pulse. In this way final temporal lens and reverse mapping from atomic-ensemble positions to light frequencies are simultaneously realized during readout. Imaging systems generally consist of lenses interleaved with free-space propagation. Analogously, temporal imaging requires an equivalent of these two basic components. Involved transformations can be viewed in temporal or spectral domain separately, or equivalently by employing a spectro-temporal (chronocyclic) Wigner function defined as W (t, ω) , where A(t) denotes the slowly varying amplitude of the signal pulse.
A pulse with spectral amplitude A(ω) = F t [A(t)](ω) propagating in a dispersive medium for a time d t acquires a parabolic spectral phase A(ω) → A(ω) exp(−i(d t /ω 0 )ω 2 ). In the language of Wigner functions, the transformation takes a form W (t, ω) → W (t , ω) with t = t + d t ω/ω 0 . Using GEM, distinct spectral components of signal light A(ω) can be mapped onto different spatial components of the atomic coherence ρ hg (z) ∝ A(βz) [46] and vice versa, where β denotes the gradient of the Zeeman shift along the propagation (z) axis. This way, spatially resolved phase modulation of spin waves (atomic coherence) stored in the GEM is equivalent to imposing a spectral phase profile onto the signal. Thus, realizing temporal equivalent of free space propagation consists of imposing onto the spin-waves a parabolic spatial phase exp(−id t /(2ω 0 )β 2 z 2 ).
The second component -temporal lens with a focal length f t acting on a pulse with a temporal amplitude A(t) imposes a parabolic phase In the language of Wigner functions, this transformation can be written as It describes a chirped pulse with a linearly increasing frequency ω(t) = ω 0 + αt where α = ω 0 /f t and ω 0 is the carrier frequency. Instead of directly manipulating the signal, we employ a more robust strategy and chirp the coupling field which interacts with the signal to create the atomic coherence in the memory. In such a case the twophoton detuning becomes time-dependent δ = αt, yet residual modulation of the coupling efficiency is negligible as ∆ + αt ≈ ∆, and thus the single-photon detuning remains constant.
Far-field imaging is typically achieved with a single lens preceded and followed by free-space propagation by the focal length; however, such a setup is equivalent to two lenses interleaved with a single propagation. In Wigner function representation, combination of two temporal lenses with focal lengths f t , separated by a temporal propagation by the time d t = f t , is described by a transformation: which represents a π/2 rotation in the phase space, exchanging temporal and spectral domains. Consequently, the output amplitude is proportional to A (αt). In practice, the finite size of the atomic cloud must be taken into account making the output amplitude proportional to is the Fourier transform of the inhomogeneously broadened absorption efficiency spectrum η 0 (ω) determined by the atomic density distribution and field gradient β, and * symbolizes convolution.
In a typical regime of operation we select the chirp α (βL) 2 to always contain the entire spectrum of the pulse within the atomic absorption bandwidth B ≈ βL. The resolution in this regime is limited by the decoherence of spin-waves caused by the control beam of light-atom interface and is given by the inverse of the atomic coherence lifetime δω/2π = 1.76/τ (see Supplement 1for derivation of the prefactor), where 1/τ = ΓΩ 2 /(4∆ 2 + 2Γ 2 ) and Γ is the decay rate of the |e state and Ω is Rabi frequency  at the |h → |e transition. At the core of our setup is a GEM based on a cold 87 Rb atomic ensemble trapped in a magneto-optical trap (MOT) over 1-cm-long pencil-shaped volume. The MOT optical depth reaches OD ∼ 70 at the |g = 5S 1/2 , F = 2, m F = 2 → |e = 5P 1/2 , F = 1, m F = 1 transition. As depicted in Fig. 2(a),we employ the Λ system to couple light signal and atomic coherence (spin waves). The interface consists of a σ + polarized strong control laser blue-detuned by ∆ = 2π · 70 MHz from the |h = 5S 1/2 , F = 1, m F = 0 → |e transition (Rabi frequency Ω = 2π · 4.7 MHz ) and a weak σ − polarized signal laser at the |g → |e transition, approximately at the two-photon resonance δ ≈ 0. GEM scheme enables mapping of distinct signal frequencies onto different positions along the atomic cloud. The SSM scheme facilitates manipulation of the spatial phase of stored spin-waves via off-resonant ac-Stark shift by illuminating the atomic cloud with a spatially shaped strong π-polarized beam, 1 GHz blue-detuned from the 5S 1/2 , F = 1→ 5P 3/2 transition. The signal emitted in |g → |e transition is filtered using Wollaston polarizer and an optically-pumped atomic filter, to be finally registered on a single photon counting module (SPCM). We finally register only 0.023 noise counts on average per single readout (see Supplement 1).
The scheme of the experiment is presented in Fig.  2. Initially, all atoms are prepared in the |g state. The control beam is chirped with an acousto-optic modulator (AOM) to have a time-dependent frequency of ω(t) = ω 0 + αt, with α = 2π · 0.04 MHz/µs. A weak signal pulse with temporal amplitude A(t) is off-resonant from |g → |e transition. The gradient of the Zeeman splitting along the z axis during signal-to-coherence conversion is β = 2π · 1.7 MHz/cm. After GEM writing, SSM laser imposes a parabolic Fresnel phase profile onto the coherence exp(−iβ 2 /(2α)z 2 ). For simplicity, during GEM readout the control beam is no longer chirped as the imposed phase would not be registered by the the SPCM. Fig. 3(c) presents exemplary measurements performed with our setup. Red dashed lines correspond to a simple theoretical model with the output signal amplitude given byA where t 0 denotes the beginning of the readout process. Blue solid lines corresponds to the full light-atoms interaction simulation. In both cases we observe good agreement between experimental results and theoretical predictions.
Figures of merit characterizing our device are bandwidth, resolution and efficiency. Those parameters are related by a formula for GEM efficiency [46] which for atoms uniformly distributed over the length L becomes ω-independent and can be approximated as where OD is the optical depth of the ensemble for ∆ = 0. Equation 2 indicates that increased bandwidth or resolution results in a drop in efficiency. In a realistic scenario atoms are non-uniformly distributed over the cloud and thus different spectral components of the input field experience different values of OD, especially at the edges of the atomic cloud. This makes the efficiency η 0 frequencydependent and leads to an operational definition of the bandwidth B as the FWHM of the η 0 (ω) profile as depicted in Fig. 4(a). Additionally, due to the decoherence induced by the coupling field during the write (and read) process the efficiency decays exponentially in time: η = η 0 Θ(t) exp(−t/τ ) as illustrated in Fig. 4(c). Therefore, to account for spectro-temporal dependencies we introduce a time-frequency averaged efficiency: (3) Fig. 4(e) illustrates measured values ofη for different B and τ . As expected from Eq. 2, we see a clear trade-off between the time-bandwidth product τ B and the average efficiencyη. Conversely, requiring a higher number of distinguishable frequency (or time) bins leads to a lower efficiency. Yet, with ∼ 10% mean efficiency we obtain τ B = 40 that simultaneously yields 20 kHz resolution and almost 1 MHz bandwidth. Notably, as the mean efficiencyη increases with OD and saturates at the value ofη = e−1 e ≈ 63%, for systems with ultra-high OD the time-bandwidth product could reach significantly higher values while maintaining near-unity efficiency for many bins.
In summary, we have introduced and experimentally demonstrated a novel high-resolution (ca. 20 kHz) farfield imaging method suitable for narrow-band atomic photon sources -a region previously unattainable. The device may also serve as a single-photon-level ultraprecise spectrometer for atomic emission, enabling characterization of spectro-tempral high-dimensional entanglement generated with atoms. In general, while temporal domain characterization and manipulation at the single-photon level is already indispensable in numer-ous quantum information processing tasks, quantum networks architectures and metrology, our device will allow those techniques to enter the ultranarrow bandwidth domain. Our method utilizes a multi-mode gradient echo memory (GEM) along recently developed processing technique -spatial spin-wave modulator (SSM) [7,44,45] -enabling nearly arbitrary manipulations on light states stored in GEM. Furthermore, our approach utilizes a quantum memory previously demonstrated [6,7] to operate with quantum states of light, and maintains the ultra low level of noise, creating novel possibilities in temporal and spectral processing of narrow-band atomic-emission quantum states of ligh. Our technique applied to systems with higher abosorption bandwidth [47] or optical depth [9] can bridge the badwidth gap to enable hybrid solidstate-atomic quantum networks operating using the full temporal-spectral degree of freedom.

Light-atoms interaction
To describe interaction between light and atomic coherence we use three level atom model with adiabatic elimination. The most comfortable coordinate system runs in time with beam t → t + z/c. We explicitly make the control Rabi frequency Ω(t) time-dependent, as this is directly controlled in the experiment. Notably, Ω(t) represents the slowly-varying amplitude of this control field. Furthermore, we write the equations in terms of demodulated zero-spatial-frequency coherencě ρ hg (z, t) = ρ hg (z, t)e iKz0z−i∆HFSt , where ρ hg (z, t) is the actual ground-state coherence, ∆ HFS ≈ 2π · 6.8 GHz is the hyperfine splitting between levels |g and |h and y − ω C /c (ω C -coupling field frequency, k x , k y -transverse spatial components of the signal beam with respect to the coupling beam; for our case k y = 0 and ck x /ω 0 ≈ 8 mrad). Then, the lightcoherence evolution is given by following coupled equations written in the frame of reference co-moving with the optical pulse: where 1/(2τ ) = |Ω(t)| 2 Γ/(8∆ 2 + 2Γ 2 ) is decoherence caused by radiative broadening, δ tot = δ 0 + δ acS + δ SSM + δ Z is total two-photon detuning including ac-Stark shift caused by control beam δ acS = |Ω(t)| 2 ∆/(4∆ 2 +Γ 2 ), SSM and spatially varying Zeeman shift δ Z = µ 0 1 2 g 1/2 B 0 + βz caused by linearly varying external magnetic field B = B 0 + 2β µ0g 1/2 z, where g 1/2 ≈ 2 is the fine-structure Landé factor and µ 0 is the Bohr magneton The atomic concentration is denoted by n(z) and we define the ensemble optical depth as OD = gn(z)dz. In practice the value of δ 0 is chosen to cancel out the light shift caused by the control field: δ 0 = −δ acS .

Spectral resolution
The spectral resolution of the device is limited by finite duration T of the measurement window combined with the exponential decay of the atomic coherence caused by the control field. One could consider that upper limit for T is given by combination of the bandwidth B and the control field chirp α by T max = B/α as for αT > B a monochromatic input fieldÃ(ω) = δ(ω) lies outside the inhomogeneously broadened absorption spectrum. However, in the usual operation regime we set α B 2 and to maintain high initial efficiency we always have τ < T max . In this regime the finite atomic coherence lifetime τ limits the available measurement time T which we set to be T = τ /2 to maintain high overall efficiencyη. To estimate the resolution accounting for both τ and T we calculate the power spectrum of a monochromatic input pulse with exponentially decaying amplitude and define the spectral resolution δω as FWHM of the power spectrum |Ã(ω)| 2 . We numerically find δω/2π ≈ 1.76/τ .

Group-delay dispersion estimate
By imposing the parabolic phase shift onto the atomic ensemble, we imitate temporal imaging setups that use group-delay dispersion in chriped fiber Bragg gratings (CFBG), or just fibers, to achieve large group delays. The temporal propagation length we achieve in our setup amounts to d t = 9500 s which corresponds to a GDD of 25 µs 2 over our 1 MHz bandwidth. To achieve such GDD, one would need 10 12 km of typical telecom fiber (GDD 25 ps 2 /km) or billions of commercially available CFBGs (GDD ∼ 10 4 ps 2 ).  Figure S1. Filtering setup. The signal separated from the coupling laser light using a sequence of far field apertures, Wollaston polarizer, near field aperture, optically pumped atomic filter and interference filter. Transmission of the signal photons through this system amounts to about 50%.

Magnetic field
To determine the quantization axis the atomic cloud is kept in external constant ∼ 1 G magnetic field along the cloud. The gradient of magnetic field for GEM can be quickly switched (0.5 G/cm/µs) to the opposite using a state-of-the-art switch based on MOSFETs. Blue lines corresponds to its integrals along y and z axis. Red area corresponds to the part of atomic cloud illuminated with signal laser. The orange line presents concentration distribution along the z axis inferred from the absorption profile in the presence of a known magnetic field gradient. Fig. S3 shows the image of atomic cloud from the side. Atoms are formed into pencil shape area with diameter of about 0.5 mm. The signal laser diameter amounts to about 0.1 mm and illuminates the middle of the atomic cloud, where the optical depth is the highest. Fig. S4 presents single photon absorption profile of the signal. Fitting the Lorentz profile, we estimated that optical depth amounts to about 76.

Filtering system and noise characterization
To minimize the noise we need to efficiently filter signal photons from the control beam and other noise. For this purpose, we have built a multi-stage filtering system (Fig.   S1). Firstly, we filter most of control beam photons using far field aperture. Next, we use the fact that they have orthogonal polarization to the signal photons and we filter them using quater-wave plate and Wollaston polarizer. After that we use near field aperture to remove photons scattered in other parts of MOT. Later, the glass cell containing Rubidium-87 pumped to 5S 1/2 , F = 1 state with 780 nm laser and buffer gas (nitrogen) is used to filter out stray control beam light while preserving the multimode nature of our device as compared to the cavity based filtering. Finally we use a 795 nm interference filter to remove other frequency photons, coming mainly from the filter pump. Fig. S2 presents noise count rate as a function of atomic coherence decay rate, which is proportional to the intensity of the control beam. The slope of the fitted line can be interpreted as average number of noise photons registered during the readout process. Note that as we increase the coupling laser intensity, we register more noise photons yet during a shorter window. This gives us a constant mean photon number per readout. Thanks to our filtering system we achieved the value ofn noise = 0.023 which means, that we register approximately 1 noise photon per 40 single experiments. Simultaneously, the transmission of the signal photons amounts to about 60%, while the detection efficiency is65%. With a typical memory process efficiency of 25%, we obtain noise per single photon sent to the device µ 1 = 0.23, which corresponds to µ 1 = 0.016 per single mode (i.e. in a single temporal mode storage experiment). The main limitation is still filtering of coupling light, as evidenced by removing the atomic ensemble and still observing the same noise level. That could be improved further by coupling the signal to a single-mode fibre or using more efficient filtering.