Abstract

Plenty of quantum information protocols are enabled by manipulation and detection of photonic spectro-temporal degrees of freedom via light–matter interfaces. While present implementations are well suited for high-bandwidth photon sources such as quantum dots, they lack the high resolution required for intrinsically narrowband light–atom interactions. Here, we demonstrate far-field temporal imaging based on ac-Stark spatial spin-wave phase manipulation in a multimode gradient echo memory. We achieve a spectral resolution of 20 kHz with MHz-level bandwidth and an ultralow noise equivalent to 0.023 photons, enabling operation in the single-quantum regime.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The temporal degree of freedom of both classical and quantum states of light enables or enhances a plethora of quantum information processing tasks [14]. 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 multimode storage and processing [58], 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—a Fourier transform—in an analogy with far-field imaging in position-momentum space. To preserve the quantum structure of nonclassical states of light, systems relying on the concept of a time lens are employed [1113]; however, presently existing physical implementations are well suited for high-bandwidth systems and involve either electro-optical phase modulation [1416], sum-frequency generation [1721], or four-wave mixing [2225] in solid-state media. Figure 1 localizes the existing schemes in the bandwidth-resolution space. Methods relying on the time-lensing concept enable spectral shaping [2628], temporal ghost imaging [2932], and bandwidth matching [33] for photons generated in dissimilar nodes of a quantum network. While those solutions offer a spectral resolution suitable for high-bandwidth photons generated in spontaneous parametric down conversion (SPDC) or quantum-dot single-photon sources, their performance is severely limited in the case of spectrally ultranarrow atomic emission ranging from few MHz to tens of kHz [3436], cavity coupled ions (below 100 kHz) [37], cavity-enhanced SPDC designed for atomic quantum memories (below 1 MHz) [38], or optomechanical systems [39,40].

In this paper, we propose and experimentally demonstrate what we believe is a novel, high-spectral-resolution approach to far-field temporal imaging that is inherently bandwidth-compatible with atomic systems, a regime previously unexplored, as seen in Fig. 1, and works at the single photon level. This approach allows the preservation of quantum correlations, manipulation of field-orthogonal temporal modes [2], and characterization of the time-frequency entanglement [41] of photons from atomic emission. Our technique uses a recently developed spin-wave modulation method combined with an unusual interpretation of gradient echo memory (GEM) [42] protocol to realize a complete temporal imaging setup in one physical system.

 figure: Fig. 1.

Fig. 1. Temporal imaging state of the art, characterized by temporal $ \delta t $ and spectral $ \delta \omega $ resolutions. Numerous implementations based on solid media (electro-optic modulators or EOMs [25,4345]; four-wave mixing, or FWM [24,46]; sum-frequency generation, or SFG [20,47]; and cross-phase modulation, or XPM [48]) are well suited for high bandwidth pico- or even femtosecond pulses, achieving spectral resolution no better than 1 GHz, with time-bandwidth products ($ \tau {\cal B} $) reaching $ 2\pi \times 2000 $. Our system–GEM & SSM–has $ {10^6} $ times better spectral resolution $ \delta \omega /2\pi \sim 20\;{\rm kHz} $, maintaining good $ \tau {\cal B} $, thus allowing exploration of a previously unattainable region. The grayed region indicates an unphysical area bounded by the Fourier limit $ \tau {\cal B}/2\pi = 1 $.

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2. PRINCIPLES OF TEMPORAL IMAGING

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 the temporal or spectral domain separately, or equivalently by employing a spectro-temporal (chronocyclic) Wigner function defined as $ W(t,\omega ) = 1/\sqrt {2\pi } \int_{ - \infty }^\infty {\rm d}\xi A(t + \xi /2){A^*}(t - \xi /2)\exp ( - i\omega \xi ) $, where $ A(t) $ denotes the slowly varying amplitude of the signal pulse.

Temporal far-field imaging is typically achieved with a single time lens preceded and followed by a temporal analog of free-space propagation. However, such a setup is equivalent to two lenses interleaved with a single propagation. In the Wigner function representation, the combination of two temporal lenses with focal lengths $ {f_{\rm t}} $, separated by a temporal propagation by time $ {f_{\rm t}} $, is described using a spectro-temporal equivalent of the ray transfer matrix,

$$\begin{split}\left[ {\begin{array}{*{20}{c}}{t^{\prime}}\\{\frac{{\omega^{\prime}}}{{{\omega _0}}}}\end{array}} \right] &= \left[ {\begin{array}{*{20}{c}}1&0\\{ - \frac{1}{{{f_{\rm t}}}}}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&{{f_{\rm t}}}\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&0\\{ - \frac{1}{{{f_{\rm t}}}}}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{t}\\{\frac{\omega }{{{\omega _0}}}}\end{array}} \right]\\& = \left[ {\begin{array}{*{20}{c}}0&{{f_{\rm t}}}\\{ - \frac{1}{{{f_{\rm t}}}}}&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{t}\\{\frac{\omega }{{{\omega _0}}}}\end{array}} \right],\end{split}$$
which represents a $ \pi /2 $ rotation in the phase space, exchanging temporal and spectral domains, where $ {\omega _0} $ is the optical carrier frequency. To visualize this concept, in Fig. 2(d) we present an equivalent of a cat state Wigner function [corresponding to the sequence presented in Fig. 3(c)] undergoing these three subsequent transformations: (1) time lens, (2) propagation, and (3) time lens.
 figure: Fig. 2.

Fig. 2. (a) Light–atom interface. Chirped control field simultaneously allows mapping of the signal optical field onto the atomic coherence $ {\rho _{hg}} $ and realizes the temporal lens. (b) Projection of signal spectral components onto atomic coherence spatial components in GEM with Zeeman splitting gradient $ \beta $. (c) During the writing process atoms are placed in a negative magnetic field gradient along the cloud (${w}$). When the writing finishes, the spatial phase of the atomic coherence is modulated with a parabolic Fresnel profile that realizes a temporal equivalent of free-space propagation. Finally, the gradient is switched to positive (${r}$) and the coherence is converted back to light, which is further registered with single photon counting module (SPCM) connected to the time tagger (TTG). (d) Evolution of the spectro-temporal Wigner function on subsequent stages of far-field temporal imaging: (1) time lens, (2) free-space propagation, and (3) time lens. The complete transformation effectively rotates the initial Wigner function of two pulses (equivalent to a Wigner function of a cat state in phase space) by $ \pi /2 $, as given by Eq. (1).

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 figure: Fig. 3.

Fig. 3. Experimental sequence for temporal imaging. (a) Time-trace of the Zeeman shift gradient $ \beta $ used in the GEM protocol, allowing two-directional mapping of signal frequencies to distinct positions in the atomic cloud. (b) Control field (red) and SSM (yellow) laser pulse sequence divided into three stages corresponding to lens–propagation–lens operations. The lens (1) is implemented during the GEM writing process by a chirped control field. (2) The 3 µs long SSM laser pulse imprints a parabolic phase profile onto the stored atomic coherence, which realizes the spectro-temporal free-space propagation. During this stage, the magnetic field gradient is reversed, allowing remapping the coherence to light. (3) Finally, the control field is turned on and the coherence is read out from the memory. Chirping the control field would implement the second lens. However, for simplicity, the control field is no longer chirped as the imposed phase would not be registered by the SPCM. (c, e) Example results for two pulses or a sine wave as inputs, respectively. Gray bins represent single photon counts. Red line corresponds to the numerical simulations. (d, f) Normalized modulus square of atomic coherence in Fourier space. The insets (i, ii) show experimentally obtained linear dependency of the time delay $ \Delta t $ (in $\unicode{x00B5}{\rm s} $) on the signal modulation frequency $ f $ (in MHz) defined on panels (c, e).

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To realize the time lens with a focal length $ {f_{\rm t}} $, one has to impose a quadratic phase $ \exp [i{\omega _0}{t^2}/(2{f_{\rm t}})] $ on the optical pulse $ A(t) \to A(t)\exp [i{\omega _0}{t^2}/(2{f_{\rm t}})] $, where $ {\omega _0} $ is the optical carrier frequency. In the language of Wigner functions, this transformation can be written as $ W(t,\omega ) \to W(t,\omega^{\prime}) $ with $ \omega^{\prime}= \omega - {\omega _0}t/{f_{\rm t}} $. This corresponds to adding to the pulse a linear chirp $ \omega (t) = {\omega _0} + \alpha t $. Typically, such a transformation is achieved by directly modulating the signal pulse using electro-optic modulators [25,4345].

The analog of free-space propagation can be understood as a frequency-dependent delay applied to an optical pulse. In the language of the Wigner function, the transformation takes the form $ W(t,\omega ) \to W(t^{\prime},\omega ) $ with $ t^{\prime} = t + {f_{\rm t}}\omega /{\omega _0} $. Equivalently, a pulse with spectral amplitude $ \tilde A(\omega ) = {{\cal F}_t}[A(t)](\omega ) $ must acquire a parabolic spectral phase $ \tilde A(\omega ) \to \tilde A(\omega )\exp [ - i({f_{\rm t}}/{\omega _0}){\omega ^2}] $. Commonly, such an operation is realized directly by propagation in dispersive media [16] or by employing a pulse stretcher/compressor.

3. TEMPORAL IMAGING USING GEM AND SSM

Our technique employs an atomic ensemble to process stored light and implement the temporal imaging operations during storage or light–atom mapping. The optical signal amplitude $ A(t) $ is mapped onto the atomic coherence in a $ \Lambda $ type system built of three atomic levels $ |g\rangle $, $ |h\rangle $, and $ |e\rangle $ [see Fig. 2(a)]. The mapping process employs a strong control field to make the atoms absorb the signal field and generate an atomic coherence $ {\rho _{hg}} $ commonly called a spin wave (SW). During the mapping (and remapping) process, the atoms are kept in a magnetic field gradient, which constitutes the basis of the GEM [42], providing linearly changing Zeeman splitting between the $ |h\rangle $ and $ |e\rangle $ levels along the atomic cloud. This means that the signal-control two-photon interaction happens with a spatially dependent two-photon detuning $ \delta $, and only atoms contained in a limited volume will interact efficiently with the signal light of a specific frequency. Therefore, distinct spectral components of the signal light $ \tilde A(\omega ) $ are mapped onto different spatial components of the atomic coherence $ {\rho _{hg}}(z) \propto \tilde A(\beta z) $ [42,49] [see Fig. 2(b)] and vice versa, where $ \beta $ denotes the Zeeman shift gradient along the propagation ($ z $) axis.

The temporal equivalent of free space propagation is realized thanks to this spectro-spatial mapping, a GEM characteristic. Spatially resolved phase modulation of a SW stored in the memory is equivalent to imposing a spectral phase profile onto the signal. Thus, by imposing onto the SWs a parabolic spatial phase $ \exp [ - i{f_{\rm t}}/(2{\omega _0}){\beta ^2}{z^2}] $, we implement the temporal analog of free-space propagation. This operation is implemented using the spatially resolved ac-Stark shift induced by an additional far-detuned and spatially-shaped laser beam, a technique we call spatial spin-wave modulation (SSM) [7,5052].

To make the time lens, we use the fact that the SW is created in coherent two-photon absorption; thus it reflects the temporal phase difference between the control and signal field. This means that chirping the signal field is equivalent to chirping the control field, as only the two-photon detuning $ \delta $ is crucial here. Hence, by changing the control field frequency, we make the two-photon detuning linearly time dependent $ \delta = \alpha t $ and impose the desired quadratic phase during the interaction time. This way the time lens is realized at the light–SW mapping stage without affecting the signal field directly. Yet, as we chose the single-photon detuning $ \Delta \gg \delta $ residual modulation of the coupling efficiency is negligible as $ \Delta + \alpha t \approx \Delta $.

Finally, sending a signal field $ A(t) $ through a lens–propagation–lens temporal imaging system, the output amplitude is proportional to $ \tilde A( {\alpha t} ) $. In practice, however, the finite size of the atomic cloud must be taken into account, making the output amplitude proportional to $ ( [ \tilde A(\alpha t) \exp [ - i(\alpha /2){t^2}] ]*\zeta (t)*$ $\zeta (t) )\exp [i(\alpha /2){t^2}] $, where $ \zeta (t) = {{\cal F}_\omega }[{\eta _0}(\omega )](t) $ is the Fourier transform of the inhomogeneously broadened absorption efficiency spectrum $ {\eta _0}(\omega ) $ determined by the atomic density distribution and field gradient $ \beta $, and $ * $ symbolizes convolution.

In a typical regime of operation, we select the chirp $ \alpha \ll {(\beta L)^2} $ to always contain the entire spectrum of the pulse within the atomic absorption bandwidth $ {\cal B} \approx \beta L $. The resolution in this regime is limited by the decoherence of spin waves caused by the control beam of the light–atom interface and is given by the inverse of the atomic coherence lifetime $ \delta \omega /2\pi = 0.78/\tau $ (see Supplement 1 for derivation of the prefactor), where $ 1/\tau = \Gamma {\Omega ^2}/(4{\Delta ^2} + {\Gamma ^2}) $ and $ \Gamma $ is the decay rate of the $ |e\rangle $ state and $ \Omega $ is the Rabi frequency at the $ |h\rangle \to |e\rangle $ transition.

4. EXPERIMENT

The core of our setup is a GEM based on a cold $ ^{87}{\rm Rb} $ atomic ensemble trapped in a magneto-optical trap (MOT) over a 1 cm long pencil-shaped volume. After the MOT release, all atoms are optically pumped to the $ |g\rangle = 5{{\rm S}_{1/2}} $, $ F = 2 $, $ {m_F} = 2 $ state. The ensemble optical depth reaches $ {\rm OD} \sim 70 $ at the $ |g\rangle \to |e\rangle = 5{P_{1/2}} $, $ F = 1 $, and $ {m_F} = 1 $ transition. As depicted in Fig. 2(a), we employ the $ \Lambda $ system to couple the light signal and atomic coherence (spin waves). The interface consists of a $ {\sigma _ + } $ polarized strong control laser blue-detuned by $ \Delta = 2\pi \times 70\,{\rm MHz} $ from the $ |h\rangle = 5{S_{1/2}} $, $ F = 1 $, and $ {m_F} = 0 \to |e\rangle $ transition, and a weak $ {\sigma _ - } $ polarized signal laser at the $ |g\rangle \to |e\rangle $ transition, approximately at the two-photon resonance $ \delta \approx 0 $. The gradient $ \beta $ of the Zeeman splitting along the $ z $ axis during the signal-to-coherence conversion is generated by two rounded-square shaped coils connected in opposite configuration (see Supplement 1 for details). The SSM scheme facilitates manipulation of the spatial phase of stored spin waves via an off-resonant ac-Stark shift by illuminating the atomic cloud with a spatially shaped strong $ \pi $-polarized beam, 1 GHz blue-detuned from the $ 5{S_{1/2}} $, $ F = 1 \to 5{P_{3/2}} $ transition. The signal emitted in the $ |g\rangle \to |e\rangle $ transition is filtered using a 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 a $ \tau $-long detection window (see Supplement 1).

In Fig. 3, we present exemplary measurements performed with our setup. Panel (a) shows the time trace of the Zeeman shift gradient set initially to $ \beta = - 2\pi \times 1.7\,{\rm MHz}/{\rm cm} $. In panel (b), we provide the control and SSM laser sequence divided into three stages corresponding to subsequent implementations of the lens–propagation–lens operations. (1) First, a strong control field (Rabi frequency $ \Omega = 2\pi \times 4.7\,{\rm MHz} $) is used to map a weak ($ \bar n = 2.8 $) signal pulse with a temporal amplitude $ A(t) $ to the atomic coherence. The control beam is chirped with an acousto-optic modulator (AOM) to have a time-dependent frequency of $ \omega (t) = {\omega _0} + \alpha t $, with $ \alpha = 2\pi \times 0.04\,{\rm MHz}/\unicode{x00B5} {\rm s} $. This implements a time lens with focal length $ {f_{\rm t}} = 9.6 \times {10^3}{\rm s} $. (2) Next, within a 7 µs, the gradient $ \beta $ is switched to the opposite value and a parabolic Fresnel phase profile $ \exp [ - i{\beta ^2}/(2\alpha ){z^2}] $ [as depicted in Fig. 2(c)] is imprinted onto the stored atomic coherence by the 3 µs long SSM laser pulse. The linear gradient of the magnetic field only shifts the atomic coherence in the Fourier domain; therefore, phase modulation can be done simultaneously with $ \beta $ reversing. (3) Finally, the control field is turned on and the coherence is converted back to light. For simplicity, during the GEM readout, the control field is no longer chirped as the imposed phase would not be registered by the SPCM (see Supplement 1 for details).

Figures 3(c)3(f) portray the experimental results for two types of the input signal: (c, d) two peaks and (e, f) sine-wave-like waveform. Red solid lines correspond to the full light–atoms interaction simulation (see Supplement 1 for details). The density maps (d, f) below each time trace (c, e) show the simulated evolution of the atomic coherence during the experiment. For both input signal shapes, the measured efficiency amounts to about 7%. The insets (i, ii) show the experimentally obtained linear relationship between the time delay $ \Delta t $ and the signal modulation (temporal fringes) frequency $ f $ defined in panels (c, e).

We attribute the residual mismatch between experimental results and theoretical predictions to imperfect linearity of the magnetic field gradient, a decoherence caused by ac-Stark modulation and a simplification of the atomic density distribution in the simulation. However, for both exemplary measurements, we can still observe a good agreement with the theory. Notably, the simulations use independently calibrated parameters, with only the input photon number adjusted for the specific measurements.

5. TIME-BANDWIDTH CHARACTERIZATION

Figures of merit characterizing our device are bandwidth, resolution, and efficiency. Those parameters are related by a formula for GEM efficiency [49], which for atoms uniformly distributed over the length $ L $ becomes $ \omega $-independent and can be approximated as

$${\eta _0} = {\left[ {1 - \exp \left( { - 2\pi \frac{{{\rm OD}}}{{\tau {\cal B}}}} \right)} \right]^2},$$
where OD is the optical depth of the ensemble for $ \Delta = 0 $. Equation (2) indicates that increased bandwidth or resolution results in a drop in efficiency. In a realistic scenario, atoms are nonuniformly 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 $ {\eta _0} $ frequency-dependent and leads to an operational definition of the bandwidth $ {\cal B} $ as the FWHM of the $ {\eta _0}(\omega ) $ 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, so $ \eta = {\eta _0}\Theta (t)\exp ( - t/\tau ) $, as illustrated in Fig. 4(c). Therefore, to account for spectro-temporal dependencies, we introduce a time-frequency averaged efficiency,
$$\begin{split}\bar \eta &= \frac{1}{{2\tau {\cal B}}}\int_{ - {\cal B}/2}^{{\cal B}/2} \int_0^{2\tau } \eta (t,\omega ){\rm d}t{\rm d}\omega\\& = \frac{{{e^2} - 1}}{{2{e^2}{\cal B}}}\int_{ - {\cal B}/2}^{{\cal B}/2} {\eta _0}(\omega ){\rm d}{\omega}.\end{split}$$
 figure: Fig. 4.

Fig. 4. Characterization and tuning of bandwidth and resolution. (a) Efficiency spectral profile $ {\eta _0}(\omega ) $ as a function of the two-photon detuning $ \delta = \omega - {\omega _0} $ for a chosen time-bandwidth product $ \tau {\cal B} = 2\pi \times 13 $ with bandwidth $ {\cal B} $ defined as the FWHM of $ {\eta _0}(\omega ) $. The red line corresponds to a super-Gaussian approximation of the atom concentration used in the simulation. (b) Dependence of the bandwidth $ {\cal B} $ as a function of the Zeeman splitting gradient $ \beta $. Red line is a linear fit to the data. (c) Time evolution of the GEM efficiency due to incoherent scattering caused by the coupling field. The characteristic decay time $ \tau $ obtained from exponential fit (red line) limits the effective resolution $ \delta \omega /2\pi = 0.78/\tau $ (here $ \tau = 10\;\unicode{x00B5} {\rm s} $). (d) Dependence of $ 1/\tau $ as a function of the coupling field power $ P \propto |\Omega {|^2} $, along with linear fit (red line). (e) Calculated map of the average efficiency $ \bar \eta $ for varying bandwidth $ {\cal B} $ and decay time $ \tau $. The efficiency for a given time-bandwidth product $ \tau {\cal B} $ is approximately constant as expected. The star indicates the point of operation where the exemplary measurements (Fig. 3) were performed.

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Figure 4(e) illustrates measured values of $ \bar \eta $ for different $ {\cal B} $ and $ \tau $. The map is built from separate measurements of $ {\eta _0}(\omega ) $ [Fig. 4(a)] and $ \tau $ [Fig. 4(c)] for a different Zeeman shift gradient $ \beta $ [Fig. 4(b)] and the control laser power $ P \propto |\Omega {|^2} $ [Fig. 4(d)]. The parameters extracted from Figs. 4(a)4(d) are then combined to give a value of $ \bar \eta $ for each $ (\tau (P),{\cal B}(\beta )) $ point. As expected from Eq. (2), we see a clear trade-off between the time-bandwidth product $ \tau {\cal B} $ and the average efficiency $ \bar \eta $. Conversely, requiring a higher number of distinguishable frequency (or time) bins leads to a lower efficiency. Yet, with $ \sim 20\% $ mean efficiency, we obtain $ \tau {\cal B} = 2\pi \times 10 $, which simultaneously yields 100 kHz resolution and 1 MHz bandwidth. One may also choose to maximize $ \tau {\cal B} $ to reach $ 2\pi \times 40 $, with a mean efficiency $ \sim 4\% $. Notably, as the efficiency $ {\eta _0} $ saturates for a large OD, for systems with ultrahigh optical depth the time-bandwidth product could reach significantly higher values while maintaining near-unity efficiency for many bins.

6. CONCLUSIONS

In summary, we have introduced and experimentally demonstrated what we believe is a novel high-resolution (ca. 20 kHz) far-field imaging method suitable for narrowband atomic photon sources, which is a region previously unattainable. The device may also serve as a single-photon-level ultraprecise spectrometer for atomic emission, enabling characterization of spectro-temporal, high-dimensional entanglement generated with atoms. In general, while temporal domain characterization and manipulation at the single-photon level is already indispensable in numerous quantum information processing tasks, quantum networks architectures, and metrology, our device will allow those techniques to enter the ultranarrow bandwidth domain. Our method uses a multimode GEM along a recently developed processing technique called SSM [7,51,52] that enables nearly arbitrary manipulations on light states stored in GEM. We envisage that with improvement of the magnetic field gradient the GEM bandwidth can reach dozens of MHz, opening new ranges of applications such as solid-state quantum memories [53] and color centers [54]. Furthermore, our approach uses a quantum memory previously demonstrated [6,7] to operate with quantum states of light, and maintains the ultralow level of noise, creating new possibilities in temporal and spectral processing of narrowband atomic-emission quantum states of light. Our technique applied to systems with a higher absorption bandwidth [55] or optical depth [9] can bridge the bandwidth gap to enable hybrid solid-state–atomic quantum networks operating using the full temporal-spectral degree of freedom.

Funding

Narodowe Centrum Nauki (2016/21/B/ST2/02559, 2017/25/N/ST2/00713, 2017/25/N/ST2/01163); Ministerstwo Nauki i Szkolnictwa Wyższego (DI2016 014846, DI2018 010848); Office of Naval Research (N62909-19-1-2127); Fundacja na rzecz Nauki Polskiej (MAB/2018/4 “Quantum Optical Technologies”).

Acknowledgment

We thank K. Banaszek for generous support and M. Jachura for insightful discussion. Michał Parniak was supported by the Foundation for Polish Science via the START scholarship. Adam Leszczyński and Mateusz Mazelanik contributed equally to this work. The Quantum Optical Technologies project is carried out within the International Research Agendas program of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund.

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

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25. M. A. Foster, R. Salem, Y. Okawachi, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Ultrafast waveform compression using a time-domain telescope,” Nat. Photonics 3, 581–585 (2009). [CrossRef]  

26. B. Li, M. R. Fernández-Ruiz, S. Lou, and J. Azaña, “High-contrast linear optical pulse compression using a temporal hologram,” Opt. Express 23, 6833 (2015). [CrossRef]  

27. J. M. Donohue, M. Mastrovich, and K. J. Resch, “Spectrally engineering photonic entanglement with a time lens,” Phys. Rev. Lett. 117, 243602 (2016). [CrossRef]  

28. H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018). [CrossRef]  

29. S. Denis, P. A. Moreau, F. Devaux, and E. Lantz, “Temporal ghost imaging with twin photons,” J. Opt. 19, 34002 (2017). [CrossRef]  

30. S. Dong, W. Zhang, Y. Huang, and J. Peng, “Long-distance temporal quantum ghost imaging over optical fibers,” Sci. Rep. 6, 26022 (2016). [CrossRef]  

31. P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Magnified time-domain ghost imaging,” APL Photon. 2, 46102 (2017). [CrossRef]  

32. H. Wu, P. Ryczkowski, A. T. Friberg, J. M. Dudley, and G. Genty, “Temporal ghost imaging using wavelength conversion and two-color detection,” Optica 6, 902 (2019). [CrossRef]  

33. M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017). [CrossRef]  

34. L. Zhao, X. Guo, C. Liu, Y. Sun, M. M. T. Loy, and S. Du, “Photon pairs with coherence time exceeding 1 µs,” Optica 1, 84 (2014). [CrossRef]  

35. X. Guo, Y. Mei, and S. Du, “Testing the Bell inequality on frequency-bin entangled photon pairs using time-resolved detection,” Optica 4, 388 (2017). [CrossRef]  

36. P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016). [CrossRef]  

37. A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012). [CrossRef]  

38. M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, “Sub-megahertz linewidth single photon source,” APL Photon. 1, 096101 (2016). [CrossRef]  

39. S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017). [CrossRef]  

40. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012). [CrossRef]  

41. Y. Mei, Y. Zhou, S. Zhang, J. Li, K. Liao, H. Yan, S.-L. Zhu, and S. Du, “Einstein–Podolsky–Rosen energy-time entanglement of narrowband biphotons,” Phys. Rev. Lett. 124, 010509 (2020). [CrossRef]  

42. M. Hosseini, B. M. Sparkes, G. Hétet, J. J. Longdell, P. K. Lam, and B. C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009). [CrossRef]  

43. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994). [CrossRef]  

44. J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectral Fraunhofer regime: time-to-frequency conversion by the action of a single time lens on an optical pulse,” Appl. Opt. 43, 483–490 (2004). [CrossRef]  

45. H. Babashah, Z. Kavehvash, A. Khavasi, and S. Koohi, “Temporal analog optical computing using an on-chip fully reconfigurable photonic signal processor,” Opt. Laser Technol. 111, 66–74 (2019). [CrossRef]  

46. R. Salem, M. A. Foster, A. C. Turner-Foster, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “High-speed optical sampling using a silicon-chip temporal magnifier,” Opt. Express 17, 4324–4329 (2009). [CrossRef]  

47. P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016). [CrossRef]  

48. L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000). [CrossRef]  

49. B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013). [CrossRef]  

50. A. Leszczyński, M. Mazelanik, M. Lipka, M. Parniak, M. Dąbrowski, and W. Wasilewski, “Spatially resolved control of fictitious magnetic fields in a cold atomic ensemble,” Opt. Lett. 43, 1147 (2018). [CrossRef]  

51. M. Parniak, M. Mazelanik, A. Leszczyński, M. Lipka, M. Dąbrowski, and W. Wasilewski, “Quantum optics of spin waves through ac stark modulation,” Phys. Rev. Lett. 122, 063604 (2019). [CrossRef]  

52. M. Lipka, A. Leszczyński, M. Mazelanik, M. Parniak, and W. Wasilewski, “Spatial spin-wave modulator for quantum-memory-assisted adaptive measurements,” Phys. Rev. Appl. 11, 034049 (2019). [CrossRef]  

53. M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010). [CrossRef]  

54. H. Jeong, S. Du, and N. Y. Kim, “Proposed narrowband biphoton generation from an ensemble of solid-state quantum emitters,” J. Opt. Soc. Am. B 36, 646 (2019). [CrossRef]  

55. E. Saglamyurek, T. Hrushevskyi, A. Rastogi, K. Heshami, and L. J. LeBlanc, “Coherent storage and manipulation of broadband photons via dynamically controlled Autler–Townes splitting,” Nat. Photonics 12, 774–782 (2018). [CrossRef]  

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  49. B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
    [Crossref]
  50. A. Leszczyński, M. Mazelanik, M. Lipka, M. Parniak, M. Dąbrowski, and W. Wasilewski, “Spatially resolved control of fictitious magnetic fields in a cold atomic ensemble,” Opt. Lett. 43, 1147 (2018).
    [Crossref]
  51. M. Parniak, M. Mazelanik, A. Leszczyński, M. Lipka, M. Dąbrowski, and W. Wasilewski, “Quantum optics of spin waves through ac stark modulation,” Phys. Rev. Lett. 122, 063604 (2019).
    [Crossref]
  52. M. Lipka, A. Leszczyński, M. Mazelanik, M. Parniak, and W. Wasilewski, “Spatial spin-wave modulator for quantum-memory-assisted adaptive measurements,” Phys. Rev. Appl. 11, 034049 (2019).
    [Crossref]
  53. M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
    [Crossref]
  54. H. Jeong, S. Du, and N. Y. Kim, “Proposed narrowband biphoton generation from an ensemble of solid-state quantum emitters,” J. Opt. Soc. Am. B 36, 646 (2019).
    [Crossref]
  55. E. Saglamyurek, T. Hrushevskyi, A. Rastogi, K. Heshami, and L. J. LeBlanc, “Coherent storage and manipulation of broadband photons via dynamically controlled Autler–Townes splitting,” Nat. Photonics 12, 774–782 (2018).
    [Crossref]

2020 (1)

Y. Mei, Y. Zhou, S. Zhang, J. Li, K. Liao, H. Yan, S.-L. Zhu, and S. Du, “Einstein–Podolsky–Rosen energy-time entanglement of narrowband biphotons,” Phys. Rev. Lett. 124, 010509 (2020).
[Crossref]

2019 (8)

H. Babashah, Z. Kavehvash, A. Khavasi, and S. Koohi, “Temporal analog optical computing using an on-chip fully reconfigurable photonic signal processor,” Opt. Laser Technol. 111, 66–74 (2019).
[Crossref]

M. Parniak, M. Mazelanik, A. Leszczyński, M. Lipka, M. Dąbrowski, and W. Wasilewski, “Quantum optics of spin waves through ac stark modulation,” Phys. Rev. Lett. 122, 063604 (2019).
[Crossref]

M. Lipka, A. Leszczyński, M. Mazelanik, M. Parniak, and W. Wasilewski, “Spatial spin-wave modulator for quantum-memory-assisted adaptive measurements,” Phys. Rev. Appl. 11, 034049 (2019).
[Crossref]

H. Jeong, S. Du, and N. Y. Kim, “Proposed narrowband biphoton generation from an ensemble of solid-state quantum emitters,” J. Opt. Soc. Am. B 36, 646 (2019).
[Crossref]

H.-H. Lu, J. M. Lukens, B. P. Williams, P. Imany, N. A. Peters, A. M. Weiner, and P. Lougovski, “A controlled-NOT gate for frequency-bin qubits,” npj Quantum Inf. 5, 24 (2019).
[Crossref]

M. Mazelanik, M. Parniak, A. Leszczyński, M. Lipka, and W. Wasilewski, “Coherent spin-wave processor of stored optical pulses,” npj Quantum Inf. 5, 22 (2019).
[Crossref]

A. Seri, D. Lago-Rivera, A. Lenhard, G. Corrielli, R. Osellame, M. Mazzera, and H. de Riedmatten, “Quantum storage of frequency-multiplexed heralded single photons,” Phys. Rev. Lett. 123, 080502 (2019).
[Crossref]

H. Wu, P. Ryczkowski, A. T. Friberg, J. M. Dudley, and G. Genty, “Temporal ghost imaging using wavelength conversion and two-color detection,” Optica 6, 902 (2019).
[Crossref]

2018 (4)

H. H. Lu, J. M. Lukens, N. A. Peters, O. D. Odele, D. E. Leaird, A. M. Weiner, and P. Lougovski, “Electro-optic frequency beam splitters and tritters for high-fidelity photonic quantum information processing,” Phys. Rev. Lett. 120, 030502 (2018).
[Crossref]

G. Patera, D. B. Horoshko, and M. I. Kolobov, “Space-time duality and quantum temporal imaging,” Phys. Rev. A 98, 053815 (2018).
[Crossref]

E. Saglamyurek, T. Hrushevskyi, A. Rastogi, K. Heshami, and L. J. LeBlanc, “Coherent storage and manipulation of broadband photons via dynamically controlled Autler–Townes splitting,” Nat. Photonics 12, 774–782 (2018).
[Crossref]

A. Leszczyński, M. Mazelanik, M. Lipka, M. Parniak, M. Dąbrowski, and W. Wasilewski, “Spatially resolved control of fictitious magnetic fields in a cold atomic ensemble,” Opt. Lett. 43, 1147 (2018).
[Crossref]

2017 (8)

M. Karpiński, M. Jachura, L. J. Wright, and B. J. Smith, “Bandwidth manipulation of quantum light by an electro-optic time lens,” Nat. Photonics 11, 53–57 (2017).
[Crossref]

Y. F. Pu, N. Jiang, W. Chang, H. X. Yang, C. Li, and L. M. Duan, “Experimental realization of a multiplexed quantum memory with 225 individually accessible memory cells,” Nat. Commun. 8, 15359 (2017).
[Crossref]

M. Parniak, M. Dąbrowski, M. Mazelanik, A. Leszczyński, M. Lipka, and W. Wasilewski, “Wavevector multiplexed atomic quantum memory via spatially-resolved single-photon detection,” Nat. Commun. 8, 2140 (2017).
[Crossref]

S. Denis, P. A. Moreau, F. Devaux, and E. Lantz, “Temporal ghost imaging with twin photons,” J. Opt. 19, 34002 (2017).
[Crossref]

M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
[Crossref]

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Magnified time-domain ghost imaging,” APL Photon. 2, 46102 (2017).
[Crossref]

X. Guo, Y. Mei, and S. Du, “Testing the Bell inequality on frequency-bin entangled photon pairs using time-resolved detection,” Optica 4, 388 (2017).
[Crossref]

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref]

2016 (7)

P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
[Crossref]

J. M. Donohue, M. Mastrovich, and K. J. Resch, “Spectrally engineering photonic entanglement with a time lens,” Phys. Rev. Lett. 117, 243602 (2016).
[Crossref]

S. Dong, W. Zhang, Y. Huang, and J. Peng, “Long-distance temporal quantum ghost imaging over optical fibers,” Sci. Rep. 6, 26022 (2016).
[Crossref]

Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3, 100–107 (2016).
[Crossref]

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
[Crossref]

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, “Sub-megahertz linewidth single photon source,” APL Photon. 1, 096101 (2016).
[Crossref]

2015 (2)

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

B. Li, M. R. Fernández-Ruiz, S. Lou, and J. Azaña, “High-contrast linear optical pulse compression using a temporal hologram,” Opt. Express 23, 6833 (2015).
[Crossref]

2014 (2)

L. Zhao, X. Guo, C. Liu, Y. Sun, M. M. T. Loy, and S. Du, “Photon pairs with coherence time exceeding 1 µs,” Optica 1, 84 (2014).
[Crossref]

P. C. Humphreys, W. S. Kolthammer, J. Nunn, M. Barbieri, A. Datta, and I. A. Walmsley, “Continuous-variable quantum computing in optical time-frequency modes using quantum memories,” Phys. Rev. Lett. 113, 130502 (2014).
[Crossref]

2013 (3)

Y. Zhu, J. Kim, and D. J. Gauthier, “Aberration-corrected quantum temporal imaging system,” Phys. Rev. A 87, 43808 (2013).
[Crossref]

V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express 21, 196–203 (2013).
[Crossref]

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

2012 (3)

X. H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. L. Liu, B. Zhao, and J. W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nat. Phys. 8, 517–521 (2012).
[Crossref]

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref]

2010 (1)

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
[Crossref]

2009 (5)

2008 (1)

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2008).
[Crossref]

2004 (1)

2001 (1)

C. V. Bennett and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[Crossref]

2000 (1)

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[Crossref]

1999 (1)

1994 (2)

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[Crossref]

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

1989 (2)

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630 (1989).
[Crossref]

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063–5072 (1989).
[Crossref]

1988 (1)

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
[Crossref]

1974 (1)

D. Grischkowsky, “Optical pulse compression,” Appl. Phys. Lett. 25, 566–568 (1974).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063–5072 (1989).
[Crossref]

Albrecht, B.

P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
[Crossref]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063–5072 (1989).
[Crossref]

Allgaier, M.

M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
[Crossref]

Altin, P. A.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Ansari, V.

M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
[Crossref]

Aspelmeyer, M.

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref]

Azaña, J.

Babashah, H.

H. Babashah, Z. Kavehvash, A. Khavasi, and S. Koohi, “Temporal analog optical computing using an on-chip fully reconfigurable photonic signal processor,” Opt. Laser Technol. 111, 66–74 (2019).
[Crossref]

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Temporal and spectral effects of cross-phase modulation on copropagating ultrashort pulses in optical fibers,” Phys. Rev. A 40, 5063–5072 (1989).
[Crossref]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Bao, X. H.

X. H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. L. Liu, B. Zhao, and J. W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nat. Phys. 8, 517–521 (2012).
[Crossref]

Barbier, M.

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Magnified time-domain ghost imaging,” APL Photon. 2, 46102 (2017).
[Crossref]

Barbieri, M.

P. C. Humphreys, W. S. Kolthammer, J. Nunn, M. Barbieri, A. Datta, and I. A. Walmsley, “Continuous-variable quantum computing in optical time-frequency modes using quantum memories,” Phys. Rev. Lett. 113, 130502 (2014).
[Crossref]

Barthélémy, A.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[Crossref]

Bennett, C. V.

V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, “104 MHz rate single-shot recording with subpicosecond resolution using temporal imaging,” Opt. Express 21, 196–203 (2013).
[Crossref]

C. V. Bennett and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[Crossref]

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[Crossref]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[Crossref]

Berger, N. K.

Bernu, J.

Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3, 100–107 (2016).
[Crossref]

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Bielawski, S.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Blatt, R.

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Brandstätter, B.

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

Brecht, B.

M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
[Crossref]

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
[Crossref]

Bromberg, Y.

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
[Crossref]

Buchler, B. C.

Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3, 100–107 (2016).
[Crossref]

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

M. Hosseini, B. M. Sparkes, G. Hétet, J. J. Longdell, P. K. Lam, and B. C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009).
[Crossref]

Campbell, G. T.

Cao, M. T.

Casabone, B.

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

Caspani, L.

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
[Crossref]

Chan, J.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref]

Chang, D.

Chang, W.

Y. F. Pu, N. Jiang, W. Chang, H. X. Yang, C. Li, and L. M. Duan, “Experimental realization of a multiplexed quantum memory with 225 individually accessible memory cells,” Nat. Commun. 8, 15359 (2017).
[Crossref]

Chávez, M.

P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
[Crossref]

Cho, Y.-W.

Chu, S. T.

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
[Crossref]

Corrielli, G.

A. Seri, D. Lago-Rivera, A. Lenhard, G. Corrielli, R. Osellame, M. Mazzera, and H. de Riedmatten, “Quantum storage of frequency-multiplexed heralded single photons,” Phys. Rev. Lett. 123, 080502 (2019).
[Crossref]

Dabrowski, M.

M. Parniak, M. Mazelanik, A. Leszczyński, M. Lipka, M. Dąbrowski, and W. Wasilewski, “Quantum optics of spin waves through ac stark modulation,” Phys. Rev. Lett. 122, 063604 (2019).
[Crossref]

A. Leszczyński, M. Mazelanik, M. Lipka, M. Parniak, M. Dąbrowski, and W. Wasilewski, “Spatially resolved control of fictitious magnetic fields in a cold atomic ensemble,” Opt. Lett. 43, 1147 (2018).
[Crossref]

M. Parniak, M. Dąbrowski, M. Mazelanik, A. Leszczyński, M. Lipka, and W. Wasilewski, “Wavevector multiplexed atomic quantum memory via spatially-resolved single-photon detection,” Nat. Commun. 8, 2140 (2017).
[Crossref]

Datta, A.

P. C. Humphreys, W. S. Kolthammer, J. Nunn, M. Barbieri, A. Datta, and I. A. Walmsley, “Continuous-variable quantum computing in optical time-frequency modes using quantum memories,” Phys. Rev. Lett. 113, 130502 (2014).
[Crossref]

de Riedmatten, H.

A. Seri, D. Lago-Rivera, A. Lenhard, G. Corrielli, R. Osellame, M. Mazzera, and H. de Riedmatten, “Quantum storage of frequency-multiplexed heralded single photons,” Phys. Rev. Lett. 123, 080502 (2019).
[Crossref]

P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
[Crossref]

Denis, S.

S. Denis, P. A. Moreau, F. Devaux, and E. Lantz, “Temporal ghost imaging with twin photons,” J. Opt. 19, 34002 (2017).
[Crossref]

Devaux, F.

S. Denis, P. A. Moreau, F. Devaux, and E. Lantz, “Temporal ghost imaging with twin photons,” J. Opt. 19, 34002 (2017).
[Crossref]

Dietrich, P.

X. H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. L. Liu, B. Zhao, and J. W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nat. Phys. 8, 517–521 (2012).
[Crossref]

Dong, S.

S. Dong, W. Zhang, Y. Huang, and J. Peng, “Long-distance temporal quantum ghost imaging over optical fibers,” Sci. Rep. 6, 26022 (2016).
[Crossref]

Donohue, J. M.

J. M. Donohue, M. Mastrovich, and K. J. Resch, “Spectrally engineering photonic entanglement with a time lens,” Phys. Rev. Lett. 117, 243602 (2016).
[Crossref]

Drobshoff, A. D.

Du, S.

Duan, L. M.

Y. F. Pu, N. Jiang, W. Chang, H. X. Yang, C. Li, and L. M. Duan, “Experimental realization of a multiplexed quantum memory with 225 individually accessible memory cells,” Nat. Commun. 8, 15359 (2017).
[Crossref]

Dück, A.

X. H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. L. Liu, B. Zhao, and J. W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nat. Phys. 8, 517–521 (2012).
[Crossref]

Dudley, J. M.

H. Wu, P. Ryczkowski, A. T. Friberg, J. M. Dudley, and G. Genty, “Temporal ghost imaging using wavelength conversion and two-color detection,” Optica 6, 902 (2019).
[Crossref]

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Magnified time-domain ghost imaging,” APL Photon. 2, 46102 (2017).
[Crossref]

Eigner, C.

M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
[Crossref]

El Koussaifi, R.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Evain, C.

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

Everett, J. L.

Farrera, P.

P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
[Crossref]

Fejer, M. M.

Fernández-Ruiz, M. R.

Fischer, B.

Foster, M. A.

Friberg, A. T.

H. Wu, P. Ryczkowski, A. T. Friberg, J. M. Dudley, and G. Genty, “Temporal ghost imaging using wavelength conversion and two-color detection,” Optica 6, 902 (2019).
[Crossref]

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Magnified time-domain ghost imaging,” APL Photon. 2, 46102 (2017).
[Crossref]

Froehly, C.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[Crossref]

Gaeta, A. L.

Gauthier, D. J.

Y. Zhu, J. Kim, and D. J. Gauthier, “Aberration-corrected quantum temporal imaging system,” Phys. Rev. A 87, 43808 (2013).
[Crossref]

Geng, J.

Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3, 100–107 (2016).
[Crossref]

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Genty, G.

H. Wu, P. Ryczkowski, A. T. Friberg, J. M. Dudley, and G. Genty, “Temporal ghost imaging using wavelength conversion and two-color detection,” Optica 6, 902 (2019).
[Crossref]

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Magnified time-domain ghost imaging,” APL Photon. 2, 46102 (2017).
[Crossref]

Geraghty, D. F.

R. Salem, M. A. Foster, A. C. Turner-Foster, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “High-speed optical sampling using a silicon-chip temporal magnifier,” Opt. Express 17, 4324–4329 (2009).
[Crossref]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2008).
[Crossref]

Glorieux, Q.

B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, “Gradient echo memory in an ultra-high optical depth cold atomic ensemble,” New J. Phys. 15, 085027 (2013).
[Crossref]

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[Crossref]

Grazioso, F.

C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
[Crossref]

Grischkowsky, D.

D. Grischkowsky, “Optical pulse compression,” Appl. Phys. Lett. 25, 566–568 (1974).
[Crossref]

Gröblacher, S.

S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
[Crossref]

Guo, X.

Harder, G.

M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
[Crossref]

Hedges, M. P.

M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
[Crossref]

Heinze, G.

P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
[Crossref]

Hernandez, V. J.

Heshami, K.

E. Saglamyurek, T. Hrushevskyi, A. Rastogi, K. Heshami, and L. J. LeBlanc, “Coherent storage and manipulation of broadband photons via dynamically controlled Autler–Townes splitting,” Nat. Photonics 12, 774–782 (2018).
[Crossref]

Hétet, G.

M. Hosseini, B. M. Sparkes, G. Hétet, J. J. Longdell, P. K. Lam, and B. C. Buchler, “Coherent optical pulse sequencer for quantum applications,” Nature 461, 241–245 (2009).
[Crossref]

Higginbottom, D. B.

Hill, J. T.

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
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X. H. Bao, A. Reingruber, P. Dietrich, J. Rui, A. Dück, T. Strassel, L. Li, N. L. Liu, B. Zhao, and J. W. Pan, “Efficient and long-lived quantum memory with cold atoms inside a ring cavity,” Nat. Phys. 8, 517–521 (2012).
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M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
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S. Hong, R. Riedinger, I. Marinković, A. Wallucks, S. G. Hofer, R. A. Norte, M. Aspelmeyer, and S. Gröblacher, “Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator,” Science 358, 203–206 (2017).
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J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
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P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. De Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
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M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nat. Commun. 8, 14288 (2017).
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A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
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C. Reimer, M. Kues, P. Roztocki, B. Wetzel, F. Grazioso, B. E. Little, S. T. Chu, T. Johnston, Y. Bromberg, L. Caspani, D. J. Moss, and R. Morandotti, “Generation of multiphoton entangled quantum states by means of integrated frequency combs,” Science 351, 1176–1180 (2016).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Temporal imaging state of the art, characterized by temporal $ \delta t $ and spectral $ \delta \omega $ resolutions. Numerous implementations based on solid media (electro-optic modulators or EOMs [25,4345]; four-wave mixing, or FWM [24,46]; sum-frequency generation, or SFG [20,47]; and cross-phase modulation, or XPM [48]) are well suited for high bandwidth pico- or even femtosecond pulses, achieving spectral resolution no better than 1 GHz, with time-bandwidth products ($ \tau {\cal B} $) reaching $ 2\pi \times 2000 $. Our system–GEM & SSM–has $ {10^6} $ times better spectral resolution $ \delta \omega /2\pi \sim 20\;{\rm kHz} $, maintaining good $ \tau {\cal B} $, thus allowing exploration of a previously unattainable region. The grayed region indicates an unphysical area bounded by the Fourier limit $ \tau {\cal B}/2\pi = 1 $.
Fig. 2.
Fig. 2. (a) Light–atom interface. Chirped control field simultaneously allows mapping of the signal optical field onto the atomic coherence $ {\rho _{hg}} $ and realizes the temporal lens. (b) Projection of signal spectral components onto atomic coherence spatial components in GEM with Zeeman splitting gradient $ \beta $. (c) During the writing process atoms are placed in a negative magnetic field gradient along the cloud (${w}$). When the writing finishes, the spatial phase of the atomic coherence is modulated with a parabolic Fresnel profile that realizes a temporal equivalent of free-space propagation. Finally, the gradient is switched to positive (${r}$) and the coherence is converted back to light, which is further registered with single photon counting module (SPCM) connected to the time tagger (TTG). (d) Evolution of the spectro-temporal Wigner function on subsequent stages of far-field temporal imaging: (1) time lens, (2) free-space propagation, and (3) time lens. The complete transformation effectively rotates the initial Wigner function of two pulses (equivalent to a Wigner function of a cat state in phase space) by $ \pi /2 $, as given by Eq. (1).
Fig. 3.
Fig. 3. Experimental sequence for temporal imaging. (a) Time-trace of the Zeeman shift gradient $ \beta $ used in the GEM protocol, allowing two-directional mapping of signal frequencies to distinct positions in the atomic cloud. (b) Control field (red) and SSM (yellow) laser pulse sequence divided into three stages corresponding to lens–propagation–lens operations. The lens (1) is implemented during the GEM writing process by a chirped control field. (2) The 3 µs long SSM laser pulse imprints a parabolic phase profile onto the stored atomic coherence, which realizes the spectro-temporal free-space propagation. During this stage, the magnetic field gradient is reversed, allowing remapping the coherence to light. (3) Finally, the control field is turned on and the coherence is read out from the memory. Chirping the control field would implement the second lens. However, for simplicity, the control field is no longer chirped as the imposed phase would not be registered by the SPCM. (c, e) Example results for two pulses or a sine wave as inputs, respectively. Gray bins represent single photon counts. Red line corresponds to the numerical simulations. (d, f) Normalized modulus square of atomic coherence in Fourier space. The insets (i, ii) show experimentally obtained linear dependency of the time delay $ \Delta t $ (in $\unicode{x00B5}{\rm s} $) on the signal modulation frequency $ f $ (in MHz) defined on panels (c, e).
Fig. 4.
Fig. 4. Characterization and tuning of bandwidth and resolution. (a) Efficiency spectral profile $ {\eta _0}(\omega ) $ as a function of the two-photon detuning $ \delta = \omega - {\omega _0} $ for a chosen time-bandwidth product $ \tau {\cal B} = 2\pi \times 13 $ with bandwidth $ {\cal B} $ defined as the FWHM of $ {\eta _0}(\omega ) $. The red line corresponds to a super-Gaussian approximation of the atom concentration used in the simulation. (b) Dependence of the bandwidth $ {\cal B} $ as a function of the Zeeman splitting gradient $ \beta $. Red line is a linear fit to the data. (c) Time evolution of the GEM efficiency due to incoherent scattering caused by the coupling field. The characteristic decay time $ \tau $ obtained from exponential fit (red line) limits the effective resolution $ \delta \omega /2\pi = 0.78/\tau $ (here $ \tau = 10\;\unicode{x00B5} {\rm s} $). (d) Dependence of $ 1/\tau $ as a function of the coupling field power $ P \propto |\Omega {|^2} $, along with linear fit (red line). (e) Calculated map of the average efficiency $ \bar \eta $ for varying bandwidth $ {\cal B} $ and decay time $ \tau $. The efficiency for a given time-bandwidth product $ \tau {\cal B} $ is approximately constant as expected. The star indicates the point of operation where the exemplary measurements (Fig. 3) were performed.

Equations (3)

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$$\begin{split}\left[ {\begin{array}{*{20}{c}}{t^{\prime}}\\{\frac{{\omega^{\prime}}}{{{\omega _0}}}}\end{array}} \right] &= \left[ {\begin{array}{*{20}{c}}1&0\\{ - \frac{1}{{{f_{\rm t}}}}}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&{{f_{\rm t}}}\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1&0\\{ - \frac{1}{{{f_{\rm t}}}}}&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{t}\\{\frac{\omega }{{{\omega _0}}}}\end{array}} \right]\\& = \left[ {\begin{array}{*{20}{c}}0&{{f_{\rm t}}}\\{ - \frac{1}{{{f_{\rm t}}}}}&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{t}\\{\frac{\omega }{{{\omega _0}}}}\end{array}} \right],\end{split}$$
$${\eta _0} = {\left[ {1 - \exp \left( { - 2\pi \frac{{{\rm OD}}}{{\tau {\cal B}}}} \right)} \right]^2},$$
$$\begin{split}\bar \eta &= \frac{1}{{2\tau {\cal B}}}\int_{ - {\cal B}/2}^{{\cal B}/2} \int_0^{2\tau } \eta (t,\omega ){\rm d}t{\rm d}\omega\\& = \frac{{{e^2} - 1}}{{2{e^2}{\cal B}}}\int_{ - {\cal B}/2}^{{\cal B}/2} {\eta _0}(\omega ){\rm d}{\omega}.\end{split}$$

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