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High-Efficiency Coherent Light Storage for the Application of Quantum Memory
YI-HSIN CHEN, YING-CHENG CHEN,
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High-Efficiency Coherent Light Storage
for the Application of Quantum Memory

YI-HSIN CHEN, YING-CHENG CHEN, AND ITE A. YU

ABSTRACT

Photonic quantum memory is a key component for long-distance quantum communication and optical quantum computation. High storage efficiency, long storage time, and high fidelity are the required features of a quantum memory. Coherent light storage based on the effect of electromagnetically induced transparency (EIT) has the potential and is promising for the application of quantum memory, owing to the quantum nature of light-matter interaction. Previously, the storage efficiency of EIT-based memory is however less than 50% (i.e. 1 failure per 2 operations), seemingly indicating that it is not compatible with quantum information technology. Recently, we experimentally demonstrated that as long as the optical density of a medium is sufficiently high and the decoherence rate of the system is enough low, the storage efficiency of EIT-based memory can approach unity. Our work brings EIT-based memory towards practical applications in quantum information processing with photons.

INTRODUCTION

Coherent optical memory or photonic quantum memory is a device for storing wave functions of light or photons and releasing them on demand. The storage efficiency (SE) of the memory is the ratio of readout to write-in energies. The SE of the memory must be more than 50% in the no-cloning limit [1]. Furthermore, an increase of 1% in SE can shorten the transmission time for long-distance quantum communications based on quantum repeater protocol by 7-18% [2]. Information carried by photons, inert to the environment, is robust. On the other hand, information stored in matter is fragile and its signal amplitude can decay due to likely interaction with the environment. The fractional delay (FD), defined as the ratio of the storage time to the full-width-half-maximum (FWHM) pulse duration at a SE of 50%, is another important figure of merit of a memory. A larger FD means a memory can provide a delay time for more quantum operations. A practical quantum memory should have high SE, large FD, and high fidelity.

During the last decade, many studies have been devoted to fulfilling the above performance. Several types of memory devices using different mechanisms, such as EIT, gradient echo memory (GEM), atomic frequency comb (AFC), etc., were proposed and experimentally demonstrated. With the GEM storage scheme [1,3], a recent experiment utilizing laser-cooled atoms achieved a SE of 87% and a memory lifetime of up to 1 ms [4]. The AFC storage scheme allows for multiple temporal modes [5]. The up-to-date best SE of an AFC scheme was 58% [6] and the storage lifetime was around 1 ms. The EIT storage scheme was first realized with a Bose-Einstein condensate [7] and with a warm atomic vapor [8] in 2001. We achieved a SE of 78% and a storage time of 98 μs with laser-cooled atoms in 2013 [9]. More recently, we further increased SE to above 90% in a medium of high optical density (OD) in 2016 [10]. EIT-based light storage using a solid-state system can eliminate decoherence caused by atomic thermal motion [11] and, thus, a storage time up to one minute was demonstrated in 2013 [12].

A comparison of various quantum memory schemes can be found in Ref. [4]. We will present our efforts on the study of EIT-based memory in this article.

Electromagnetically induced transparency

The EIT effect results from the destructive interference among different transitions pathways. The effect is a powerful mechanism not only to greatly eliminate linear susceptibility (i.e. one-photon absorption), but also to significantly enhance nonlinear susceptibility (i.e. two-photon coherence). The EIT system in the Λ-type configuration consists of two ground states (|1> and |2>) and one excited state (|3>), driven by a weak probe field and a strong coupling field as shown in Fig. 1(b). Because states |1> and |2> are immune from spontaneous decay, the two-photon (or Raman or ground-state) coherence established by the probe and coupling fields can be robust and persistent in the system for a long duration.

To illustrate the EIT phenomenon, we start with a two-level system as shown Fig. 1(a). Consider that the probe field (denoted as εp) has the resonant frequency of a transition from a ground state |1> to an excited state |3> and propagates through a medium. The probe field suffers an attenuation determined by the optical density (OD) of the medium. The OD (denoted as α) is equal to nσL, where n is the atomic density, σ is the atomic absorption cross section, and L is the medium length. After propagating through the medium, the probe power or energy decays to e. Taking α = 100 as an example, the output probe transmission is less than 10-43. On the other hand, when the strong coupling field, driving the transition from a ground state |2> to the same excited state |3>, is present in the system as depicted in Fig. 1(b), the absorption of the probe field can be completely inhibited or greatly suppressed. In other words, the presence of the coupling field makes the medium change from complete opacity to nearly 100% transparency for the probe field. Figure 1(c) is a representative EIT spectrum in a medium with an OD of about 750. The inset shows a very narrow and high contrast transparency window. Note that EIT physics can work in any 3-level system such as all kinds of atomic vapors, laser-cooled, cryogenic, room-temperature and heated samples, solids, quantum wells/dots, etc.

Fig. 1: (a) In a two-level system, a probe field εp drives the transition from a ground state |1> to an excited state |3>. The output transmission of the probe field is reduced by e, where α is the OD of the system. (b) In a three-level EIT system, a coupling field Ωc drives the transition from a ground state | 2> to the same excited state |3>. The coupling field suppresses the absorption of probe field, making the medium transparent for the probe field. (c) A typical EIT spectrum shows the probe transmission versus the probe detuning in units of the spontaneous decay rate of the excited state, Γ. The circles are experimental data. The solid line is the best fit, determining the OD of the system to be approximately 750. The inset enlarges the very narrow spectral profile of the EIT transparency window in the central part of the main plot. The transparency width is about 0.2Γ.

Slow light

The width of the transparency window in the EIT spectrum is determined by the coupling Rabi frequency Ωc and the OD of the system. A narrow as well as high-contrast spectral width indicates a rapid change in refractive index of the medium, resulting in a dramatic reduction of the group velocity of a light pulse. In addition to the suppression of probe absorption, this narrow EIT transparency width provides the supplementary control that can slow down a light pulse and spatially compress it inside the medium.

The motion of a light pulse traveling in a dispersive EIT medium has a temporal delay (td), mainly depending on the coupling Rabi frequency (Ωc) and the OD (α), i.e. td ≈ αΓ / Ωc2 where Γ is the spontaneous decay rate of the excited state |3>. Figure 2(a) shows data for slow light under the constant presence of the coupling field. The probe pulse is delayed by 3.5 μs in an atom cloud of 9 mm length, implying the group velocity of the pulse is less than c /105. Furthermore, this 1 km-long probe pulse in free space is compressed inside the 9 mm-long atom cloud.

Light storage/quantum memory

The slowly moving probe pulse can be stopped in the EIT medium by adiabatically switching off the coupling field. This stopping process is actually a mechanism that stores the probe photons in the form of Raman coherence (i.e. the coherence between two atomic ground states) in the medium. By switching on the coupling field again, one can read out the stored photons. Such EIT light storage is a coherent process, involving exchange of wave functions between photons and atoms and leading to application as quantum memory.

Figure 2(b) shows the data of the storage and retrieval of a probe pulse in the EIT medium. The black and blue circles are the input probe pulse and the recalled probe pulse, respectively. The green dashed line represents the timing of the coupling field. After the coupling field was switched off, the detector received no probe signal at all for about 4 μs. During this 4 μs, the dashed red line is a copy of the red-color data in Fig. 2(a), illustrating that the probe signal should appear if there was no storage. After the storage time of 4 μs, we switched on the coupling field and read out the stored probe pulse as shown by the blue circles. Because the coherence time in our system is much longer than the storage time, the retrieved probe pulse has almost no attenuation at all.

EXPERIMENTAL SETUP

We carried out the experiment in a cigar-shaped cloud of cold 87Rb atoms produced by a magneto-optical trap (MOT). Typically, we can trap 1.0×109 atoms with a temperature of about 300 μK. The dimensions of the atom cloud are about 9.2×1.8×1.8 mm3. The schema and photo of experimental setup are shown in Fig. 3. We employed the temporally dark and compressed MOT for about 7 ms to increase OD before performing each measurement. In addition, we optically pumped all the population to a single Zeeman ground state such that the atomic system became a simple three-state system depicted in Fig. 1(b). Details can be found in Refs. [13-16].

Fig. 2: Slow light in (a) and stored-and-then-recalled light in (b). Black, red, and blue circles are the input probe pulse, the output probe pulse under the constant presence of the coupling field, and the stored-and-then-recalled probe pulse, respectively. Green dashed lines represent the coupling field. Solid lines are the theoretical predictions. (a) Slow light under the constant presence of the coupling field. The propagation delay time of the probe pulse is about 3.5 μs. (b) Switch off the coupling field when the probe pulse is completely inside the medium. Dashed red line is the copy of the red-circle data in (a), but there was no actual signal received by the detector because the probe pulse was stored inside the medium. After a storage time of about 4 μs, the stored probe pulse was released by turning on the coupling field. OD =156 and Ωc = 1.1Γ are the calculation parameters in the theoretical predictions.

QUANTITIES OF A GOOD MEMORY

Storage efficiency

We sent the incident probe pulse, which had a Gaussian profile with a FWHM of 1.5 μs, to the cold atoms. The EIT medium behaves like a frequency filter as demonstrated by the spectrum in Fig. 1(c), meaning only the frequency of the probe pulse well within the EIT bandwidth can pass through the medium without energy loss. When the slowly moving probe pulse was compressed completely within the atom cloud, we adiabatically switched off the coupling field so that the probe pulse is stopped in the EIT medium. The incoming pulse is mapped into a long-lived atomic wave function (i.e. the Raman or ground-state coherence). The stored signal was later read out by switching on the coupling field again.

Fig. 3: (a) Schematic experimental setup. Both probe and coupling beams propagated along the major axis of the cigar-shaped cloud of cold atoms. The bottom part is the setup of a beat-note interferometer, used for the phase measurement of the probe field [16]. PD1, PD2, and PD3: photodetectors; PMF: polarization-maintained optical fiber; CPL: optical fiber coupler; BS: cube beam splitter; PBS: cube polarizing beam splitter; λ/4: zero-order quarter-waveplate; L1-L3: lenses with the focal length f specified in units of millimeters. M: mirror; AOM: 80-MHz acoustic optical modulator. (b) Photo of the experimental setup.

The experimental data of input probe pulse, output probe pulse under the constant presence of the coupling field, and stored-and-then-recalled probe pulse are shown by the open circles in Fig. 2. The solid lines are the theoretical predictions obtained by numerically solving the Maxwell-Schrödinger equation of the light pulse and the optical Bloch equation of the atomic density-matrix operator. The numerical calculation with OD = 156 and Ωc = 1.1Γ can well fit the experimental data. We obtained an SE of about 69% after 4 μs storage time, which already exceeded previous results for EIT-based storage.

Fig. 4: SE as functions of OD in the forward (blue) and backward (red) retrievals. Squares and circles are the experimental data and solid lines are the theoretical predictions. In the measurements of forward retrieval, the coupling Rabi frequency was varied from 0.3Γ to 1.1Γ with increasing OD to maximize SE. In the measurements of the backward retrieval, both the Rabi frequencies of the forward and backward coupling fields were fixed to 1.2Γ.

A system with a small OD can hardly obtain sufficiently high SE since the medium cannot store the entire probe pulse. We measured SE as a function of OD shown by the blue squares in Fig. 4. For each value of OD, the power of the coupling field was optimized for maximizing the SE. Because the EIT bandwidth is equal to Ωc2 / ( Γ), for a higher OD of the medium we apply a stronger Ωc to achieve an optimum SE. In the entire measurement, Ωc was varied from 0.3Γ to 1.1Γ. The solid blue line is the theoretical prediction. At higher ODs (with stronger Ωc), the data points are slightly below the predicted line because a nearby excited state degrades the SE.

The stored information can be retrieved either in the forward or backward direction. With the time-space-reversing method in a backward-retrieval configuration proposed by Ref. [17], we can further improve the SE. The probe pulse first propagated into the atoms in the forward direction and was then stored. After a storage time, we retrieved the probe pulse in the backward direction by switching on a backward-propagation coupling field. The main idea of the time-space-reversing method is to generate a best probe profile, leading to low energy loss during pulse propagation. The expected shape of the incident pulse has a slowly varying front and a sharp-edge rear shown in the inset of Fig. 5. The front part or the slowly varying part, with frequency components well within the EIT bandwidth, propagated through the atoms and suffered little energy loss. The rear part or the sharp-edge part, corresponding to high-frequency components, was stored at the entrance of the atom cloud, and then retrieved without propagating through the atoms.

Similar to the forward retrieval, we measured the SE as a function of OD as shown by the red circles in Fig. 4. We did not vary the coupling Rabi frequency because it only affects the optimum pulse profile but influences SE very little. At the same coupling Rabi frequency, a larger OD required a wider probe pulse. The solid red line in Fig. 4 represents the theoretical prediction. The measured data and the predicted values both indicated that SE increases with OD and approaches an asymptotic value. However, the experimental data were systematically less than the theoretical predictions. This discrepancy is due to a non-negligible phase mismatch during backward retrieval in the system.

Nearby excited states can induce the unwanted process of photon switching, which degrades SE and fidelity. The proper choice of the EIT transition scheme is a way to reduce detrimental influences from nearby excited states. We built another system of cold cesium (Cs) atoms to further enhance the SE. The OD of this system can be larger than 1,000. In addition, we selected the Cs D1 transition lines, which is nearly free from the photon switching effect because of large energy separations from nearby excited states. We achieved a SE larger than 90% in this system [10].

Storage time

Storage efficiency will decrease with storage time due to relaxation mechanisms such as atomic thermal motion, stray magnetic fields, collisions between atoms, etc. Figure 5 shows SE as a function of storage time. The circles represent the experimental data, and the solid line is the best fit. The major relaxation in our system is the randomization of the spatial phase pattern caused by atomic thermal motion. Hence, the fitting function is a Gaussian decay, i.e. exp (-t2 coh2). The best fit gives a coherence time τcoh of 92 μs. Combining the storage time at 50% SE with a FWHM of the input pulse of 0.88 μs, we obtain a FD of 74. In our Cs system, we achieved a coherence time of 325 μs, corresponding to a FD of 1,200 at 50% SE [10]. This high SE and long FD are the up-to-date best record for all kinds of possible schemes for the realization of quantum memory.

Fig. 5: SE as a function of storage time. The solid line is the best fit with the Gaussian-decay function, determining the coherence time to be about 92 μs. The inset shows the input probe pulse with a FWHM of 0.88 μs and, thus, the FD is 74±5 at 50% SE.

Fidelity

For practical applications, a memory must ensure that the information remains intact. The EIT storage is a coherent process and provides a method for exchange of wave functions between photons and atoms. In our case, using classical light, the fidelity of the retrieved signal is the wave-function similarity (both their amplitude envelopes and phase evolutions) between the input and output pulses. Because the retrieval process maps both amplitude and phase of the stored atomic wave function to the readout light pulse, the readout signal should have a high fidelity.

We measured the phase and amplitude of light waves by the method of beat-note interferometer. Details can be found in Ref. [16]. From the analysis of beat-note signals, we obtain fidelities of 0.94 and 0.90 after 7 μs and 55 μs storage time, respectively [18]. The theoretical prediction indicates a fidelity of 0.97, due to the probe pulse width being broadened during the propagation in the atoms. The measured fidelity is limited by the signal-to-noise ratio in the beat-note signal. We also demonstrated that fidelity was nearly independent of the storage time.

CONCLUSION

In our system of cold Rb-87 atoms, we successfully achieved a SE of 78% in an EIT-based memory in 2013. At 50% SE, we obtained a FD of 74. Recently, we achieved a SE larger than 90% and a FD of about 1,200 in cold Cs atoms. With the phase measurement, we verified that the fidelity of the recalled pulse was 0.94 and nearly independent of the storage time. Such excellent SE, long FD and high fidelity bring the EIT-based quantum memory toward the practical application of quantum information manipulation.

Acknowledgement: This work is a part of the research project "Applications of EIT-based Photonic Storage in Quantum Information Manipulation", supported by the Science Vanguard Research Program of the Ministry of Science and Technology, Taiwan.

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Yi-Hsin Chen is an assistant research fellow at National Tsing Hua University (NTHU), Taiwan. After receiving her PhD degree in physics from NTHU in 2011, she worked in the same university as a postdoctoral researcher from 2012 to 2014. At Stuttgart University, Germany, she was a Humboldt Postdoctoral Research Fellow from 2014 to 2016. She joined the Department of Physics, NTHU in 2016. Her research field includes nonlinear optics, strong photon-photon interactions via the Rydberg blockade effect, generation of single-photon sources, and quantum information science.

Ying-Cheng Chen is an associate research fellow at the Institute of Atomic and Molecular Sciences (IAMS), Academia Sinica, Taiwan. After receiving his PhD degree in physics from National Tsing Hua University of Taiwan in 2002, he worked at Rice University, US as a postdoctoral fellow until 2004. He became a faculty member of IAMS in2005. His research interests include cold atoms and molecules, quantum optics and quantum information science. His current research focuses on EIT-based quantum memory, single-photon-level quantum manipulation, and cooperative radiation phenomena.

Ite A. Yu is a Distinguished Professor of National Tsing Hua University, and a Fellow of the Physical Society of R. O. C. Taiwan. After receiving his PhD degree from Massachusetts Institute of Technology in 1993, he became a faculty member of the Department of Physics, National Tsing Hua University in1995. His research interests include experiment and theory of cold atoms, quantum optics, and quantum information manipulation. His group pioneered the research of low-light-level nonlinear optics via stored light in 2006, and currently focuses on studies of EIT-based quantum memory and photon-photon interaction.