Collective excitation in a large ensemble of cold atoms 

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Collective excitation in a large ensemble of cold atoms

Developing coherent protocols that allow storage and manipulation of quantum states carried by light is crucial for quantum information science, including quantum networks and communications. Because of their long-lived coherences, atomic spin states are precisely good candidates for this purpose. As many protocols require the use of single photons, such quantum memory protocols have to work at the single-photon level, a not-so-easy to reach regime.
Intuitively the simplest approach for reversible transfer is to map single-photon states into single atoms. Thus, a single-atom coherently absorbs and emits later on-demand a single-photon. Strong interaction between single photons and atoms are required to achieve good storage and read-out efficiency, but the absorption of an individual atom is very small. An elegant solution to this problem, the well-known cavity QED framework [Miller et al., 2005], consists in placing the atom into a high finesse resonator in order to enhance its effective cross section. The photon now travels many times in the cavity, and can interact with the atom with a larger probability. A lot of work has been done in this direction over the past few decades by different groups, leading to the recent textbook demonstration of polarization qubit storage in a single-atom quantum memory [Specht et al., 2011]. However it is still technically challenging to reach the very strong coupling regime and to control the motion of individual atoms at the same time.
On the other side, atomic ensembles containing a large number of atoms can interact strongly with light. The key feature relies on collective effects, which allow to achieve easily strong and controllable coupling between many-atom systems and photons. The use of optically thick atomic ensembles was thus motivated by the simplicity and the potential efficiency compared with single-atom settings. The last ten years have seen a tremendous activity in this direction, both theoretically and experimentally [Lukin, 2003, Lvovsky et al., 2009, Simon et al., 2010, Sangouard et al., 2011].
In atomic ensembles the coherent and reversible mapping of photonic quantum states can be achieved via the electromagnetically induced transparency (EIT) phenomenon. The atoms are driven by an auxiliary laser field that enables to control the photon group velocity, and in consequence to stop and trap them inside an atomic cloud [Fleischhauer and Lukin, 2002]. Another protocol, which was seminal for the study of protocols based on atomic ensembles and quantum repeater architectures, is the DLCZ scheme [Duan et al., 2001]. This is a measure-induced protocol. Based on Raman scattering this protocol allows for instance the generation of probabilistic but heralded entanglement between remote atomic ensembles [Chou et al., 2005, Laurat et al., 2007a]. This entanglement can be transferred from atoms to photonic modes on demand, so this scheme represents an elementary link of a quantum repeater. The building block of the DLCZ scheme can be also used as an heralded single photon source.
In this chapter we first motivate the choice of cold atoms as a medium for quantum memory implementation. Then we remind the principle of collective excitation, and in particular we focus on the two quantum memory protocols introduced above, which rely on collective enhancement, the dynamic EIT protocol and the DLCZ scheme. Finally critical parameters to achieve such quantum memory protocols in atomic ensembles are presented.

Why using cold atoms ?

The use of cold gases instead of warm vapors in order to build a quantum memory is motivated by several aspects, in particular the need to reach the single-photon level and the necessity of avoiding Doppler broadening. We discuss in the following these two different aspects.

Down to the single-photon level

Single-photon has become a basic resource in quantum information science, where a great variety of protocols involving single photons have been proposed. As stated before, the implementation of reliable quantum memories at the single-photon level is thus truly wanted but still experimentally challenging. In order to detect these photons single-photon counters, such as avalanche photodiode (APD), are very convenient because they are sensitive to the single-photon level, easy to use and now reaching reasonable efficiency, typically > 50% at the cesium wavelength, 852 nm, with low dark count rate. Nevertheless, strong laser pulses must illuminate the atomic ensemble to control the interaction, whether it be for the dynamic EIT or the DLCZ protocol. Let us give a first indicating number. The bright coupling pulse contains a huge number of photons, e.g. 2 108 in a 0.5 µs pulse with a power of 100 µW at 852 nm. Rejection by at least 80 to 100 dB is thus required. To increase the difficulty, EIT or DLCZ quantum memory schemes were proposed in a configuration where the coupling beam co-propagating along with the interesting signal. It is therefore essential to efficiently separate the single photons from the coupling field, which mostly overlap in time, in order to collect single photons with good signal-to-noise ratio for characterization or further applications. Going down to the single-photon level is not a trivial task in many practical implementations. The first experimental demonstrations of nonclassical correlations between photon pairs generated in atomic ensembles by the first step of the DLCZ protocol were indeed realized in collinear geometry [Kuzmich et al., 2003, Wal et al., 2003]. One of the main experimental challenges was to separate the photons from the classical pulses, and that is why classical fields were filtered in polarization and also in frequency, using optically pumped atomic cells. Values for the normalized correlation function g1,2 , which characterizes quantum correlations between the two photons, slightly above the classical limit of 2 were obtained [Kuzmich et al., 2003]. However, the authors were still strongly limited by the write and read field contamination in each detection path. The solution was then to use an off-axis configuration, namely to put an angle between the driving fields and the ones to detect for spatial filtering [André, 2005].
First experimental uses of an off-axis configuration were reported in the classical regime in 2004 by the authors of [Braje et al., 2004] and in the quantum regime in 2005 [Matsukevich et al., 2005]. In the latter the authors reached g1,2 ≈ 300 with this additional spatial filtering, and high-quality heralded photons were thereafter generated and values of g1,2 as large as 600 were obtained [Laurat et al., 2006]. Two similar experiments of EIT storage of a single photon generated by the aforementioned method give an illustration of the effect of this spatial filtering. One has been performed in warm atoms and with collinear configuration [Eisaman et al., 2005], while the other has used an ensemble of cold atoms and the off-axis configuration [Chanelière et al., 2005]. The off-axis configuration allowed to achieve 20 times better g1,2 than the collinear one does at the single-photon creation step.
Although coherence times of few milliseconds have been demonstrated in warm atomic ensembles in the collinear configuration, using another protocol [?], working in off-axis configuration kills the effective coherence time in warm atomic ensembles because of motional dephasing. Indeed, whether it be EIT or DLCZ, both of these protocols lead to the formation of a spin-wave grating in the atomic ensemble with a spatial period that can be much smaller than the size of the whole sample. The interfringe depends on the difference |ΔK| between the K vector of the two fields (either control and signal in EIT or write and field 1 in DLCZ), which form a small angle θ, Λ = 2π ≈ λ (1.1) |ΔK| sin θ
Here the wavelength of both fields are supposed to be identical and equal to λ, a very good approximation in most experimental cases.
This grating is written during the writing process and has to remain until the reading process otherwise the information about the optical excitation is partially lost and the efficiency decreases. If atoms fly by a distance comparable to its spatial period within the storage time, the grating is destroyed. As a typical value, one can evaluate the average time for an atom to fly over an interfringe of the interference pattern because it limits the maximum achievable storage time. A realistic experimental example would be the following: a control and a signal fields at 852 nm propagate in an ensemble of cesium atoms with an angle of 2◦ . As a result the interference pattern period is 25 µm. So, it takes only 170 ns for 330 K atoms to travel across Λ, while it takes 100 µs for 1 mK atoms and 300 µs for 100 µK atoms. In this configuration the dephasing time due to atomic motion will be more than 500 times longer in cold atomic ensembles than in warm ones. Collisional effects also induce decoherence in warm vapors. The authors of [Manz et al., 2007] investigated the dephasing on collective state in the DLCZ scheme due to collision and concluded that buffer gas in particular configuration should help to achieve longer coherence time, in addition to beam with large diameters. This was confirmed by a study of the effect of the angle between coupling and signal beams in a rubidium vapor heated at 78◦ C, leading to the observation of 10 µs coherence time with an angle θ = 2◦ [Jiang et al., 2009].
Another issue related to warm atomic ensembles is the required power for the coupling fields. First, if the laser needs to be off-resonance it must be set with a detuning comparable to the Doppler broadening, about some hundreds of MHz. To compensate this large detuning it is essential to operate at higher power of the coupling beams, and filtering becomes more difficult. In addition, enlarging the size beams in order to avoid decoherence effects contributes also to the increase of the power needed. That is why several hundreds of milliwatts in the coupling path may be sent in warm vapors and co-propagate with the single-photon level signal. Thus, filtering in this scheme is very difficult and requires especially frequency filters such as cascaded Fabry-Perot etalons. With high power pulsed lasers, a quantum memory operating at the single-photon level has been recently demonstrated in a warm atomic ensemble [Reim et al., 2011]. However, the memory time was limited to 1.5 µs, with a signal-to-noise ratio around one.
In summary, although very interesting works have been realized in warm vapors, cold gases appear as a better platform to achieve the single-photon regime and keep reasonable coherence time at the same time.

Avoiding Doppler broadening

Historically, the LKB quantum optics group studied the behavior of warm and cold atomic ensembles and first implemented EIT-based quantum memories in ensembles of warm atoms [Cviklinski et al., 2008, Ortalo et al., 2009]. In warm vapors, the Doppler broadening is the main component of inhomogeneous broadening. Each atom sees both control and signal fields with different frequencies depending on its velocity class. For a pure 3-level system in EIT, the resulting effect is a narrowing of the transparency window. However the presence of hyperfine structure in the excited level associated with the Doppler broadening, which is of the same order of magnitude as the separation in the excited state, destroyed almost totally the transparency effect. Studies have been done in that direction in our group and it has been demonstrated that depumping some velocity classes enables to enhance EIT in warm cesium atoms [Mishina et al., 2011, Scherman et al., 2012]. Cooling and trapping atoms in a magneto-optical trap strongly suppresses the Doppler broadening. This enables to see deep transparency window and consequently to implement EIT-based quantum memory with better performances in ensembles of cesium atoms.
Drawbacks of cold atoms Even though there are many advantages of using cold atomic ensemble, as seen before, to implement quantum memories it still remains drawbacks. Here are the main ones:
1. The vacuum chamber, the ion pump and all the other vacuum components are much bulkier than a simple ambient-temperature cell.
2. It requires at least two frequency-stabilized lasers just to cool and trap the atoms.
3. When based on magneto-optical trap, the trapping magnetic field must be switched off during the memory implementation (see chapter 2). Thus, the experiment runs in cycles, with a loading stage and an experiment stage, and this has two consequences. First the storage time is intrinsically limited by the cloud expansion. That is why a vertical configuration has been developed in [Bao et al., 2012], storage times of few milliseconds were reported. Secondly the repetition rate of the memory implementation is also limited. In fact, the time required to cool and trap the atoms usually represents the main part of the timing, specially for ultracold atoms experiments.
One way to avoid the latter drawbacks is to use dipole traps [Chuu et al., 2008, Zhao et al., 2009]. Thus, the atoms are always trapped, even during the memory implementation stage, and once the differential light shifts induced by the dipole traps across the entire lattice are compensated, long storage times can be obtained. Very recently, optical storage in such a medium during 16 s has been observed in a classical regime [Dudin et al., 2013].
In conclusion, despite of some drawbacks, cold atoms enable to realize quantum memories at the single-photon level, with both large signal-to-noise ratio and long coherence time. Moreover, Doppler broadening is limited, and this allows to increase the memory performances. In the following section, we introduce the memory protocols that we implement in an ensemble of cold atoms during this PhD work, the EIT-based memory, and the DLCZ building block. The common point between them is the generation of collective excitations, which we first discuss.

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Collective excitation

Collective states are of major interest because they can lead to collective enhancement, enabling the retrieval of an excitation stored in a large atomic ensemble with an efficiency close to unity, in a well defined spatiotemporal mode. In order to achieve long storage time it is also important to operate with long-lived states, such as hyperfine sublevels of the ground state of alkali atoms.
In a large ensemble of N identical atoms optically pumped in a ground state |g , a collective excitation corresponds to the transfer of one atom among the N atoms to another ground state |s , but it is absolutely impossible to know which one is concerned (see fig. 1.1). The spin-flip is thus said delocalized over the atomic ensemble. This state is therefore a coherent superposition of all the possible terms with N − 1 atoms in |g and one atom in |s and it can be written as the following spin symmetric state
|1= sˆ† |0 = sˆ† |g1 , g2 , g i, g N
1 N = √ |g1 , g2 , s i, g N , (1.2)
N i=1
where sˆ† is the creation operator for one atomic excitation in |s . Note that in practice, the amplitudes of each term may vary, depending on the laser beam profiles or on the shape of the atomic ensemble.
Now we describe two memory protocols, the EIT-based memory and the DLCZ building block, which rely on such collective excitations.

The dynamic EIT protocol

Via the electromagnetically induced transparency phenomenon, an initially strongly absorptive sample can become transparent for a given signal by the mean of an additional control field [Harris, 1997]. This effect is accompanied by a reversible reduction of the signal group velocity, which enables to build an optical memory. We explain in the following how the control field opens a transparency window, and how a light pulse can be mapped into an ensemble of atoms by dynamically changing the control field intensity.

Electromagnetically-induced transparency

We consider an ensemble of N identical atoms with a Λ-type configuration as shown in figure 1.2: two ground states, |g and |s , and one excited state. Initially all the atoms are prepared in |g . A control field drives the |s → | e transition with a Rabi frequency Ω while the signal field probes the |g → | e transition.
In the regime where Ω is much larger than the Rabi frequency of the signal field we first consider the two states |s and |e interacting with the control field which is represented here by a Fock state with n photons. In the basis {|s | n , |e | n − 1 the Hamiltonian associated with one atom is H n = 0 Ω , (1.3) assuming = 1 for simplicity, as in the rest of the manuscript. The dressed states |s | n and |e | n − 1 have the same energy that corresponds to a null coefficient on the diagonal of the Hamiltonian and they are coupled to the control field related to the Rabi frequency Ω. The eigenstates of H n,
|ψ+n 1 = √ (|s | n + |e | n − 1 ) ,
|ψ−n 1 (1.4) = √ (|s | n − | e | n − 1 ) ,
are associated with the eigenvalues ±Ω. The eigenstates of the system {atoms+field} correspond to superpositions of the dressed states |s | n and |e | n − 1 . The energy gap between these two states is proportional to the Rabi frequency of the coupling field. For a large enough intensity of this coupling field a transparency window for the signal appears, with a width scaling with Ω, as shown in figure 1.3. In the same time, the dispersion of the medium becomes very large, that leads to drastically reduce the signal group velocity. In these conditions, « slow light » was observed experimentally in 1999, with an impressive reduction of the light speed down to 17 m/s in a Bose-Einstein condensate [Hau et al., 1999]. Nevertheless, the EIT phenomenon is still widely studied, e.g. in optomechanical systems [Safavi-Naeini et al., 2011].

The DLCZ building block

Introduced by Duan, Lukin, Cirac and Zoller in 2001 [Duan et al., 2001], the seminal DLCZ protocol constitutes a scheme for quantum repeater implementation. The building block can be seen as the generation of photon pairs temporally separated by a user-defined delay and so is very interesting by itself for various purposes in quantum information science, and enables to work with atomic ensembles in the single-excitation regime.
As for the EIT setting, the DLCZ building block requires a large ensemble of N identical atoms with a Λ-type atomic configuration. As shown in figure 1.5, a weak write pulse detuned from the |g → | e transition induces spontaneous Raman scattering into a photonic mode called field 1 or Stokes photon, transferring one atom into the state |s . There is no preferred direction of emission for the field 1, the total emission probability corresponds to the sum of the emission probability for each photon, but we focus on one particular direction for both field 1 photon and atomic excitation. If the write pulse power is low enough so that two excitations are unlikely to occur, the system {atoms + field 1} is described by the following state |ψ = |0 a |0 a |1 1 + O(χ) , (1.8) where χ = 4g2 N L/c (Ω/Δ)2 tw is the small Raman scattering probability, L the length of the medium, Ω the Rabi frequency, Δ the detuning, and tw the duration of the write pulse [Lukin, 2003]. The index a denotes the atomic state while the index 1 is for the field 1 state. The collective state |1 a is similar to the one presented above (equation 1.2), or in the EIT part (equation 1.5). It corresponds to a symmetric spin excitation in the atomic ensemble, with a phase term ei(Kw −K1 ) • Ri |g1 , g2 , s i, g N ,
|1 a = √ (1.9)
N i=1
where Kw and K1 are the K vectors of the write field and the field 1 photon, and Ri is the position of the ith atom. Equation 1.8 indicates that the atomic spin-flip and the emission of the field 1 photon are strongly correlated. Practically, the field 1 is filtered in polarization, in frequency, and spatially in the case of an off-axis configuration. The frequency filtering is very important to guaranty the creation of a spin-flip. Consequently, in the very low excitation regime, the detection of a photon in the field 1 mode projects the ensemble into a non-classical state with a single excitation delocalized among the whole ensemble. The writing process is probabilistic, but heralded. It is an example of measured-induced protocol.
After a programmable delay a read pulse is sent on resonance with the |s → | e transition and enables to transfer the atomic collective excitation into a second photonic mode, the field 2 or anti-Stokes photon. Due to collective effect, this read out process can become very efficient, but some supplementary phases arise, leading to a phase matching condition. Indeed, the term corresponding to the emission of a field 2 photon is proportional to ei(Kw −K1 ) • Ri ei(Kr −K2 ) • R′ i |0 a |1 1 |1 2 , (1.10) i=1 where Kr and K2 are the K vector of the read field and the field 2 photon, and R′ i is the position of the ith atom at the read time. We immediately see that the condition for constructive interference depends on the atomic motion during the storage time.
• If atoms are moving, constructive interference arises only for K1 = Kw and K2 = Kr , and we recognize what has been mentioned above in section 1.1.1, that the collinear geometry is strongly recommended with warm atomic ensembles.
• If there is no atomic motion, constructive interference occurs whenever the phase-matching condition K1 + K2 = Kw + Kr (1.11) is fulfilled. In consequence the probability amplitude for the field 2 emission in the Kw + Kr − K1 direction is large.
For very large atomic ensembles the emission in the direction that satisfies the aforementioned conditions can completely dominate all the others and leads to a high efficiency collection of the field 2 photon thanks to many-atom interference effect, namely collective enhancement. In the ideal case the final state of the system {field 1 + field 2} is |φ =|0 1|0 2+√ |1 1|1 2+O(χ) (1.12)
Photon numbers for fields 1 and 2 are correlated. In fact the two modes are entangled, as it is for two-mode squeezed states in parametric down conversion or four-wave mixing processes. This DLCZ building block based on Raman scattering is thus a source of photon pairs with a delay, and can also be used as a heralded single photon source [Chou et al., 2004, Laurat et al., 2006]. Entangling atomic ensembles The DLCZ scheme enables the generation of entanglement between two remote atomic ensembles [Chou et al., 2005, Laurat et al., 2007a]. In a synchronous manner two write pulses illuminate two atomic clouds, denoted left (L) and right (R) and separated by a long distance. The scattered photons in field 1 are collected and interfere in an indistinguishable manner on a 50/50 beam splitter, outputs of which are detected by two single-photon counters (see figure 1.6). The detection of one and only one field 1 photon heralds a unique spin excitation in one of the two ensembles, but knowing in which one is impossible from a fundamental point of view. The two atomic ensembles are thus projected in a maximally entangled state |Ψ LR = √ (|1 L|0 R±|0 L|1 R) , (1.13) 2 where the ± sign depends on the output where a photon is detected. Finally, the heralded entanglement between the two atomic ensembles can be transferred to photonic entanglement by applying two read pulses simultaneously. Thus, the two field 2 paths are entangled. More details about the entanglement swapping and about quantum repeaters in general can be found in [Sangouard et al., 2011, Laurat et al., 2007b].

Table of contents :

Introduction 
Part I Theoretical and experimental keys for quantum memory implementations
1 Collective excitation in a large ensemble of cold atoms
1.1 Why using cold atoms ?
1.1.1 Down to the single-photon level
1.1.2 Avoiding Doppler broadening
1.2 Collective excitation
1.2.1 The dynamic EIT protocol
1.2.1.1 Electromagnetically-induced transparency
1.2.1.2 Reversible mapping of a photonic excitation
1.2.2 The DLCZ building block
1.3 Critical parameters
1.3.1 Optical depth
1.3.1.1 Definition and general considerations
1.3.1.2 Optical depth in the DLCZ building block
1.3.1.3 Optical depth in an EIT medium
1.3.2 Decoherence of collective excitation and timescales
1.3.2.1 Motional dephasing
1.3.2.2 Differential light shift
1.3.2.3 Residual magnetic field
2 Tools for the atomic ensemble preparation
2.1 Large cloud of cold cesium atoms
2.1.1 Magneto-optical trap
2.1.2 Timing
2.1.3 Optical paths for memory implementation
2.1.4 Measuring the optical depth
2.2 Cancellation of the magnetic field
2.2.1 Probing the residual magnetic field by microwave spectroscopy
2.2.2 Magnetic field compensation
2.3 Optical pumping in mF = 0
2.3.1 Principle
2.3.2 Experimental implementation
2.3.3 Results and discussion
Part II Experimental implementations of quantum memory protocols 
3 Experimental investigation of the transition between the Autler-Townes splitting and the electromagnetically induced transparency models
3.1 From the EIT to the ATS models
3.1.1 Atomic susceptibility for a three-level system
3.1.2 Electromagnetically induced transparency model
3.1.3 Autler-Townes splitting model
3.2 Experiment
3.2.1 Preparation of the atomic medium and timing
3.2.2 Signal and control fields
3.2.3 Absorption spectra
3.2.4 Rabi frequency of the control
3.3 Fitting of the absorption profiles
3.3.1 Akaike weights
3.3.2 Per-point Akaike weights
3.4 Theoretical simulations
3.4.1 Multilevel structure
3.4.2 Doppler broadening
4 Quantum memory for orbital angular momentum of light
4.1 Optical memory for twisted photons in the single-photon regime
4.1.1 Experimental setup
4.1.1.1 Mode generation
4.1.1.2 Implementation of the reversible mapping
4.1.1.3 Detection at the single-photon level
4.1.2 Experimental results
4.2 Quantum memory for OAM encoded qubit
4.2.1 Measuring the qubit coherence
4.2.1.1 Interferometry technique
4.2.1.2 Accessing the interferometer phase
4.2.1.3 Observing the interference fringes
4.2.2 Quantum state tomography
4.2.3 Benchmarking
4.2.3.1 Fidelity with the ideal state
4.2.3.2 Fidelity with the input state
4.2.3.3 Classical limit
5 Generation of non-Gaussian state of light from atomic ensembles
5.1 Characterizing the DLCZ building block
5.1.1 Conditional retrieval efficiency
5.1.2 Normalized intensity cross-correlation function
5.1.3 Conditional autocorrelation function
5.2 Experimental implementation of the DLCZ building block
5.2.1 Experimental setup
5.2.2 Characterization
5.2.2.1 Dependence with the excitation probability
5.2.2.2 Dependence with the optical depth
5.3 Tomography of the retrieved single-photon state
5.3.1 Principle
5.3.1.1 Single-photon state in phase space
5.3.1.2 Quantum state tomography
5.3.2 Experiment
5.3.2.1 Experimental setup
5.3.2.2 Temporal mode
5.3.3 Preliminary results
Conclusion 
A Phase-frequency locking
A.1 Lasers
A.2 Operation
B Labview managed spectroscopy
B.1 Frequency scans
B.2 Absorption measurement
C Analysis of single-photon detector outputs
C.1 EIT-based memory experiment
C.1.1 Histograms
C.1.2 Efficiency calculation
C.1.3 With interferometry
C.2 DLCZ building block experiment
C.2.1 Timing
C.2.2 Parameter calculation
D From Laguerre-Gaussian to Hermite-Gaussian modes
D.1 Generalities
D.2 Laguerre-Gaussian modes
D.3 Hermite-Gaussian modes
D.3.1 Horizontal and vertical modes
D.3.2 Diagonal and antidiagonal modes
E Effects of detection imperfections on the fidelity
E.1 For LG states
E.2 For HG states
F Atomic filter for the DLCZ experiment
G Homodyne detection
G.1 Principle
G.2 Details about the experimental setup
Bibliography

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