Time- and frequency-domain polariton interference
G. T. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, B. C. Buchler
aa r X i v : . [ qu a n t - ph ] M a y Time and Frequency Domain Polariton Interference
G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B.C. Buchler
Centre for Quantum Computation and Communication Technology, Department ofQuantum Science, The Australian National University, Canberra, Australia
Abstract.
We present experimental observations of interference between an atomicspin coherence and an optical field in a Λ-type gradient echo memory. The interferenceis mediated by a strong classical field that couples a weak probe field to the atomiccoherence through a resonant Raman transition. Interference can be observed betweena prepared spin coherence and another propagating optical field, or between multipleΛ transitions driving a single spin coherence. In principle, the interference in eachscheme can yield a near unity visibility. ime and Frequency Domain Polariton Interference ime and Frequency Domain Polariton Interference Δ>>γ,δγ δ ( a )( b ) ε p α B - s w it c h B - s w it c h Coupling field Ω c |1> |2> ε s ε ε ε p ε s T =0R =1 T , R Time T =0R =1 z Atomic ensemble |1>|2> ε ε ε p ε s Ω c1 = ε c Ω c2 = ε c e -i δ t + φ Ω c1 = ε c Ω c2 = ε c B-switch T i m e z τ τ Figure 1. (a) Schematic representation of atom-light interference in the memory. Theprobe pulse, E p , is fully absorbed in the atomic spin coherence ( α ). The second steeringpulse, E s , enters the memory at the precise time that the first echo is being emitted sothat it can interfere with the recalled light. The interference is determined by relativephase of the pulses and the effective beamsplitter ( T , R ), which is controlled by thestrength of the Raman coupling field. The remaining atomic coherence can be recalledlater as E . (b) Left: Double-Λ level structure and optical fields used for interferenceof two Raman absorption paths of signal fields (probe and steering) with differentfrequencies. Both Λ transitions drive the same coherence. Right: The procedureused for observing double-Raman interference. The probe and steering pulses are sentinto the memory each with a corresponding coupling field. Interference between theunabsorbed probe pulses, E , and the atomic coherence, which is recalled from thememory as E , can be observed by varying the relative phase of the two Λ transitions. ime and Frequency Domain Polariton Interference k being the spatial frequency) of the optical field ˆ E ( t, k ) and the coherence,ˆ σ ( t, k ), between the atomic energy levels | i and | i . The normal mode of the two-level GEM is defined as ˆ ψ ( t, k ) = k ˆ E ( t, k ) + N ˆ σ ( t, k ) [12] and propagates in the( t, k ) plane where N is the linear atomic density. Like the normal mode in EIT[3],ˆ ψ ( t, k ) is a combination of atomic polarisation and optical field and can be considered apolariton. The velocity at which the polariton propagates in k -space is proportional tothe atomic frequency gradient and, by switching the sign of the magnetic field gradient,the evolution of the polariton can be reversed. When the polariton again reaches k = 0the atomic coherence is rephased and the polariton is recalled as an optical field.A similar polariton equation, ˆ ψ ( t, k ) = k ˆ E ( t, k ) + N Ω∆ ˆ σ ( t, k ), can be defined for theΛ-GEM system by replacing N with N Ω c ∆ , where Ω c is the coupling field Rabi frequencyand ∆ is the Raman detuning from the excited state. In the three-level case σ ( t, k )describes the atomic spin coherence.The polariton propagates in the ( t, k ) plane following ∂∂t − η ( t ) ∂∂k − i gN Ω c k ∆ ! ˆ ψ ( t, k ) = 0 , (1)where g is the atom-light coupling strength and η is the detuning gradient that resultsfrom the applied magnetic field [12]. The evolution of the three-level GEM is similar tothat of the two-level GEM but the addition of the coupling field results in additionalflexibility: the strength and phase of the coupling between the probe field and thememory can be optically controlled.Our first experiment investigates interference of light pulses with a mode storedin the atomic memory. Following on from Ref. [13], this effect can be thought of asa time-delayed beamsplitter system. The effective optical depth (OD) of a Λ-GEM isdefined as β = gNη ( Ω c ∆ ) , (2)where g is the coupling strength and η is the linear broadening from the magnetic field.For the writing stage the transmissivity, T ( β ), of the effective beamsplitter is thefraction of the input light field that is leaked through the memory so that T ( β ) = e − πβ ,while the fraction of the light written into the memory is given by the reflectivity R ( β ) = 1 − T ( β ). For the reading stage, the R ( β ) will be the fraction of the polaritonthat is converted into a recalled optical field while T ( β ) will be the fraction that remainsin the memory. Since T ( β ) and R ( β ) are defined by the strength of the coupling field,one can tune the transmittivity of the beam-splitting through the power of the couplingfield. A series of reading and writing events, as shown in Fig. 1(a), can then be describedusing appropriate reflectivites. The amount of light recalled in the first echo is given by E = √ R R e − γ τ E p + e iθ √ T E s , where E p is an initial probe pulse and E s is a secondpulse, which we label the steering pulse, that enters the medium at the time that the ime and Frequency Domain Polariton Interference E p , at a rate γ ,during the storage time τ and the phase θ can be chosen at will. This equation showsthat interference can arise between recalled fraction of E p and E s and, in particular, if √ R R e − γ τ = √ T and θ = π then E can be fully suppressed. This simple analysisignores other details such as the matching of the temporal modes of the pulses. Otherfactors that limit ideal interference will be discussed later when we analyse the resultsof our experiments.We use the polariton picture to visualise the dynamics of the time-domainbeamsplitting operation. As the probe pulse is absorbed and the atomic coherencedephases, the polariton evolves from to higher spatial frequencies and becomes primarilyan atomic spin-wave. The position of the mode in k-space represents how much theatoms in the spin wave have dephased. A higher atomic frequency gradient means thereis a larger dephasing and therefore faster progression to higher values of k. When themagnetic field gradient is reversed, the polariton propagates back towards k = 0 atwhich point the coherence is rephased and the polariton is converted into an opticalfield which exits the memory. The k = 0 crossing is analogous to the interface of abeamsplitter. At this point the Λ transition couples the optical mode of the probefield, propagating in the ( t, z ) plane, and the spin coherence of the atomic ensemble,propagating in the ( t, k ) plane. T i m e ( μ s ) z ( c m ) k ( c m - )
20 4 6 80 5 10 05-5 ε p ε s ε ε Figure 2.
Numerical simulation showing interference between the electric field, plottedon the z-t plane, and the atomic coherence, plotted on the k-t plane, where the secondlight pulse is out of phase from the first echo field. The parameters used in thesimulations are: gN L/γ = 40, Ω c / ∆ = 0 .
75, Ω c ( t = 4 µs ) = 0 . c ( t = 2 µs ) and φ E s − φ E p = π . The magnitudes of the electric field and the atomic coherence areplotted. ime and Frequency Domain Polariton Interference -1 1 Time (1/ γ ) Time (1/ γ ) z ( c m ) ε (a.u) α (a.u) ε p ε s ε ε -1 1 Figure 3.
The real parts of the electric field (left) and the atomic coherence (right) fora simulation of the time-domain interference experiment. The insets above the plotsshow the electric field and atomic coherence at the input of the ensemble.
Figure 2 shows a numerical simulation of the time-domain interference scheme usingthe methods described in [12]. The simulation shows the evolution of the electric fieldin real space (the ( t, z ) plane) and the atomic spin coherence in Fourier-space (the ( t, k )plane). In this numerical simulation, the phase of the steering pulse is chosen suchthat a suppression of the echo from the probe pulse is observed. The intensity of thecoupling field is chosen so that the effective splitting ratio is roughly 0.5. Constructiveinterference into the atomic polarisation occurs and the resultant polariton is recalledin field E after the second gradient switch.Figure 3 shows the real components of the electric field and the atomic coherenceto illustrate the interference for the same simulation parameters. The phase of theamplitude transmission and reflection coefficients in our beamsplitter analog can beextracted from the Maxwell equation, which gives k E ( t, k ) = N Ω c ∆ ˆ σ ( t, k ) [12]. Thephase of the recalled probe field depends on both the phase of the coupling field, Ω c ,and the phase of the atomic coherence, ˆ σ ( t, k ). This provides optical control over therelative phase between the two inputs to the interference. Note that when the normalmode crosses k = 0 the relative phase of E ( t, k ) and ˆ σ ( t, k ) changes by π . This isanalogous to the π phase shift experienced on one side of an optical beamsplitter andis a requirement for the conservation of energy. ime and Frequency Domain Polariton Interference Rb atoms and a linearswitchable varying magnetic field as described in Ref. [9]. The coupling and probefields are passed through acousto-optic modulators (AOMs) that allow us to create therequired pulse sequences by driving them with appropriate RF signals. To generate theprobe and steering pulses, the RF signals were created using separate, but phase locked,arbitrary waveform generators and were combined together before the AOM (details arein the supplementary material). In this manner, the frequency, phase and amplitude ofthe coupling, probe and steering fields can be independently controlled. The couplingfield power used for maximum coupling between the optical and atomic modes was 330mW and was adjusted to control the coupling. The probe and steering pulses have thesmae frequency and were on the order of few µ W. The coupling field, blue detuned by3 GHz from the S / , F = 2 → P / , F ′ = 2 transition, is Raman resonant with theprobe and steering pulses which are blue detuned from the S / , F = 1 → P / , F ′ = 2transition.We stored a 4 µs probe pulse in the memory and recalled it after a storage time of τ = 10 µs . The steering pulse was injected just as the atomic coherence excited by theprobe returned to k = 0. We label the light detected at this time as E , and integratethe detector signal over the pulse duration to obtain a value for the pulse energy. Thepolariton that remains in the atomic medium after the first recall is itself recalled afterstorage time τ = 10 µs . We detect it in the same manner as E and label it E . Figure1(a) shows the sequence of pulses that are stored, interfered and retrieved along withthe coupling field intensity for each step.The energies of the recalled pulses, E and E , were measured as a function of therelative phase of the probe and steering pulses. The phase of the atomic coherencedepends on the relative phase of the coupling and probe fields. It is therefore possibleto control the phase of the interference by scanning the phase of either the steering pulseor the corresponding coupling field. Fig. 4(a) shows interference fringes for E (blue,dashed line) and E (red, solid line) obtained by varying the phase of the coupling fieldcorresponding to the steering pulse. This was accomplished by varying the phase of theRF signal that drives the coupling field AOM during the interference event relative to itsphase during the storage of the probe pulse. For this data, the powers of E p and E s wereequal and the coupling field power during the interference event was tuned to find themaximum fringe visibility on E , which was found to be 68%. The visibility of E echo,23%, is substantially lower due to the power mismatch of the steering pulse and therecalled atomic coherence required to optimise the interference in E . The reflectivitycorresponding to the recall of E is 37%.Control over the effective beamsplitter ratio is demonstrated in Fig. 4 (b). It can beseen that by varying the coupling field power the effective splitting ratio can be tuned tofind a maximum in the interference. For this data, the power contained in the steeringpulse was adjusted to provide good visibility for both E and E . It is interesting to notethat for strong coupling fields, one optical pulse is written into memory while another isbeing recalled with little interference between the two, analogous to a high-reflectivity ime and Frequency Domain Polariton Interference Phase (a) V i s i b ilit y Normalized control field power (b) ε ε π 2π N o r m a li ze d i n t e n s it y ε N o r m a li ze d i n t e n s it y ε Figure 4. (a) Atom-light interference fringes at different times resulted frominteraction of the steering pulse with echo generated from the probe pulse. The firstarm of the interferometer which is in the optical mode leaves the memory(blue data)and the second arm is stored as an atomic coherence that is transformed back to thelight field after re-switching the B-field (red data). The dashed blue and solid redlines are sinusoidal fits to the corresponding data. The red and blue data yield a fringevisibility of 68% and 23%, respectively. (b) Visibility of fringes for two pulses separatedin time at the first (blue points) and second (red points) reading stage as a functionthe normalised coupling field power. beamsplitter. For a weak coupling field, on the other hand, the effective beamsplitterbecomes fully transmissive, again meaning no interference between the pulses as thesteering pulse passes straight through without storage and the probe pulse remainstrapped in the atomic coherence.Now we consider the second experiment, in which the interference results fromdriving a single atomic coherence with multiple two-photon transitions as depictedin Fig. 1(b). In this case, the probe and steering pulses are co-propagating andenter the medium simultaneously but are separated in frequency by more than thememory bandwidth. In the far-detuning and adiabatic regimes, this double-Λ systemis equivalent to a quasi-two-level system interacting with two fields of different Rabifrequency (see Fig. 1 (b)). The interference between the two Λ transitions will changethe response of the medium to the probe and steering pulses. When they interferedestructively, the absorption of the probe and steering fields is suppressed and bothpulses are transmitted through the medium. When the two Λ transitions are in-phase,both pulses are coherently absorbed and can be recalled later on-demand.As with our first experiment, the properties of the interference can be controlledthrough the coupling fields. The relative intensity and phase of the two coupling fieldscontrol the superposition of the probe and steering pulses that is transferred to theatomic coherence. This effect has been explored in EIT experiments [14, 15]. UnlikeEIT, however, the optical modes that are not coupled to the atomic coherence in theΛ-GEM scheme propagate through the atomic medium with little loss.The frequency difference between the probe and steering fields was set to 1 MHz,which was larger than the memory bandwidth of 300 kHz to avoid overlap between ime and Frequency Domain Polariton Interference N o r m a li ze d i n t e n s it y ε N o r m a li ze d i n t e n s it y ε Phase
2π 4π3ππ
Figure 5.
Interference fringes from E (blue) and E (red) that resulted frominterference between double-Λ transitions created from the probe and steering pulsesof different frequency stored simultaneously in the memory. The dashed blue and solidred lines are sinusoidal fits to the corresponding data. The red and blue data yield afringe visibility of 73% and 25%, respectively. two broadened Raman lines. Each of the probe and steering fields has a correspondingcoupling field which is tuned to Raman resonance. The pulse length, 4 µs , was chosen togive a bandwidth slightly less than the memory bandwidth. Fig. 5 shows the interferencefringe obtained by varying the relative phase between the two Raman absorption lines.This was done by sweeping the phase of one of the coupling fields. The powers of thecoupling fields are equal, 160 mW each, and remain constant throughout the storageand retrieval process. E is the portion of the probe and steering pulses that does not getstored in the memory and E is the portion that is retrieved from the memory after a 10 µs storage period. The energies of E and E are measured by integrating the detectorsignal over the pulse period.From an operational standpoint, this second experiment can be thought of as thefrequency-domain counterpart to the first. While the first experiment demonstrated abeamsplitting operation between two pulses separated in time, the second demonstratesa beamsplitting operation between simultaneous pulses separated in frequency.In both the time and frequency domain interference experiments we attributethe less-than-unity fringe visibility primarily to spatial and temporal mode mismatchbetween the probe polariton and the steering pulse. We believe that this is mainly due tothe atomic motion and non-zero transverse magnetic field, which affects the echo signalfor long storage times. This can be justified by the larger visibility measured in thefrequency-domain interference scheme, where interference occurs between pulses thatsimultaneously interact with the atomic coherence. During the storage time, atomicdiffusion can change the spatial mode of the coherence and as a result, the echo signalwill have a slightly different mode compared to the input signal. This effect is negligiblefor shorter storage times. The presence of a transverse magnetic field can induce an ime and Frequency Domain Polariton Interference k x and k y ) during storage. The transverse k vector is imprintedto the echo signal at the readout diverting the output optical field slightly compareto the steering pulse. In the beamsplitter analogy, this amounts to a poorly alignedinterferometer. An inhomogeneous longitudinal magnetic field can alter the shape of theecho signal compared to its input leading to temporal mode mismatch. We anticipate,therefore, that the visibility could be improved by increasing the buffer gas pressure orusing a cold atomic sample in order to increase the time of flight of the atoms and takingextra care with the magnetic environment to prevent pulse deflection and distortion. Forthe time-domain interference, numerical simulations (see supplementary material) revealthat, in the limit of large OD, the interference visibility of the system can approach unityfor both interfereometer outputs.In summary, we have demonstrated interference effects between propagating opticalfields and a collective atomic spin coherence. Fringe visibilities of 68% and 73%were observed for time-domain and frequency-domain interference schemes, respectively.These schemes may have relevance to manipulating optical quantum information. Unlikeprevious schemes, interference in a gradient echo memory could offer dynamic, opticallyaddressable linear operations on optical qubits. These gates could operate on eithertime-bin or frequency multiplexed qubits or even a combination thereof. The time-delayed beamsplitter scheme can also be used for optical quantum state engineering[16, 17] and also for optimal Gaussian purification of coherent states from severalimperfect copies [18]. The ability to construct this type of interferometer is also ofinterest in building a coherent all-optical switch [19, 20].This research was conducted by the Australian Research Council Centre ofExcellence for Quantum Computation and Communication Technology (project numberCE110001027). [1] K. Bergmann, H. Theuer, and B. W. Shore. Coherent population transfer among quantum statesof atoms and molecules. Rev. Mod. Phys. , 70:1003–1025, Jul 1998.[2] Michael Fleischhauer, Atac Imamoglu, and Jonathan P. Marangos. Electromagnetically inducedtransparency: Optics in coherent media.
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