Spin Wave Diffraction Control and Read-out with a Quantum Memory for Light

3.6 MB·English

by  Gabriel Hétet

Spin Wave Diffraction Control and Read-out with a Quantum Memory for Light
Gabriel H´etet1 and David Gu´ery-Odelin2,3
1 Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure-PSL Research University,
CNRS, Universit´e Pierre et Marie Curie-Sorbonne Universit´es,
Universit´e Paris Diderot-Sorbonne Paris Cit´e, 24 rue Lhomond, 75231 Paris Cedex 05, France
2 Universit´e de Toulouse ; UPS ; Laboratoire Collisions Agr´egats R´eactivit´e, IRSAMC ; F-31062 Toulouse, France and
3 CNRS ; UMR 5589 ; F-31062 Toulouse, France
Aschemeforcontrolandread-outofdiffractedspinswavestopropagatinglightfieldsispresented.
Diffraction is obtained via sinusoidally varying lights shifts and ideal one-to-one mapping to light
is realized using a gradient echo quantum memory. We also show that dynamical control of the
diffracted spin waves spatial orders can be implemented to realize a quantum pulse sequencer for
temporal modes that have high time-bandwidth products. Full numerical solutions suggest that
bothco-propagatingandcouterpropagatinglightshiftgeometriescanbeused,makingtheproposal
5 applicable to hot and cold atomic vapours as well as solid state systems with two-level atoms.
1
0 PACSnumbers:
2

n
Macroscopic spin superpositions in atomic ensembles
a
J have been studied for decades and have found use in 2π/kR
many areas of physics. They are essential for quantum a) + - 0
6 -2 2
optical information storage and retrieval [1] and for sen- θ -1 1
] sitive magnetometry [2]. Spin superpositions containing
h quantized excitations are prepared either by performing Input Lbiegahmt sshift - + Retrieved
p probe spatial orders
quantum non-demolition measurements on the scattered
-
t light [3–6], or by directly mapping quantum states of
n
a light [7–10] onto the atoms. Using appropriate schemes b) i) + - c) (x105)
u furthermore enables the actual shape of a spin wave to 1600 i) ii) iii) -3-2 0 23
q be controlled to a high degree. Andr´e et al. [11] pro- - + 1200 -5-4 -1 1 45
[ posed a scheme to perform Bragg scattering of admix-
ii)
1 turesofspinandlightcomponentswithcounterpropagat- θ k 800
v ing lasers and electromagnetically induced transparency
400
4 (EIT).Twocounterpropagatingpolaritonmodesarecou- - +
iii)
9 pled, enabling light fields to be localised within the 0
11 medium with far reaching prospects in quantum non- + z - 5 15 25 35
linear optics [12] and quantum simulations [13, 14]. A Time (t)
0
signatureofpolaritonicBraggdiffractionwasobservedin
.
1 the fluorescence induced by the stationary light [15] but
FIG.1: a)Gradientechomemory(GEM)schemewithalon-
0 observing the diffracted modes is currently challenging gitudinaldiffractiongratingappliedduringstorage. b)i)Cre-
5
due to the drastic requirements on the atomic ensemble ation of a narrow k-space distribution of spin waves by map-
1
temperature and the lack of direct read-out procedure. ping a pulse of light in the memory. ii) Diffraction of the
:
v Thesemethodsarealsonotapplicablefortwo-levelatom polaritonbyapplyingapairofintensedetunedlaserswithan
Xi based quantum memories [16, 17]. angleθforatimeτ. iii)Thedetunedlasersareturnedoffand
theread-outofthediffractionpatternisdonebyflippingthe
r An alternative for realising Bragg scattering of spin gradient sign. c) 2D-map of the atomic coherence amplitude
a
waves is through the use of a controlled inhomogeneous ink-spaceasafunctionoftime,demonstratingKapitza-Dirac
broadening, and a prominant technique is the gradient diffraction of the spin coherence wave.
echo memory (GEM) [18, 19]. GEM was already shown
to allow light storage at the quantum level with two
[16] and three atomic levels [20], to demonstrate efficient specific durations of the grating imprint, the coupling of
pulse re-ordering [21] and spectral manipulation of light the spin wave to the output field is efficiently canceled
pulses that have large time-bandwidth products [22]. and that diffraction can actually be temporally reversed
Here,wedemonstratethatGEMcanbeusedinconjunc- by displacing the grating nodes. These two effects to-
tion with a grating to enable efficient coherent mapping gether realize a two-level atom coherent pulse sequencer
between diffracted modes and propagating light fields. forlightpulsesthathavelargetime-bandwidthproducts.
For this, we use a theory that includes all spatial orders To perform longitudinal diffraction of a spin wave one
inscatteringtheoryandshowthatone-to-onemappingis creates a spin wave with longitudinal momentum spread
possibleeveninaroomtemperaturevapourcell, beyond ∆kbywritingaweaklaserfieldexcitationinanoptically
the secular approximation [23]. Last, we show that for dense atomic ensemble. Using GEM, an effective two-
2
level atomic ensemble is first subjected to a controlled to the atoms for a time τ yields the following polariton
linearly varying gradient η(t)z as depicted in Figure 1. evolution
After an input pulse with temporal width ∆t enters the
+∞
medium, polaritonic modes with increasing momentum (cid:88)
Ψ(k,t +τ) = inJ (ντ)Ψ(k+nk ,t ), (3)
k over time are created. When the gradient is turned 0 n R 0
off, polaritons remain trapped in the ensemble with a n=−∞
longitudinal momentum extension ∆k = η∆t. Here, we
where L is the sample length and J is the n-th order
suppose that the effective two-level atoms form a quasi n
Bessel function of the first kind. We thus expect the ini-
one-dimensional ensemble along a direction z. The cor-
tial wave to be splitted into multiple spatial orders with
respondingoperatorsareσj (z,t ),where|1(cid:105)and|2(cid:105)are
12 0 a weight given by the Bessel functions. iii) The grat-
the two atomic states and j labels atoms within a slice
ing is finally turned off at a time t = 20 and the gradi-
with length δz. The locally averaged coherence opera-
ent is turned on with a negative slope to complete the
tors read σ (z,t ) = 1 (cid:80)Nz σj (z,t ), where N is
12 0 Nz j=1 12 0 z gradient echo protocol. The result of numerical simula-
the number of atoms in a volume given by the cross sec-
tionsisdisplayedontherightpanelofFigure1-c),show-
tion area of the interacting light field over a slice δz. We ing the evolution of |Ψ(k,t)|2 as a function of rescaled
now suppose that the atomic superpositions do not de-
space and time. For the grating parameters, we choose
phase over the course of the experiment. In particular, ν =ν/g =2, k =k l =150×105 which, with g =3
0 R R 0
this means that we can ignore the Langevin operators
kHz, corresponds to a grating step of 16 mm. We also
and treat the operators σ12 as c-numbers. use η = 7.106 and an optical depth 2πN2/η = 5.4. Nu-
Torealizediffractionofthespinwave,apairofintense merical simulations show that in the mapping phase the
pulses with an angle difference θ, with a Rabi frequency polariton evolves to higher k until the gradient is turned
Ω and detuning ∆ from the two-level atomic transition off. When the grating is turned on Bragg diffraction is
arethenappliedtotheatomicsamplepreparedinaspin clearly observed. The zeroth order mode is depleted and
superposition. If the detuning is larger than the decay higher order modes are populated as a function of time,
rate from the excited state, the atoms will thus inter- as expected from Eq. (3). The light-shift δ(z,t) thus
act with a standing wave that produces a sinusoidally leads to an evolution of the collective spin wave that is
varying light shift δ(z,t) = ν(t)cos(kRz) on the atomic analogous to the Kapitza-Dirac diffraction [25].
coherences, where ν(t) = Ω(t)2/∆. k is related to the
R At t = 20 the grating is turned off and the gradient
angle between the two lasers that generate the optical
turned on. As expected, the light field is reemitted at
grating and must be greater than ∆k [32]. In rescaled
k = 0. One specific feature of GEM here is that, as is
time and space units (normalized by l = c/g and 1/g
0 0 manifest in Fig. 1-c), the energy of the spin wave modes
respectively where g is the vacuum Rabi frequency in
0 is released from the memory at times T = T +nk /η,
0 R
the light mode and c the vacuum speed of light), the
wherendenotesthemodeorderandT thetimeatwhich
0
Maxwell-Bloch equations read:
the zeroth order mode comes out. The polariton spatial
modes are here in the positive range before switching
∂ √
E = i Nσ so they are all released from the memory with a period
∂z 12
T = k /η = 2.2. In typical diffraction grating experi-
∂ (cid:104) (cid:105) √ R
ments with light, electrons [25], degenerate atomic gases
σ = i η(t)z+ν(t)cos(k z) σ +i NE, (1)
12 R 12
∂t [26, 27] or molecules [28], the field onto which a periodic
phase profile is imprinted is let free to expand in two-
with η = ηl/g , ν = ν/g and N is the atomic density.
0 0 dimensions to realize the equivalent of a Fourier trans-
TransformingtheseequationsintothespatialFourierdo-
formation. This operation maps the local phase onto a
main, one gets an evolution equation for the spin wave
transverse profile that displays the discrete set of spatial
inside the sample [24]
modes. We demonstrated here that GEM exactly real-
izes such a Fourier transform but in the time domain,
(cid:16)∂ ∂ N(cid:17)
−η(t) +i Ψ(k,t) allowing efficient read-out of the diffracted longitudinal
∂t ∂k k modes in this time-of-flight like sequence.
(cid:2) (cid:3)
=ν(t) Ψ(k+kR,t) + Ψ(k−kR,t) , (2) Wenowturntothedemonstrationofaquantumpulse
sequencer. At this point, we shall distinguish two op-
where Ψ = kE +Nσ . In this picture, absorption or erating regimes for diffraction read-out: the regime I
12
emissionoflightoccurswhenthepolaritonreachesk =0, where n(cid:48)k > k = η(t − t ) and regime II where
R 0 0 in
i.e. when the phase matching condition is met. n(cid:48)k < η(t −t ), where n(cid:48) = (cid:100)ντ(cid:101) is the number of
R 0 in
Letusnowconsiderthetime-of-flight-likesequencede- populated modes for a grating duration τ. In regime I,
picted in Fig. 1-b). i) A pulse of light is sent to the sam- n(cid:48) spatial modes exactly occupy both the positive and
ple at t = 1.5 with a positive gradient slope and with negative k-values. In regime II, n(cid:48) spatial modes only
in
ν = 0. ii) The gradient is turned off and the grating occupy the positive k-space, as per Figure 1-b).
turned on at a time t = 15. Assuming that the optical Simulations are now realised with the sequence shown
0
fieldisfullytransferedtotheatoms,applyingthegrating in the inset Figure 2-a) and in the regime I, reached by
3
amplitudes of stationary atoms. It is therefore in princi-
a) plereversibleprovidedtheystayinapurestatethrough-
i)
1
y 1Input Output outthecouplingtothegrating. Spatialmoderefocussing
ii) z val efficienc000...468 Normalized probe energy i) iτi) iii)WgWgirrviiaatt)tthhiinnoggut ctpoahcnaosble,eoaafscohsnhieeovweodnf tqfihugeiutaerersmi2ms-bpo)lfy:tuhfseoinlglgorwaatiinsnpggindi-sieffcsrhhaoicfttlieiokdne,bpytrhoπe-
e at a time t = 11 to exactly cancel the phase acquired
iii) etri0.2 τs 0 5 Tim10e 15 20 by each cohserence. This can readily be done experimen-
R
tallyusingacousto-opticmodulators. AsshownFig. 2-b)
0
iv) 0 1 2 3 4 5 6 7 8 9 10 (rightpanel)theatomiccoherenceevolutionindeedrefo-
Grating duration ντ
cusses so that the zeroth order spatial mode is recovered
withunitefficiency,demonstratingthereversibilityofthe
b)
Δϕ=0 (x105) diffraction process.
i) I
i) t=ts ii) The effects presented Fig 2 can also be observed using
2000
othermemoryprotocols. Forinstancewithhomogeneous
three-levelΛatomicmemoryschemessuchasEIT[29]or
k
0 Ramancoupling [30], the gratingcan beturnedon when
ii) I Δϕ=π the control fields are off, that is when the optical field
is transferred to the atomic coherences. The GEM con-
−2000
veyor belt in k-space is however essential for diffraction
Δϕ
imaging. Furthermore,thelasttwopropertiesofthepre-

8 9 10 11 12 1 3 sented spin wave diffraction have direct implications for
z
Time t
precision manipulation of optical pulses and single pho-
tonsusingGEM.ItwasalreadyshownthatGEMenables
FIG. 2: a) Simulations of the gradient echo memory (GEM) efficientstorageofpulsesthathavelargetime-bandwidth
scheme with a diffraction grating applied during storage. productswithouttherequirementforhighopticaldepths
Light read-out efficiency as a function of grating duration [24]. Using a Raman transition and a time dependent
τ. Inset : Input and output probes with and without the coupling to the ground state, pulse re-ordering was also
grating. b) Refocussing of the diffraction pattern. At a time demonstrated[21]. Forinstance,inputpulseswereshown
t=t ,thegratingpositionisdisplacedbyλ/2byshiftingone
s to come out first-in-first-out (FIFO) by turning off the
of the arms phase by π (left panel), producing rephasing of
coupling to the ground state at the first k = 0 crossing
thediffractionpatterntothezerothorderspatialmode(right
andbyswitchingthegradienttwiceusingaΛ-GEM.Ex-
panel).
tending these effects to two-level atomic memories is im-
portantasitwouldenlargetherangeofmaterialsystems
where random access memories can be implemented. A
simply increasing the grating momentum by a factor of
combination of diffraction intensity and refocussing con-
3.3 and reducing the gradient by the same factor. Fig-
trol in the regime I actually provides the ingredients to
ure 2-a) shows the retrieval efficiency of the zeroth order
do so very efficiently as we now show.
mode as a function of grating duration in a time window
t = 18±3. As the duration of the interaction with the The idea is shown Fig. 3-a). Three pulses are input
gratingisincreased, thezerothordermodegetsdepleted to the GEM memory. At t = 10, one turns on a grat-
leading to a drop in retrieval efficiency. This is because ing on for a time τs such that the zeroth order mode is
the polariton keeps the energy in higher order spatial completely depleted, that is at the first node of the ze-
modesthatremaintrappedintheatomicensemble. The rothorderBesselfunction. Thegradientisthenreversed.
first positive spatial mode will only be phase matched Since the zeroth order mode is depleted, when the k =0
to the optical field a time k /η after the zeroth order condition is reached, there is no emission of light and
R
mode is emitted. The energy of the zeroth order mode all the pulse energy is mapped onto higher order spatial
releasedfromthememoryfollowsthezerothorderBessel modes. One can however retrieve this energy back into
function, meaning that the energy is coherently sloshing the zeroth order mode by turning off the gradient, ap-
back and forth between stationary spatial modes inside plying the grating again during a time τs but with one
theensemblebeforebeingoutput,asignatureofthewave arm of the grating shifted by π. The key point is that,
natureofthespincoherences. Remarkably,alltheenergy duringtherefocussing,thefirstordermodesdonotcross
remains trapped inside the sample at a time τ =2.4 cor- k =0. Turning the gradient on again therefore realizes a
s
responding to the first node of the zeroth order Bessel similarFIFOread-outasintheΛ-GEM:thepulsescome
function. Aswewillshow,thisoffersanewusefuldegree out without time-reversal but now using only two-level
of freedom for sequencing light pulses. atoms and this purely dispersive diffraction control.
Another important ingredient for pulse reordering is One can even re-arrange the order of pulses. For in-
refocussing of the spatial modes. The presented diffrac- stance,onecanmakeasequenceofpulses1-2-3comeout
tionoriginatesfromasuperpositionoftheelectrondipole in an order 3-1-2 using a similar combination of diffrac-
4
the temperature should be low enough so that atomic
a) b) motion does not wash out the higher order modes of
First in-first out (FIFO) Pulse re-ordering
the spin wave. This is especially important when one
Probe intensity0000....24168 12 3 12 3 0000....12468 1 2 3 3 1 2 uiinnsgdeeswecdiollufibnnetdeortn-hpatrthopeaasogsruadmteirningogffihaeulgdnrsdartweindhg’esrseotfetphneaonfgor4ma0t0eitnnegrmss.paWancde-
0 Mangez bon, c’est bien 5 10 15 20 25 30 35t 00 5 10 20 25 30 35 t a storage time of 0.1 ms, atomic temperatures of less
k than 1 µK are required for the atoms to feel the valleys
FIFO
of the standing wave and thus for the atomic coherences
to be diffracted, favouring quasi-copropagating geome-
Holding
tries where magneto-optical traps or even warm vapours
can be used. Conversely, provided the light grating does
t t
notinfluenceatomicmotion,thisdiffractioneffectcanbe
used as a very sensitive quantum thermometer for ultra-
cold atoms. The narrow two-level atomic transitions in
(x105)
rare earth doped crystals [16] are also ideal for the effect
η 500
to be observed experimentally. The sinusoidal detuning
-500
ν 0 π 0 π can be realized simply by using a light shift applied di-
τ τ t τ τ t rectly close to the optical transition and the M˝ossbauer
s s s s
Diffracting Rephasing effect prevents the standing wave to induce atomic mo-
tion. The light fields should only be tuned so that stim-
FIG.3: Re-orderingofpulsesusingtwolevelatoms. Fromtop ulated and spontaneous emission are negligible. Finally,
to bottom : probe intensity as a function of time, evolution being able to control the two-level atom coupling to a
of the energy of the polariton in k-space and gradient slope given mode in rare earth doped materials can be useful
andgratingamplitudesusedinthenumericalsimulations. a) for turning delay lines into memories without the noise
First in first out scheme. b) Pulse reordering. induced by π-pulses [17, 31] thus widening the range of
materials available for quantum memory applications.
In conclusion, we present a mechanism akin to diffrac-
tion and refocussing techniques. This is demonstrated
tion of optical fields or electron motion, where coher-
Fig. 3-b). First, the third pulse is left to come out after
ent superpositions of spins are diffracted and efficiently
a standard GEM sequence. Then, the FIFO sequence is
read-out. Applying far-detuned laser pulses along an
implementedtoreorderthepulses1and2. Thegradient
array of atomic spin coherences is shown to generate
combinedwithsinusoidalshiftsisthereforeusefulforen-
an atomic diffraction pattern in momentum space that
gineering quantum memories and specifically to control
can be rephased, engineered to control the coupling to
pulsesoflightinatomicensemblesthatlackaΛscheme.
light and detected via a gradient echo mapping to opti-
Let us now discuss how the above mentioned effects
calfields. Besidestheproposedcoherentpulsesequencer
can be demonstrated using current technologies. Cold
for two-level atom memories, the very high sensitivity
gases ofalkali atomscanbe usedtostore lightefficiently
of diffraction to atomic temperature has implications for
using GEM [20] but due to the short lifetime of their ex-
tunable and non-invasive cold atom thermometry.
cited states, one shall use the Λ-GEM [24]. The phase
gratingcanthenbeinducedbyapairofcircularlypolar- We would like to acknowledge useful discussions with
ized lasers tuned to the transition used to Raman cou- B.C.Buchler,P.K.LamandfinancialsupportfromANR
ple in order to induce a differential light shift to the retour post-doctorants SMEQUI (ANR-13-PDOC-0024-
ground states. With cold atoms as a medium however, 01).
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