Hamiltonian design in readout from room-temperature Raman atomic memory

Hamiltonian design in readout
from room-temperature Raman atomic
memory
Michał D ˛abrowski, Radosław Chrapkiewicz,∗ and Wojciech
Wasilewski
Institute of Experimental Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
∗radekch@fuw.edu.pl
Abstract: We present an experimental demonstration of the Hamiltonian
manipulation in light-atom interface in Raman-type warm rubidium-87
vapor atomic memory. By adjusting the detuning of the driving beam we
varied the relative contributions of the Stokes and anti-Stokes scattering
to the process of four-wave mixing which reads out a spatially multimode
state of atomic memory. We measured the temporal evolution of the readout
ﬁelds and the spatial intensity correlations between write-in and readout as
a function of detuning with the use of an intensiﬁed camera. The correlation
maps enabled us to resolve between the anti-Stokes and the Stokes scat-
tering and to quantify their contributions. Our experimental results agree
quantitatively with a simple, plane-wave theoretical model we provide.
They allow for a simple interpretation of the coaction of the anti-Stokes and
the Stokes scattering at the readout stage. The Stokes contribution yields
additional, adjustable gain at the readout stage, albeit with inevitable extra
noise. Here we provide a simple and useful framework to trace it and the
results can be utilized in the existing atomic memories setups. Furthermore,
the shown Hamiltonian manipulation offers a broad range of atom-light
interfaces readily applicable in current and future quantum protocols with
atomic ensembles.
© 2014 Optical Society of America
OCIS codes: (270.5585) Quantum information and processing; (020.0020) Atomic and molec-
ular physics; (290.5910) Scattering, stimulated Raman; (190.4380) Nonlinear optics, four-wave
mixing.
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(C) 2014 OSA 20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.026076 | OPTICS EXPRESS 26076

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1. Introduction
On-demand retrieval of pre-stored photons can be accomplished using quantum memories [1].
This is an indispensable technique for long distance quantum communication networks [2, 3]
and for the enhanced generation of multi-photon states [4]. Such states find applications in a
linear quantum computing scheme [5] or quantum simulators using linear optics [6,7].
Implementations of quantum memories in room-temperature setups are among the most
auspicious in terms of possible future applications due to their robustness. Until now room-
temperature quantum memories have been realized in solid state systems [8, 9] and in warm
atomic ensembles [1] such as gradient echo memory [10, 11], Raman memory [12, 13] or EIT
memory [14–17].
In this paper we focus on Raman-type atomic memory implemented in warm rubidium-87
vapors [18–20]. In such memory photons are stored in atomic collective excitations called spin-
waves. They are interfaced to photons via off-resonant Raman transitions. Storage times up to
30 ms in a single spin-wave mode was demonstrated [21]. Multimode storage, both temporal
[10] and spatial [11,19], is also feasible. With the use of multiple transverse spin-wave modes
images can be stored and retrieved [22, 23]. The storage time of the multimode memory is
limited by diffusional decoherence and thus can be prolonged by increasing the beam size
[20,23] at the expense of laser intensity.
Here we demonstrate experimentally the theoretical concept of “Hamiltonian design” pro-
posed in [24]. We implement the Hamiltonian design at the readout from Raman atomic mem-
ory. Ideal readout from Raman atomic memory relies on a pure anti-Stokes scattering process
which maps the spin-wave state onto photons state. In real systems additional Stokes scattering
at the readout is virtually inevitable and it is a source of extra noise [13, 25]. By modifying
the interaction Hamiltonian we are able to control the relative contributions of anti-Stokes and
Stokes scattering processes. In the particular setting of our experiment this enables parametric
amplification of the readout, albeit with extra noise.
In this paper we use simple theoretical model describing the coaction between anti-Stokes
and Stokes scattering in readout. Previous approaches to describe theoretically the readout of
spin-wave excitations in Raman scattering were focused only on pure anti-Stokes process. The-
oretical descriptions of readout from Raman-type memory have already been given for a num-
ber of cases, including spatio-temporal evolution with losses [26], temporal eigen-modes [27],
optimized retrieval [28], and spatial modes [29]. However, in those papers emphasis was put
on the pure anti-Stokes scattering. In the present paper we use the extended description of the
readout process [30], taking into account the four-wave mixing that includes the Stokes scatte-
ring.
In the experiment we detect scattered light with spatial resolution and temporal gating. This
enables directly relating the experimental results to the theoretical predictions for temporal

evolution of scattered light. Stokes and anti-Stokes light contributions can be distinguished
through intensity correlation measurements and quantified via careful post-processing.
We provide evidence for a possibility of engineering a wide range of atom-light interfaces
which can be described theoretically as simultaneous readout and parametric amplification.
This may enable enriching purely optical quantum metrology [31] or communication [32]
schemes with quantum storage capabilities.
This paper is organized as follows: In Sec. 2 we introduce a single-mode, theoretical model of
temporal evolution of Stokes and anti-Stokes in readout, Sec. 3 describe experimental details, in
Sec. 4 we present the phenomenological model utilized for data analysis and the results; finally
Sec. 5 concludes the paper.
2. Theory
We begin by synthesizing a simple, plane-wave theoretical model of the readout by four-wave
mixing from a Raman memory to qualitatively support our experimental results. The spin-wave
excitation in the atomic medium can be created by different means. Light can be absorbed
in a two-photon Raman transition to store previously prepared state of light [13, 25]. In our
experiment we induced spontaneous Stokes scattering using write pulse with a wave vector kW
to populate the spin-waves, i.e. collective atomic excitations from the levels |0 to |1 as in
Fig. 1(a). Both approaches results in the spin-wave excitation, however the latter which we use
D
(a) (b)
(c)
write-in readout
Fig. 1. Atomic levels and phase matching in Λ-scheme Raman scattering induced by clas-
sical laser field EW and ER. (a) In the spontaneous write-in process Stokes photons ( â†
WS
- mode) and spin-wave excitation (ˆb† - mode) are created pairwise. (b) Four-wave mixing
in readout consists in simultaneous anti-Stokes and Stokes scattering into modes â†
RA and
â†
RS. (c) Phase matching condition or momentum conservation dictates the wave vectors
of single photons coupled to a spin-wave excitation with a certain wave vector Kb. Write
beam with kW wave vector is scattered as Stokes photon with kWS wave vector, while read
beam with kR wave vector either scatters Stokes or couples anti-Stokes of the respective
wave vectors kRS, kRA.

does not require special efforts to match the photons from the external source to the memory
bandwidth. Ideally, the number of created spin-wave excitations nb with a certain wave vector
Kb equals the number of scattered Stokes photons with wave vector kWS = kW −Kb. We were
able to estimate those numbers in each single iteration of the experiment.
Here we focused on the retrieval stage at which the spin-wave excitations are converted to
photons in four-wave mixing induced by the read laser pulse depicted in Fig.1(b). The read
laser is assumed to be plane-wave with a wave vector kR. Spin-wave with a certain wave vector
Kb are coupled to anti-Stokes and Stokes fields with wave vectors kRA = kR + Kb and kRS =
kR − Kb respectively. In the experiment those weak light fields illuminated distinct pixels of
the camera which was located in the far field. They were also shifted with respect to the initial
Stokes photons with the wave vector kWS = kW −Kb due to different direction of the write
beam kW. We summarize the phase matching condition relating the wave vectors in Fig. 1(c).
To describe the interaction at the readout stage we use bosonic operators of the weak field
light modes âRA and âRS. The spin-wave can be described with the Holstein-Primakhoff approx-
imation by a bosonic annihilation operator ˆb which removes one excitation from spin-wave with
the wave vector Kb [33]. The Hamiltonian describing both the Stokes and anti-Stokes scattering
at the readout stage can be obtained by eliminating the excited level adiabatically [30,34]:
ˆHR = i¯hχ â†
RA
ˆb+i¯hξ â†
RS
ˆb†
+H.c. (1)
where χ and ξ are the coupling coefficients for anti-Stokes and Stokes Raman transitions [35].
The coupling coefficients squared χ2 or ξ2 can be calculated [36] as ΓdΩ2
p/∆2 where d is
optical depth at resonance, Ωp is the pump Raman frequency and ∆ is the detuning of the pump
from transition from level 1 or 0 to excited state for anti-Stokes coupling χ2 or ξ2respectively.
Such a Hamiltonian (Eq. (1)) has been used in several papers describing experiments on room-
temperature atomic vapors, i.e. teleportation between distant atomic objects [34].
The above Hamiltonian leads to coupled Maxwell-Bloch equations, which can be integrated
as e.g. in [30,33]. This yields input-output relations linking the operators of the incoming light
fields and the initial spin-wave state to the outgoing fields and the final spin-wave state.
We used the input-output relations to compute the quantities observed in the experiments, that
is the mean number of scattered photons as a function of time in anti-Stokes â†
RA(t) âRA(t) and
Stokes fields â†
RS(t) âRS(t) respectively. With the assumption that the incoming light fields are
in a vacuum state and the mean number of initial spin-wave excitations equals nb = ˆb†(0)ˆb(0) ,
the result is:
â†
RA(t) âRA(t) = χ2
et(ξ2−χ2
)
GRA(t)
nb +
χ2ξ2
ξ2 − χ2
(et(ξ2−χ2
) −1)
SRA(t)
, (2)
â†
RS(t) âRS(t) = ξ2
et(ξ2−χ2
)
GRS(t)
nb +
ξ2
ξ2 − χ2
(ξ2
et(ξ2−χ2
) − χ2
)
SRS(t)
, (3)
The underbraced terms GRA(t) and GRS(t) represent the readout gains, that is the number
of scattered photons per each input spin-wave excitation. In turn, the terms SRA(t) and SRS(t)
represent the spontaneous scattering into anti-Stokes and Stokes quantum fields, which is inde-
pendent of the initial state of quantum fields. Just as expected, they appear only in the presence
of Stokes scattering ξ = 0. Note that both coupling coefficients χ and ξ appear in both formulas
(2) and (3), confirming that anti-Stokes and Stokes scattering cannot be treated separately and
should be viewed together as a full four-wave mixing process.
Note that the above results were derived neglecting all decoherence. In the experiment, the
most important sources of decoherence are diffusion of atoms and spontaneous emission from

the excited state in random direction. Both can be neglected provided the optical depth is large
and the duration of the interaction short.
Let us proceed to examining the evolution of scattered anti-Stokes and Stokes light in a few
typical cases.
For ξ 0 there is virtually no Stokes scattering while the anti-Stokes scattering appears
instantaneously and decays exponentially as depicted in Fig. 2(a). There is virtually no spon-
taneous noise SRA 0 and the time integrated gain reaches unity ¯GRA =
´
GRAdt = 1. In the
ideally case of ξ = 0 this leads to the readout Hamiltonian of form ˆHR ∼ â†
RA
ˆb + H.c. which
represent purely anti-Stokes scattering process.
In real systems Stokes interaction is typically unavoidable. The case where χ > ξ > 0 is
depicted in Fig. 2(b). Here the readout appears in both the anti-Stokes and Stokes fields with
comparable intensity and decays over time. Time-integrated gain may reach over unity ¯Gi =´
Gidt > 1, yet with spontaneous noise that slowly builds up Si > 0, for i = RA, RS. This is the
setting utilized in [37].
A situation where Stokes interaction dominates, ξ > χ, is depicted in Fig. 2(c). In this case
readout goes predominantly to Stokes field with exponentially increasing gain. Yet, the noise
intensity virtually equals the gain Gi Si for both fields. In the limit of ξ χ we can put χ = 0
which leads to the readout Hamiltonian ˆHR ∼ â†
RS
ˆb† + H.c. consists only the contribution of
Stokes scattering process.
It is instructive to integrate gains GRA(t) and GRS(t), and noises SRA(t) and SRS(t) over the
time of interaction and inspect them as functions of coupling coefficients. Coupling coefficients
χ and ξ are inversely proportional to the detuning of the read laser from the excited level
0.0 0.2 0.4 0.6 0.8
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Δ
ξ=0.2,χ =0.8
(a) (b)
(c) (d)
0 2 4 6 8 10
0.01
0.02
0.05
0.10
0.20
0.50
ξ=0.55,χ =0.45ξ=0.99,χ =0.01
0 2 4 6 8 10
10- 8
10- 6
10- 4
0.01
1
t t
0 2 4 6 8 10
0.01
0.1
1
10
100
t R
GRA - anti-Stokes gain GRS - Stokes gain
SRA - anti-Stokes noise SRS - Stokes noise
timeintegratedgain/noise
gain/noisegain/noise
gain/noise
Fig. 2. (a) to (c) - temporal evolution of gains Gi and spontaneous noises Si building up the
light fields. Data plotted for different values of coupling coefficients χ and ξ correspond to
detuning parameters ∆R= 0.3, 0.6 and 0.8. (d) Integrated readout gains ¯Gi and spontaneous
noises ¯Si versus detuning ∆R.

∆R [33,35]. One coupling coefficient can be increased at the expense of the other by tuning the
frequency of the read laser when it is in between the |0 ↔ |e and |1 ↔ |e resonances as in
Fig. 1(b).
In Fig. 2 (d) we plot the time integrated gains ¯Gi =
´
Gidt and noises ¯Si =
´
Sidt as a function
of the detuning ∆R from |0 ↔ |e transition in the units of ground state splitting. We assumed
χ ∝ 1/∆R and ξ ∝ 1/(1−∆R). Different characters of anti-Stokes and Stokes interaction causes
asymmetry of the plot in Fig. 2 (d). The integrated readout gain in the anti-Stokes domination
regime χ ξ remains equal to unity ¯GRA = 1 and the noise is suppressed ¯SRA 1 . These are
the conditions for perfect unamplified readout.
On the contrary, in the Stokes domination regime χ ξ the integrated gain ¯GRS varies almost
exponentially with the Stokes coupling ξ entailing elevated noise ¯SRS. In this domain four-
wave mixing enhances the anti-Stokes emission and ¯GRA rises although the relative contribution
¯GRA/ ¯GRS diminishes. Note that this is the regime we use to populate the spin-waves by inducing
spontaneous Stokes scattering with write pulse.
Finally, let us consider a single realization of the four-wave mixing process. Assume that the
number of initial spin-wave excitations nb has been measured by counting the Stokes photons
nWS scattered during prior write-in, ideally nWS = nb. In this particular iteration the number of
photons in the anti-Stokes and the Stokes field equals
ni = ¯Ginb + ˇSi, i = RS, RA (4)
where ˇSi is the noise treated here as an independent random variable. ˇSi is the time integrated
spontaneous scattering and its mean is calculated based on Eqs. (2), (3) ˇSi = ¯Si. Later on we
refer to the correlation of the number of anti-Stokes nRA or Stokes photons nRS with the initial
spin-wave excitation nb. The above formula links the correlated part to the integrated readout
gain ¯Gi and the uncorrelated noise to the integrated spontaneous contribution ¯Si.
3. Experiment
We implemented Raman-type atomic memory in rubidium-87 vapors. We applied spontaneous
write-in process into memory similarly as in previous works [18,38]. The spontaneous write-in
process had been proven to efficiently create the spin-wave excitations, thus we could focus
on the retrieval characteristics. Here we extended the setup and measurements schemes applied
and explained in detail in our previous papers [19,20].
We used a 10 cm glass cell containing pure 87Rb isotope with krypton as a buffer gas under
pressure of 1 torr. The cell was heated by bifilar windings to 90oC equivalent to the optical
read
read
Stokes
6,835GHz
′
′
anti-Stokes
write
Stokes
pump
(D1-line,
(795nm)
Fig. 3. Configuration schemes of 87Rb hyperfine energy levels at D1 line (795nm) using
write-in and readout respectively. Laser beams and scattered light detunings in both pro-
cesses are also depicted.

density of 135. The cell was magnetically shielded to avoid decoherence produced by stray
magnetic field and the main source of decoherence was due to diffusion [20].
We used three external cavity diode lasers: pump, write and read laser, operating at D1 line of
87Rb (795 nm). The frequencies of all laser beams and the scattered light are sketched against
the 87Rb level scheme in Fig. 3. The pump laser was resonant to F=2 →F’=2 transition, while
the write and read laser were detuned to the red from transition F=1→F=1’. The detuning of the
write laser was ∆W = 1.77 GHz and it was kept constant throughout all measurements while
the detuning of the read laser ∆R varied. The pump and write lasers were frequency locked
using the DAVLL setup [39]. The exact values of the write and read lasers detunings were
set repeatedly and measured precisely using Doppler-free saturated absorption spectroscopy
inside an auxiliary rubidium cell and their reference beat-note signal was measured on a fast
photodiode. Inside the cell write, read and pump lasers had the power of 6.8 mW, 4.5 mW and
75 mW respectively. During the experimental sequence the lasers power fluctuated at a level of
5% and the frequencies were changing by no more than 50MHz.
The simplified schematic of the setup is shown in Fig. 4(a). All of the three horizontally
polarized beams overlapped inside the cell where their diameters were 6 mm for the pump
beam and 4 mm for both write and read beams. The write and read beams were tilted at the
angle θ = 2 mrad. Acousto-optic modulators were used to shape rectangular pulses with rising
time of 1 µs from laser beams. The pulse sequence applied in the experiment is depicted in
Fig. 4(b). The sequence was initiated by optical pumping of rubidium atoms into the ground
state of 87Rb S1/2 F=1. The 700 µs long rectangular pulse of 75 mW power yielded a pumping
efficiency of 98%. Then we applied a 10 µs long rectangular write pulse to create spin-wave
excitations between F=1 and F=2 levels together with the Stokes scattering. The rectangular
read pulse started right after the end of the write pulse and its duration was set to 40 µs. The
read pulse generated both anti-Stokes and Stokes scattering. The total time duration of write and
read pulse corresponds to the average atomic displacement of c.a. 0.5 mm due to diffusion [20]
which was small enough to keep the atoms in the range of the pump beam size.
We separated the horizontally polarized laser beams from the vertically polarized scattered
pump write read
WS
RS+RA
read
intensifier
gate
Pump
Write Read
Rb cell87
intensified
camera
85
Rb filter
(a)
(b) (c)
RS+RA
Fig. 4. (a) Experimental setup: write and read laser beams propagate forward to the inten-
sified camera and the pump beam propagates backward. 87Rb - atomic memory cell, 85Rb
- absorption filter, blue arrows correspond to scattered light. (b) Pulse sequence with long
intensifier gate covers the whole write and a part of the read pulse (both rectangular). Ex-
ponential shapes in front of write and read pulses are typical time-resolved intensities of
the scattered light observed on the intensified camera, WS and RS stand for Stokes while
RA for anti-Stokes scattering respectively. (c) Scheme of the measurement of the temporal
evolution of the readout light using short gate duration τ = 250 ns.

(a)
−0.5 −0.25 0 0.25 0.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−0.5 −0.25 0 0.25 0.5
X (mrad)
Y(mrad)
(b)
X (mrad)
(c) (d)
X (mrad)
0−0.35 0.35
X (mrad)
0−0.35 0.35
Fig. 5. Representative images of the retrieved field, write-in and read-out in upper and lower
parts respectively. (a) Intensity map in a single shot obtained using a long gate in the linear
regime of camera operation and (b) the average over 104 frames. (c) Photon positions in a
single shot [40] obtained using a short gate positioned in the readout stage and (d) the total
number of photons per sCMOS pixel summed over 2000 frames.
fields by polarization and spectral filtering, which yielded the total attenuation factor for laser
beams that exceeded 109. For spectral filtering we applied a 85Rb absorption filter at 130◦C
placed in magnetic field which increase the absorption by broadening and shifting the 85Rb
spectral lines. The attenuation of laser beams inside the 30 cm length absorption filter was at
least 104. We also measured the transmission of the filter at frequencies corresponding to the
scattered light generated at write-in and readout stages. They were found to be 12% for Stokes
scattering in write-in and 76% for all applied frequencies of anti-Stokes and Stokes in readout.
In our system we generated and retrieved spatially multimode light as described in detail
in [19, 33]. This light was detected in the far field by a gated image intensifier coupled to
sCMOS camera [40]. We used the intensified camera in two different operational schemes.
In the first scheme we applied a long gate pulse which covered both the write and the read
pulse as in Fig. 4(b). We set the gain of the intensifier to a low value. Then the response of
the camera system was proportional to the intensity of scattered light. We calibrated the excess
noise contributed by the image intensifier and made sure that it was insignificant as compared to
the shot-to-shot intensity fluctuations of scattered light. In the second scheme we used a short,
delayed gate of τ = 250 ns duration as depicted in Fig. 4(c). Here we set the image intensifier
to high gain and the camera system was sensitive to single photons [40]. We also utilized an
intensified camera as a photon number resolving detector [41] and we counted all photons with
a quantum efficiency of 20% in the region of readout scattering. Representative images: single
shots and averaged intensities obtained typically in those regimes are depicted in Fig. 5. Note
that in Fig. 5(a) the number and localization of speckles are random so they average to the
smooth intensity profile as shown in Fig. 5(b). Both measurement regimes are phase insensitive
although spatially resolved, homodyne-type detection was also reported [37] and then it could
be used to directly measure phase noise and squeezing properties of the generated light.
4. Results
4.1. Temporal evolution of readout intensity
At first we measured the temporal evolution of the mean number of photons emitted during
readout stage. For this purpose we counted the scattered photons using an intensified camera
with a gate duration of τ = 250 ns. We collected all photons in the specified circular camera
region around the point corresponding to the center of the read beam i.e. wave vector kR. Thus,

0 5 10 15 20 25 30 35 40
0
10
20
30
40
50
gateRdelay δTRwithRrespectRtoRtheRreadRpulseRstartR6µs7
numberRofRphotonRcountsRperR250ns
∆R
=4.19GHz
∆R
=3.84GHz
∆R
=3.52GHz
∆R
=1.67GHz
Fig. 6. Average number of photon counts detected at a large solid angle around read beam
per a gate duration of 250 ns for different readout laser detunings ∆R from 87Rb F=2 →
F’=2 resonance. The curves represent exponential fits to the first and last data points. The
size of errorbars is comparable with marker size.
we measured the anti-Stokes and the Stokes scattering together, both their spontaneous and
stimulated parts. In Fig. 6 we present the results of measurement for mean number of photons
versus gate delay with exponential fits. Each data point was obtained by averaging 2000 frames
in the area of angular diameter 0.5 mrad and the time separation between the data points was
1 µs. The sequence was repeated for different detunings ∆R of the read beam from F=2→F’=2
resonance.
The results plotted in Fig. 6 vividly depict a transition from decay to growth of the readout
scattered light intensity. For a small detuning ∆R = 1.67 GHz the read beam is close to the
F=2→F’=2 resonance and we expected the anti-Stokes coupling to dominate. Indeed, we ob-
served an exponential decay as predictied. At the opposite extreme ∆R = 4.19 GHz, in turn,
the read beam is much closer to the F=1→F’=1 transition and here it was the Stokes field that
we expected to dominate. Again, the total number of scattered photons increases as expected,
compare Fig. 2(c). Nonetheless, in the experiment the growth starts after the first 4 µs of the
evolution during which the number of scattered photons slightly decreases. In the intermediate
cases we observed initial decay followed by growth. These cases might be associated with the
case presented in Fig. 2(b), where stimulated field contributions decay while the spontaneous
contributions rise. Moreover, in the observed evolution the spin-wave decoherence certainly
plays an important role, which, however, is not accounted for by our simplified model.
In Fig. 7 we plot decay and growth coefficients as functions of detuning ∆R of the read beam
from F=2→F’=2 resonance. The coefficients were calculated by fitting exponential decay or
growth to the first or the last 10 µs, as depicted in Fig. 6. Fitting entire 40 µs with a single
exponential fails because the significant influence of decoherence and multimode character of
scattering on this time scale is not capured by our simple thoeretical model. The dependence
of the growth and decay coefficients on the detuning ∆R agree qualitatively with predictions of
the theoretical model as they rise along with ∆R. However, there is a discontinuity between the
growth and decay coefficients at ∆R = 3.8 GHz that may be attributed to spin-wave decoherence
influence.
4.2. Spatial resolving of anti-Stokes and Stokes scattering
The analysis of intensity correlations between light scattered in various directions provides
valuable insight into the four-wave mixing at the readout. To set the stage we shall first recall
the relations between scattered photons directions. The scattering generated during the write
pulse is emitted around the write beam and falls upon a circular region of angular diameter

growth
decay
0.5 1 1.5 2 2.5 3 3.5 4 4.5
−140
−120
−100
−80
−60
−40
−20
0
20
40
60
readout’laser’detuning ∆
R
from
87
Rb’F=2→ F’=2’yGHz/
Decay/growth’coefficientγykHz/
Fig. 7. Exponential decay and growth coefficients for scattered light found by fitting data as
in Fig. 6 versus read laser detuning ∆R. For triangle points the size of errorbars are smaller
than the marker size. Inset: 87Rb D1 line absorption spectrum with read and write laser
frequencies marked.
0.5 mrad in the upper part of the camera. Similarly, the readout light illuminates the lower part
of the camera of angular diameter 0.5 mrad placed in the far field. Both regions are contained
in a small central part of the camera sensor. The angle between the write and read beams equal
to θ = 2 mrad inside the 87Rb cell is precisely the distance between the centers of both circular
regions on the camera. Due to phase matching illustrated in the Fig. 1(c) the scattered fields
coupled to the same spin-wave with a wave vector Kb propagate in different directions. They
are detected in the far field on three different pixels of the camera.
For a Kb oriented rightward a pixel located left of the center in the write region gathers
Stokes-coupled light of an intensity proportional to the number of photons nWS. Another pair
of pixels, right and left of the center in the read region, gathers anti-Stokes and Stokes cou-
pled light of intensities proportional to nRA and nRS respectively. Note that the latter pair of
pixels also gathers light coupled to the spin-wave with an opposite wave vector −Kb, and their
role is reversed. That is, the left and the right pixels gather anti-Stokes and Stokes scattering
respectively. These contributions add extra noise when correlated with Stokes scattering nWS
generated in the write-in process.
In order to perform correlation measurements the camera intensifier was operated in linear
regime and long gate pulses were used. We detected the light scattered during both the write-in
and the readout processes. In each iteration of the experiment we registered entire, randomly
varying camera images. The signal Ii at the i−th pixel of the camera was proportional to the
intensity of the incoming light. We calculated the correlation coefficient between intensities of
light registered by any two pixels, i−th and j−th, of the camera:
Cij = corr(Ii,Ij) =
IiIj − Ii Ij
(∆Ii)2 (∆Ij)2
. (5)
In Fig. 8(a) we present the maps of correlation coefficients Cij as in Eq. (5) between one
pixel marked by a cross referred as i = WS and all other pixels j in the image. The pixel
i = WS corresponds to the emission of Stokes field coupled to the spin-wave with the wave
vector |Kb| = 45.8 cm−1. This value was set, so as to spatially resolve anti-Stokes and Stokes
correlation regions. Each correlation map was calculated basing on 105 single-shot images. We
limited our calculations to the circles marked in the maps. They correspond to the regions where
the scattering was registered, centered around the directions of the write and read beams. In the

(a)
−0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−0.5 −0.25 0 0.25 0.5
Xofmradb
Yofmradb
∆R
=1.17GHz ∆
R
=2.75GHz ∆R
=3.28GHz
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CWScRA
C
WScRS
(b)
Xofmradb Xofmradb
RS
RA
WS
RS
RA
WS
RS
RA
WS
1 1.5 2 2.5 3 3.5
30
40
50
60
70
80
readoutolaserodetuning ∆R
from 87
RboF=2→ F’=2ofGHzb
correlationocoefficientoC
WScRA
coC
WScRS
fnb
Fig. 8. (a) Spatial correlation maps for different read laser detunings from 87Rb F=2 →
F’=2 resonance. Red and green circles mark the respective areas where photons scattered
at the write and read stages fell. (b) Correlation coefficients at the peaks corresponding to
anti-Stokes CWS,RA and Stokes CWS,RS versus detuning ∆R with (quadratic) trend.
upper red circles we registered the Stokes scattering emitted in the write-in process whereas in
the bottom green circles light from readout was observed.
In each map in Fig. 8(a) three spots appear. The maxima of the spots marked by the crosses
on pixels i = WS, RA, RS correspond to the emission of scattered fields coupled to the same
spin-wave with the wave vector Kb and their positions agree with the phase matching conditions
illustrated in Fig. 1(c).
As we discussed in conclusion of Sec. 2, the correlation between the write scattered light and
two other fields scattered at the readout stage is mediated by the spin-wave, which caries ran-
dom number of excitations nb in each iteration of the experiment. This number is proportional
to the signal on the write pixel IWS and contributes to the signals IRA and IRS. The signals on
pixels far away are correlated to the excitations of other spin-waves, which vary independently.
We verified that correlation maps displayed characteristic three-spot pattern for arbitrarily cho-
sen reference points except when two spots in the readout merge. This guarantees that, indeed,
we resolve the Stokes and anti-Stokes scattering in the readout. Correlation measurements are a
practical alternative to spectral filtering in other experiments with quantum memories [37,42].
Now we can proceed to examine how the anti-Stokes and Stokes scattering contributions
change with the read laser detuning ∆R. In Fig. 8(a) we depicted three correlation maps for dif-
ferent read laser detunings ∆R. Transition from the anti-Stokes to Stokes scattering domination
can be achieved by adjusting of the detuning, here from ∆R = 1.17 GHz to ∆R = 3.28 GHz. To
quantify the effect, we calculated the correlation between the Stokes scattering in write-in and
the anti-Stokes scattering — CWS,RA or the Stokes scattering in readout — CWS,RS. The values
of CWS,RA, CWS,RS were taken as the marked maxima of the respective anti-Stokes/Stokes re-
gions for various detunings. Fig. 8(b) summarizes the results and illustrates the transition.

4.3. Effective gains of anti-Stokes and Stokes readout
The measured correlation values together with mean intensities and their variances can be used
to extract information about time integrated readout gains for the anti-Stokes ¯GRA and Stokes
¯GRS separately. This is accomplished by tracing the origins of signal observed at respective
pixels.
The signal IWS registered at pixel i = WS was calibrated to the number of photons. It consists
of the part proportional to the number of photons nWS generated during the write-in to the spin-
wave of the wave vector Kb and the camera system read noise fWS:
IWS = tWSnWS + fWS, (6)
where tWS is the transmission of the write-in Stokes photons through the filter system. The read
noise fWS is random and its variance depends only on the intensity on the pixel (∆fi)2 ∼ I2
i .
We made the calibration of the noise prior to the measurements, which allowed us to exactly
determine the variance of the read noise at each pixel.
The signal in the readout at pixel i = RA can be decomposed into the following contributions:
IRA = tRAnRA +tRSnRS + fRA, (7)
where nRA is the number of photons generated in the anti-Stokes readout from the spin wave
of the wave vector Kb and nRS is the number of photons generated in the Stokes readout from
the other spin wave with the opposite wave vector −Kb, illuminating the same pixel. tRA and
tRS are respectively the transmissions of the anti-Stokes and the Stokes photons and fRA is the
camera system read noise at the pixel i = RA.
We can write an analogous formula for the components of the signal registered at the pixel
i = RS:
IRS = tRSnRS +tRAnRA + fRS. (8)
In the actual experiment the number of created spin-wave excitations nb is smaller than the
number of scattered photons nWS due to decoherence at the write-in stage nb = ηWnWS. That
factor ηW is write-in efficiency. In turn, the decoherence at the readout stage leads to a limited
efficiency ηR during this process. We include those efficiencies in the formula Eq. (4), which
yields the phenomenological formulas for the number of retrieved photons:
nRA = ηWηR
¯GRAnWS + ˇSRA, (9)
nRS = ηWηR
¯GRSnWS + ˇSRS, (10)
where ¯GRAand ¯GRS are the time integrated gains and ˇSRAand ˇSRS is the spontaneous emis-
sion generated at the readout introduced in the theoretical model. The factors ηWηR
¯GRA and
ηWηR
¯GRS are the effective gains of the anti-Stokes and the Stokes readouts in the experiment.
Eqs. (7)-(10) and simple observations that corr(nWS,ηWηR
¯GRAnWS) = 1 and corr(nWS, ˇSi) =
corr(nWS,ni) = corr(nWS, fi) = 0 allow us to calculate the effective gains from the measured
quantities:
ηWηR
¯GRA =
tWS
tRA
CWS,RA (∆IWS)2 (∆IRA)2
(∆IWS)2 − (∆fWS)2
, (11)
ηWηR
¯GRS =
tWS
tRS
CWS,RS (∆IWS)2 (∆IRS)2
(∆IWS)2 − (∆fWS)2
. (12)
The above formulas can be verified by evaluating the numerators of the right hand sides.
In Fig. 9 we present the results of the effective gains of the anti-Stokes scattering ηWηR
¯GRA
and the Stokes scattering ηWηR
¯GRS for different read laser detunings ∆R. The efficiency due to

1 1.5 2 2.5 3
10
15
20
25
30
35
40
45
50
55
readoutGlaserGdetuning ∆
R
from
87
RbGF=2→ F’=2GSGHz)
effectiveGgainGSv)
η
W
η
R
G
RA
ηW
ηR
GRS
Fig. 9. The effective gains of the anti-Stokes scattering ηWηR
¯GRA and the Stokes scattering
ηWηR
¯GRS. For the smallest detuning ∆R absorption in rubidium cell starts to contribute.
decoherence in the write-in ηW is fixed. The efficiency of the readout ηR is mostly determined
by the decoherece during this process, thus it also weakly depends on the detuning ∆R. Thus
we expect that the experimentally determined effective gains vary mostly due to the change of
the time integrated gains ¯Gi and can be compared with the theoretical predictions.
Our results show that in our configuration the Stokes scattering during readout is an impor-
tant contribution to the scattered light. The highest observed efficiency of the anti-Stokes re-
trieval was ηWηR
¯GRA =22% for ∆R = 1.37 GHz whereas the efficiency of the Stokes scattering
ηWηR
¯GRS grew with ∆R. Note that the observed dependence of the anti-Stokes and Stokes con-
tributions on ∆R agrees qualitatively with our simple theoretical model for the integrated gains
¯Gi presented in Fig. 2(d). Note that for large detunings the anti-Stokes scattering would decay
if there was no Stokes contribution. We suspect that the efficiencies values ηWηR
¯GRA > 10%
for the anti-Stokes for ∆R > 2 GHz originated from the enhancement due to the full four-
wave mixing process. The switch from domination of anti-Stokes to Stokes scattering occurs at
∆R 1.5 GHz.
In our setup we were able to manipulate the gate duration and thus check how the values of
the effective gains depend on the time of interaction. We measured the effective gains for two
different gate durations covering 10 µs write pulse and a part of 40 µs read pulse selected by a
gate duration τ = 25 µs or τ = 50 µs, as we depicted in Fig. 4(b).
We measured the effective gains for two different detunings ∆R and for the two values of
the gate duration τ. We checked that the anti-Stokes effective gain ηWηR
¯GRA = 12% remained
independent of the interaction time which indicates that light that has been correlated with the
stored spin-waves is always retrieved in the beginning stage of the readout. On the other hand,
the efficiency of the Stokes scattering varies, as we summarized in Table 1. As we can see,
longer interaction time increases the effective gain ηWηR
¯GRS and in particular we can obtain
more correlated light than we did at the write-in stage ηWηR
¯GRS > 100%. This corresponds
to the case where the integrated gain ¯GRS significantly exceeded unity and overcompensated
Table 1. The effective gains of the Stokes scattering ηWηR
¯GRS for different gate durations.
∆R
ηWηR
¯GRS
τ = 25 µs τ = 50 µs
2.0GHz 18% 31%
3.8GHz 93% 122%

for the losses, albeit bringing a large amount of accompanying noise ¯SRS. Such a high gain of
the obtained signal can find an application in various operations with quantum memories. For
instance the dominant Stokes scattering process can be utilized for testing if the memory is
empty. No photons retrieved from the memory at the readout may suggest that memory was
initially empty with a certain probability. This probability is much higher for amplified readout
with Stokes interaction dominant as opposed to conventional pure anti-Stokes readout process
of limited efficiency.
5. Conclusions
In conclusion, we have presented an experimental demonstration of manipulation of the Hamil-
tonian in the readout from Raman-type atomic memory. We measured the temporal evolution of
the readout light and the spatial correlations between the Raman scattering in the write-in and
the readout. Our measurements confirm the adjustability of coupling parameters corresponding
to the anti-Stokes and Stokes scattering. The results match our simple theoretical model of a
full four-wave mixing process. We resolved the anti-Stokes and the Stokes scattering contribu-
tions to the readout thanks to the phase matching in the atomic vapor which dictated directional
correlations with the Stokes photons during write-in.
Our results provide a very simple framework for interpretation of extra noise in experiments
on storing light in atomic vapor. When anti-Stokes scattering is used to map the spin-wave states
onto the states of light, the accompanying Stokes scattering creates unwanted random photons
and atomic excitations. Our results show that, though inevitable, this contribution can be esti-
mated by our model and perhaps suppressed by adjusting the coupling light frequency to the
other side of the atomic resonance. There is also an optimal duration for the anti-Stokes interac-
tion. Beyond the optimum, the spontaneous noise contribution increases. It may be favorable to
switch to noncollinear configuration where control and quantum fields enter the atomic medium
at a small angle. Then the Stokes scattering photons will become directionally distinguishable
from the anti-Stokes, which may lower the noise in some experiments.
The design of the Hamiltonian we demonstrated can be implemented in many types of quan-
tum memories at little or no extra cost. The amplification of readout signal by Stokes scattering
may be very useful in some applications especially if extra noise is not crucial. This is the case
when we use detectors of small quantum efficiency or we are focused on other properties of
retrieved light e.g. in retrieval of stored images [23, 43]. For instance the amplification in the
readout can be utilized as a robust single-shot projective test to see whether the atomic memory
is in the ground (empty) state. A complete absence of Stokes signal on an inefficient detector
is a relatively rare occurrence if the extra Stokes gain overcompensates for the losses at the
detection stage. Notably then, complete absence of signal ensures us that the memory was in
the ground state.
Our results can be useful for suppressing unconditional noise floor in readout of Raman-type
quantum memories at the single photon level [13, 25, 44]. We provide evidence for a possibil-
ity of engineering a wide range of atom-light interfaces which can be described theoretically
as simultaneous readout and parametric amplification e.g. quantum non-demolition interaction
while coupling coefficients are equal. The facility to continuously tune the Hamiltonian co-
efficients sets the scene for developing new quantum protocols in room-temperature atomic
memories.
Acknowledgments
We acknowledge J. Nunn for the insightful discussion and C. Radzewicz and K. Banaszek for
generous support. The project was financed by the National Science Centre projects no. DEC-
2011/03/D/ST2/01941 and DEC-2013/09/N/ST2/02229.

Hamiltonian design in readout from room-temperature Raman atomic memory

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Hamiltonian design in readout from room-temperature Raman atomic memory