Two-photon dynamics in coherent Rydberg atomic ensemble

0.46 MB·

by  Bing He

Two-photon dynamics in coherent Rydberg atomic ensemble
Bing He,1,2 A. V. Sharypov,3,4 Jiteng Sheng,2 Christoph Simon,1 and Min Xiao2,5,∗
1Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada
2Department of Physics, University of Arkansas, Fayetteville, AR 72701, USA
3Kirensky Institute of Physics, 50 Akademgorodok, Krasnoyarsk, 660036, Russia
4Siberian Federal University, 79 Svobodny Av., Krasnoyarsk, 660041, Russia
5National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China
We study the interaction of two photons in a Rydberg atomic ensemble under the condition of
electromagneticallyinducedtransparency,combiningasemi-classicalapproachforpulsepropagation
andacomplete quantumtreatment for quantumstate evolution. Wefindthat theblockaderegime
4 isnotsuitableforimplementingphoton-photoncross-phasemodulationduetopulseabsorptionand
1 dispersion. However,approximatelyidealcross-phasemodulationcanberealizedbasedonrelatively
0 weak interactions, with counter-propagating and transversely separated pulses.
2

r
a Strong nonlinearity at the single-photon level is desir- a complete solution of the problem rather challenging.
M abletotherealizationofall-opticalquantumdevices. En- With the combination of a semi-classical approach for
sembles of highly excited Rydberg atoms under electro- pulsepropagationandacompletequantumapproachfor

2
magneticallyinducedtransparency(EIT)conditioncom- pulsequantumstateevolution,wefindarealisticpicture
1
binetheadvantagesofstrongatom-fieldcouplingwithout for the dynamical process by showing the concerned fig-

] significant absorption and non-local atomic interaction, ures of merit. We show that our proposed setup outper-
h and haveattractedintensive experimental[1–7] andthe- formsthepreviouslyconsideredRydbergblockaderegime
p
oretical studies [8–17] recently. The strong correlation [20]clearlyintermsofmuchlowerphotonabsorptionand
-
t directly between single photons inside Rydberg atomic negligible group velocity dispersion.
n
ensemblewasobserved[7],andtheformationofaWigner The detailed two-photon XPM via Rydberg EIT is
a
u crystalofindividualphotonsisalsopredicted[15]. When as follows. One respectively couples the far-away input
q such interaction is applied to implement the cross-phase photons to cold Rydberg atoms under the EIT condi-
[
modulation(XPM)betweentwoindividualphotonswith tion to form the light-matter quasi-particle called dark-

2 a non-zero relative velocity as in Fig. 1 [18–21], a main state polariton (DSP) [22]. The spatial distribution of
v difference from a single probe beam propagation in Ry- the pulses necessitates a quantum many-body descrip-
0 dberg EIT medium [1–17] is that no steady state exists tion of the process. The prepared DSPs are in the
4 for the pulses, because their interaction varies with the state 1 = d3xf (x)Ψˆ†(x)0 for l = 1,2, where
15 relative distance, pulse velocity that changes pulse sizes, fl(x) a|riel theirRsnapslhots lwith| i d3xfl(x)2 = 1, and
| |
. as well as the absorption in medium. The realistic time- Ψˆ(x)=cosθˆ(x) sinθSˆ(x)asthResuperpositionofelec-
1 dependence in the inherent nonlinear dynamics makes tromagneticEfield−operator ˆ(x) and Rydberg spin-wave
0
4 fieldoperatorSˆ(x)istheDSEPfieldoperator. Themany-
1 body version of the atom-field Hamiltonian
: ∆
v a (cid:1) ∆ |r> 1
Xi Ω (cid:2) HAlF = −2Z d3x(cid:8)ΩcSˆl†(x)Pˆl(x)+g√NEˆl†(x)Pˆl(x)
r (a) (cid:4)
a ∆ |e> + H.c. +Z d3x∆1Pˆl†(x)Pˆl(x)
(cid:3) (cid:9)
γ
g = d3x ω+Φˆ† Φˆ (x)+ω−Φˆ† Φˆ (x) (1)
−Z +,l +,l −,l −,l
|g> (cid:8) (cid:9)
(b) (c)
(~ = 1) also involving the polarization field Pˆ(x)
FIG.1: (coloronline)(a)Twosinglephotonpulsespropagate for the excited level e can be diagonalized in terms
in a Rydberg atomic ensemble. (b) Two pulses propagate in of two bright-state p|oilariton (BSP) fields Φˆ (x) =
+
two parallel waveguides filled with Rydberg atoms. For the sinθsinφˆ(x)+cosφPˆ(x)+cosθsinφSˆ(x)andΦˆ (x)=
−
insignificant diffraction or the propagation in (b), the pulse sinθcosφEˆ(x) sinφPˆ(x)+cosθcosφSˆ(x), where their
profilesfor thenumericalsimulations inFigs. 3and 4canbe E −
spectrum ω± = 1(∆ ∆2+g2N +Ω2) is a function
approximatedasone-dimensional. (c)Atomiclevelschemefor 2 1± 1 c
the system. Without pulse interaction there is ∆1+∆2 =0 of the input photon detupning ∆1, pump beam Rabi fre-
under the EIT condition. Here ∆1 = ωeg −ωp, and ∆2 = quency Ωc and atom density N. The combination co-
ω −ω (ω is the input pulses’ central frequency, and ω efficients for the polariton field operators satisfy the re-
re c p c
is the frequency of the pump beam). The Rydberg level is lations tanθ = g√N/Ω and tan2φ = g2N +Ω2/∆
c c 1
shifted by ∆R dueto theinteraction with another pulse. with g as the atom-field coupling constpant. When the
2
DSPs get close to each other, the interaction n-1 vg(cid:144)c
0.4 HaL 2.5´10-7 HbL
HI = Z d3xd3x′Sˆ1†(x)Sˆ2†(x′)∆(x−x′)Sˆ2(x′)Sˆ1(x)(2) 0.2 DR-=-D91Γ(cid:144)Γ 1.25.´´1100--77
-5 -4 -3 -2 -1 1
betweenthepulsestakeseffect. HereweconsidertheVan 1.´10-7
derWaals(VdW)potential∆(x)= C6/x6inRydberg DR=-0.5Γ DR=0 -0.2 -DR(cid:144)Γ
atomic ensemble. Such interaction,−howev|e|r, also causes -0.4 2 4 6 8 10 12
the transition of DSP to BSPs containing Pˆ(x) compo-
l FIG. 2: (color online) (a) Shift of refractive curves with in-
nentsdecayingattherateγ. ThedecayofthePˆ(x)field
l creasing ∆ < 0. Here n−1 = 1/2 Re{χ(1)} [25] is ob-
R
is described by [23]
tained from the approximated solution to (4a)-(4b) in slow
lightregime,where∆ changesslightlyduringthedecaytime
R
HDl = i√γZ d3x Pˆl(x)ξˆl†(x,t)−Pˆl†(x)ξˆl(x,t) ,(3) in the order of 1/γ. We take the photon detuning ∆1 = 2γ
(cid:8) (cid:9) and the pump detuning ∆2 = −2γ under the EIT condi-
withthewhite-noiseoperatorsofthereservoirssatisfying tion with ∆R = 0. The pulses’ initial group velocity under
[ξˆ(x,t),ξˆ†(x′,t′)]=δ3(x x′)δ(t t′). Theevolvedpulse the EIT condition is set as vg = 10m/s (vg = c/ng with
qulantumlstate under all a−bove m−entioned factors should ng = n + ωp∂n/∂ωp), while the pump Rabi frequency is
Ω =2γ. The excited level |ei is 5P of 87Rb. Thedashed
be close to the ideal output eiϕ 1 1 (ϕ is a uniform c 1/2
1 2 curveisthatforthetwolevelsystemofthecorrespondingpa-
| i | i
one) for realizing a photon-photon XPM. rameters. The minus sign of the horizontal axes label comes
Before studying the input’s quantum state evolution, from our definition of ∆ . (b) Group velocity v vs ∆ with
1 g R
one needs to ascertain the pulses’ propagation in the thesamepulseandpumpdetuningasin(a). Thethickersolid
medium, sothat their interactiontime shouldbe known. curveisforΩc =2γ,whilethethinnerisaboutΩc =4γ. The
dashed line is the group velocity of the corresponding two-
Theabsorptionanddispersionofthepulsescanbefound
level system.
inasemi-classicalapproach[24,25]thattreatsthe input
pulses as the classical fields E (x), which are equivalent
l
to the averages ˆ(x) of the quantum fields (up to a
l
hE i When two pulses approach each other, one phe-
constant). In this framework the atom-field coupling is
nomenonthat could happen is knownas Rydberg block-
described by the following equations for the atomic den-
ade. For the red-detuned photons (∆ > 0) the rising
sity matrix elements [24]: 1
magnitude of negative ∆ constantly shifts the refrac-
R
γ µ Ω
eg eg c tiveindexcurvegoingthroughtheEITpointatacertain
ρ˙ = ( +i∆ )ρ +i E (ρ ρ ) i ρ ,
eg 1 eg l ee gg rg
− 2 2 − − 2 detuning ∆ toward that of the corresponding two-level
1
(4a)
system. In the limit ∆ γ the system will virtually
R
| | ≫
γ Ω µ turn into a two-level one; see Fig. 2(a). One signature
rg c eg
ρ˙ = +i(∆ +∆ +∆ ) ρ i ρ +i E ρ ,
rg − 2 1 2 R rg− 2 eg 2 l re of Rydberg blockade is a platform of nearly unchang-
(cid:0) (cid:1) (4b) ing group velocity shown in Fig. 2(b). In the blockade
where µij are the transition dipole matrix elements and regime the pulse group velocity asymptotically tends to
γij thedecayratesoftherelevantlevels. Theinteraction that of the corresponding two-level system; only those
withanotherpulseshiftstheenergylevelof r andhence with ∆ 0.5γ in Fig. 2 can reach the speed of light c
adds an extra term ∆R(x,t) = sin2θT(t)| id3x′∆(x − with gr1ow≤ing negative ∆R.
x′)hΨˆ†3−lΨˆ3−li(x′,t) to the detuning ∆2 oRf the pump The pulses will enter the superluminal regime charac-
beam, where T(t) is the time-dependent transmission terized by anomalous dispersion [25], which is accompa-
rate. This practice of reducing the interaction effect to nied by huge dissipation, if the interaction induced de-
a c-number detuning ∆R is equivalent to a mean field tuning ∆R of the positive sign is gradually added to the
treatment for the spin-wave fields in (A-2). One has the pump beam of the system in Fig. 2. Equivalently this
time-dependent solution phenomenon happens to the blue-detuned single-photon
(cid:18)ρρergg((tt))(cid:19) = −i21µegZ0tdτeRτtdt′Mˆ(t′)(cid:18)E0l (cid:19) (5) pTshuholisuselsddaibnnegteahrveoopidfreecdsoeminncpeplerotaefcltytihcede.aamttpriancgtivteheVidnWputpoptheonttoianls.
to(4a)-(4b)under the weakdriveapproximation[24,25] We thereforefocus onred-detuned photons coupledto
for single photons, where ensemble and propagating toward each other under the
(γ +i∆ ) iΩc attractive interaction. As the pulses get closer, they will
Mˆ(t)=(cid:18)− 2 iΩc 1 γrg i(∆−+2∆ +∆ (t))(cid:19) expand spatially because the characteristic size of their
− 2 − 2 − 1 2 R distributions Ψˆ†Ψˆ (z,t) is proportional to the average
h l li
with γ = γ. It is straightforward to obtain the time- of the distributed group velocity v (z,t) over the pulses.
eg g
dependent refractive index and decay rate from the sus- This modifies the ∆ calculated with the relative dis-
R
ceptibility χ(1)(t)= 2Nµ ρ (t)/(ǫ E) based on (5). tanceandabsorptionofthepulses,whichconstantlykeep
eg eg 0 l
−
3
tion has been harmful to the survival of the interacting
photons (see Figs. 3(b) and 3(c)). The pulse absorp-
tion rate and groupvelocity in the blockade regime tend
to those for a two-level system with the corresponding
system parameters, so the only way to reduce the pulse
loss in the blockade regime is using a higher photon de-
tuning ∆ . However, one trade-off for doing so is to
1
require a narrower pulse bandwidth (correspondingly a
longer pulse size) to fit into the smaller EIT window,
incurring a more prominent effect measured by the ra-
tio δ (Z)= v (Z,σ(Z)) v (Z,0) /v (Z,0) shown in
v g g g
|{ − } |
Fig. 3(d). Here v (Z,σ(Z)) is the group velocity at the
g
locationofthe characteristiclongitudinalsize σ(Z) from
FIG. 3: (color online) Numerical simulation for pulse mo- the pulse center, and v (Z,0) is that at the pulse cen-
g
tion. Here we adopt the relative distance coordinate Z
ter. The non-uniform group velocity distribution over
as a substitute for the time scale. We use |gi = 5S ,
1/2 pulses (in the co-moving coordinate with the pulse cen-
|ei = 5P , and |ri = 82D of 87Rb, with the VdW co-
1/2 3/2 ters)indicatedbytheratioisequivalenttoagroupveloc-
efficient |C | = 8500 GHz·µm6 [26] and γ = 2π·5.75 MHz.
6
The system parameters are chosen as Ng2/Ω2 = 0.75·107 ity dispersion that could make the pulses totally disap-
c
(v (−L) = 10 m/s), Ω = 2γ, and ∆ = −∆ = 2γ. The pear even without absorption. Another disadvantage for
g c 1 2
input Gaussian shaped pulses with f(z) = (cid:0)√1πσ(cid:1)12e−12z2/σ2 lsapragteiaplluylsdeisstizriebσutiesdthpautlstehse(pdreotupnoirntigonvaalluteo∆1/Rσf6rofomr tthhee
have the initial size σ = 11.1 µm, with the corresponding
VdWpotential)willbebelowthemagnitudeforasignif-
bandwidth well fitted into the EIT window. The dashed
curves are about the transverse separation a = 0.58 σ, the icant XPM. Our results thus show that in the blockade
thicker solid ones for a = σ, and the thinner solid ones for regime considered in [20] the imperfections due to ab-
a = 1.5 σ. The iterative step size for the numerical sim- sorptionandothersareactuallymuchmoreproblematic.
ulation is 0.005 σ. (a) Interaction induced ∆ (Z) at pulse
R
centers. The insertion is the refined plot for a = 1.5 σ. (b) The next target is to understand the real-time evolu-
Transmission rate T(Z) = exp(−kpRZLdyIm{χ(1)(y)}). (c) tion of the DSP state |1i1|1i2 given before (1). Under
Group velocity v (Z) = c/n (Z) at p−ulse centers. The most the perfect EIT condition, there is the approximation
g g
transverselyadjacentsituationshowsavelocityplatformnear σˆ = µ E /Ω (σˆ = g r) or its quantum many-
gr eg l c gr
h i − | ih |
Rydbergblockade. (d) Group velocity deviation ratio δv(Z). body version Sˆl(z) = (g√N/Ωc)ˆl(z) after neglecting
ForthesameZ,moreextendingpulseshavehigherδ dueto − E
v thenon-adiabaticcorrectionsforthenarrow-bandpulses,
more spatially inhomogeneous interaction.
implyingthe identicalpropagationofthe quantizedDSP
fieldwiththe electromagneticfield treatedasclassicalin
(4a)-(4b) [25]. In the suitable weak interaction regime
changingaswell. We use a numericalalgorithmto simu-
we find for the two-photon process, such as the most
late this dynamical process. From the coordinate origin
transversely separated pulses in Fig. 3 (corresponding
Z = 0 situated on the center of one pulse, the longitu-
to the refractive curves close to that of ∆ = 0 in Fig.
dinal relative distance L Z L to the other pulse’s R
− ≤ ≤ 2(a)), this approximation still holds with a small ratio
center throughout their motion is divided into n grids.
d ∆ /Ω . The kinetic Hamiltonian for the slowly mov-
The detuning ∆ at the i-th (0<i n 1) position is R c
R ≤ d− ing DSPs in the weak interaction regime can, therefore,
calculatedwiththepulsesizeandtransmissionrateatthe
be constructed as H = iv (t) dzΨˆ†(z)∂ Ψˆ (z),
i 1-th position. Together with the obtained numerical K − l g,l l z l
va−lues of ∆ at the previous positions of 0 k i 1, where the group velocity vgP,l(t) is deteRrmined with (4a)-
R ≤ ≤ − (4b). Meanwhile, for a slow light with cosθ 1, the
it is plugged into (5) for the numerical integral to find ≪
BSPs interact very slightly with the DSPs and among
the susceptibility χ(1). In the same way the updated
themselves because they contain negligible Rydberg ex-
group velocity and transmission rate from the suscepti-
citation. Theirquickdecouplingfromthesystemandde-
bility at the position i will be used to calculate the ∆
R caying into the environmentallow one to treatthe BSPs
atthe i+1-thposition. Running the iterativeprocedure
as motionless oscillations, though their group velocities
withsufficientlylargegridnumbern approachesthereal
d can be read from their spectrum in (1).
pulse motion.
Figure 3 illustrates an example of pulse motion found Our method for pulse state evolution is to adopt
by the above mentioned numerical method. As shown the joint evolution U(t,0) as the time-ordered exponen-
in Figs. 3(a) and 3(b), the greater interaction between tial Te−iR0tdτ{H(τ)+HD(τ)} on the initial state ψin =
| i
moretransverselyadjacentpulsesisinseparablewiththe 1 1 0 as the product of the input pulse state and
1 2 c
| i | i | i
more significant pulse losses. In where Rydberg block- the reservoir vacuum state 0 . Tracing out the reser-
c
| i
ade starts to manifest, the accumulated pulse absorp- voir degrees of freedom in the evolved state U(t,0)ψ
in
| i
4
gives the evolved system state. We have three non-
commutativeitems(H ,H andH )inH(t),aswellas
K AF I
the dissipationHamiltonianH (t)of(A-4), forthe joint
D
evolution operator U(t,0). Directly applying U(t,0) on
the DSP operators in ψ is impossible, as it is equiv-
in
| i
alent to analytically solving a nonlinear Langevin equa-
tion. One technique to circumvent the difficulty is the
FIG.4: (coloronline)(a)Fidelityandcrossphaseofphoton-
factorization of an evolution operator into the relatively
photon XPM for two counter-propagating pulses with the
tractable ones [27]. For our problem we have U(t,0) = transverse separation a = 1.5 σ in Fig. 3. L is the medium
UK(t,0)UAF(t,0)UI(t,0)UD(t,0) [28]. Among the fac- size. The system parameters are the same as in Fig. 3. The
torized processes U (t,0) = Texp i tdτH˜ (τ) , for insertiondescribesanimaginedsituation byreducingtheini-
X = K,AF,I andXD, H˜ and H˜{− aRr0e indXiffere}nt to tial pulse velocity to 10−2 m/s. (b) Fidelity and cross phase
K D for two pulses propagating together along two tracks sepa-
their original form H and H , respectively. The op-
K D rated by a = 1.5 σ. Due to pulse absorption, their group
erator UD(t,0) takes no effect on ψin , but the non- velocity is not stable in such co-propagation (for example, it
| i
commutativity of HD with HAF makes the BSP field dropsfrom 11.007 m/s to 11.002 m/s from L=2σ to 5σ).
operators in H become those in H˜ as follows:
AF AF
Φˆ±,l(z) Ξˆ±,l(z,τ)=e−φ2±γ(t−τ)/2Φˆ±,l(z)
→
t In Fig. 4 we plot the fidelity and cross phase for the
√γφ± dt′e−φ2±γ(t′−τ)/2ξˆl(z), (6) most transverselyseparatedpulses in Fig. 3. Due to the
± Z
τ
steepdecayoftheVdWpotentialatlongdistances,both
where φ = cosφ(sinφ). A sufficiently large γ ap- fidelity and cross phase for the counter-propagation in
+(−)
proximates the commutator [Ξˆ (z,τ ),Ξˆ† (z′,τ )] = Fig. 4(a) quickly converge to fixed values with increas-
±,l 1 ±,l 2
e−γφ±|τ1−τ2|δ(z z′)asvanishingforτ1 =τ2. Underthis ing medium size. A cross phase of π rad that still keeps
− 6 close to unit F could be achieved if the VdW coefficient
approximation the BSP operators in U (t,0) also take
I
the forms in (6), hence the evolved state Uˆ (t,0)ψ C6 , for example, is lifted by about nine times with a
I in | |
| i different Rydberg level. Contrary to a widely held no-
Z dz1dz2f(z1)f(z2)e−ic43R0tdτ∆(z1τ−z2τ)Ψˆ†1(z1)Ψˆ†2(z2) thiiognh,ficoduenlittyer;-psereop[3a0g]a.tiTonhedoinessenrtoitonauotfomFiagt.ic4a(llay)esnhsouwres
(cid:8)
2 t the fidelity for animagined motionof two pulses passing
− ic33 Z dτZ dz1dz2f(z1)f(z2)e−ic43R0τdt′∆(z1t′−z2t′) each other very slowly. The same propagation geometry
Xl=1 0 indicates that the degrading fidelity in the slow motion
∆(zτ zτ)(c Ξˆ† +c Ξˆ† )(z ,τ)Ψˆ† (z ) 0 , (7) comesfromthegrowingpulseentanglementoveralonger
× 1 − 2 1 +,l 2 −,l 1 3−l 2 | it interaction time. In comparison we also study the co-
(cid:9)
(unnormalized) to the first order of cosθ, where the no- propagating pulses in Fig. 4(b). The co-propagationex-
tations c1 = cosθsinφ, c2 = cosθcosφ, c3 = sinθ, hibitsconsiderabletrade-offbetweenF andϕ,andwould
zτ = z + 0τ dτ′vg,l(τ′), and |0it = |0i|0ic are used to be unfavorable for making large phases of good quality.
simplify thRe result. The detailed procedure for deriving
Insummary,wehavestudiedtheprocessoftwo-photon
the evolved state is given in [28]. The succeeding opera-
interaction via a Rydberg atomic ensemble. Our ap-
tion U only affects the BSP components in (7), while
AF
proach based on the complete dynamics for both sin-
U displaces the coordinate of Ψˆ†(z ).
K l l gle atoms and ensemble enables a more realistic descrip-
The interaction potential ∆(z z ) renders the DSP
1− 2 tion of the situation without steady state. The previ-
part in (7) no longer factorizable with respect to z
1 ously considered regime near Rydberg blockade is found
and z . This entangled piece deviates from the ideal
2 to be short of the favorable figures of merit for photon-
output state eiϕ 1 1 with a uniform phase ϕ. We
| i1| i2 photon XPM. We also prove that approximately ideal
measure the degrees of such deviation by comparing the
XPM creating considerable nonlinear phase can be real-
real output ψ = U(t,0)ψ with a reference state
| outi | ini izedwithcounter-propagatingandtransverselyseparated
ψ0 = U (t,0)U (t,0)U (t,0)ψ . In the absence
| outi K AF D | ini pulsesthatweaklyinteractwitheachother. Thephoton-
ofU (t,0) this reference keeps to be inthe product state
I photon XPM we have discussed can be the basis for an
1 1 0 , even if the amplitude f (z ) in the output
| i1| i2| ic l l all-optical deterministic quantum phase gate.
photon state 1 could be lowered due to any residual
l
| i
absorption. The output’s fidelity F with the ideal one B. H. and C. S. acknowledge the support by AITF
and the associate cross phase ϕ can thus be found from and NSERC. M.X. acknowledges the supports in part
the overlap √Feiϕ = ψ0 ψ , where the two output by NBRPC (Grant No. 2012CB921804)and NSFC (No.
h out| outi
states are normalized. Similar definitions for F and ϕ 11321063). A. V. S. was supported by RFBR 12-02-
can be found in [29, 30]. 31621.
5
Supplementary Information for “Two-photon dynamics in coherent Rydberg atomic ensemble”
A. Decomposition of Joint Evolution Operator
The system Hamiltonian in the concerned problem consists of three parts. The atom-field coupling Hamiltonian
for the ensemble is
2
H = dz ω+Φˆ† Φˆ (z)+ω−Φˆ† Φˆ (z) , (A-1)
AF − Z +,l +,l −,l −,l
Xl=1 (cid:8) (cid:9)
where ω± = 1(∆ ∆2+g2N +Ω2), and the bright-state polariton (BSP) fields are defined as
2 1± 1 c
p
Φˆ (z)=sinθsinφˆ(z)+cosφPˆ(z)+cosθsinφSˆ(z),
+
E
Φˆ (z)=sinθcosφˆ(x) sinφPˆ(z)+cosθcosφSˆ(z).
−
E −
The polarization field Pˆ(x) in the above is the continuous average g e/√∆N of the flip operators
g i e for the atoms inside a small volume ∆V around x, which contaPinis∈∆∆VN| iih |1 atoms. So is the definition
| i h | g r/√∆N for the spin-wave field Sˆ(x). The second part that describes t≫he pulse interaction process is
i∈∆V | iih |
P
H = dz dz′Sˆ†(z)Sˆ†(z′)∆(z z′)Sˆ (z′)Sˆ (z)
I Z Z 1 2 − 2 1
2
1
= dz c Φˆ† (z)+c Φˆ† (z)+c Ψˆ†(z) c Φˆ (z)+c Φˆ (z)+c Ψˆ (z) Vˆ(z), (A-2)
2Z 1 +,l 2 −,l 3 l 1 +,l 2 −,l 3 l l
Xl=1 (cid:0) (cid:1)(cid:0) (cid:1)
whereVˆ(z)= dz′∆(z z′)Λˆ† Λˆ (z′,τ)withΛˆ =c Φˆ +c Φˆ +c Ψˆ . Hereweusethenotationsc =cosθsinφ,
l − 3−l 3−l l 1 +,l 2 −,l 3 l 1
c2 =cosθcosφR, and c3 =sinθ from the main text. The third part is the kinetic Hamilton
2
H = i v (t) dzΨˆ†(z)∂ Ψˆ (z), (A-3)
K − g,l Z l z l
Xl=1
for the DSPs, where v (t) is found with a semi-classical treatment of the atom-field coupling in the main text.
g,l
SimilarlytheBSPkinetic Hamiltoniancanbe constructedwiththeirgroupvelocities1/2c(1 ∆ / ∆2+g2N +Ω2)
± 1 1 c
from the spectrum in (A-1). In a slow light regime considered in the main text, the BSPs go mupch faster than and
interactveryslightlywith the DSPs,while they decayintothe environment. Suchquick decouplingofthe BSPs from
the systemallows one to approximatethem as motionless oscillations, andthis simplifies the coordinatesfor the BSP
field operators in most equations below. In addition, the coupling between the polarization fields Pˆ and reservoir
l
that leads to the dissipation is described by
2
H = i √γ dz Pˆ(z)ξˆ†(z,t) Pˆ†(z)ξˆ(z,t)
D Z l l − l l
Xl=1 (cid:8) (cid:9)
= i √γ dz cosφ Φˆ (z)ξˆ†(z,τ) sinφ Φˆ (z)ξˆ†(z,τ) H.c. , (A-4)
Z +,l l − −,l l −
Xl (cid:8) (cid:9)
where the random-variable noise operators satisfy [ξˆ(z,t),ξˆ†(z′,t′)] = δ(z z′)δ(t t′). The infinitesimal action of
l l − −
the joint evolution U(t,0)=Te−iR0tdτ(H(τ)+HD(τ), where H =HK +HAF +HI, on the field operators and the joint
quantum state of the system and reservoir gives rise to the exact Langevin equation about the system operators and
the exact master equation about the system state, respectively [23]. The solution to these equations are difficult to
find in the presence of the nonlinear term in (A-2).
HerewepresentadifferentapproachtofindthetransformationU(t,0)Ψˆ (z)U†(t,0)byfactorizingthejointevolution
l
operator U(t,0) into relatively tractable processes. First, we separate the kinetic part out of the total evolution
operator as follows:
t t t
Texp i dτH(τ) = Texp i dτH (τ) Texp i dτU† (τ,0) H +H +H U (τ,0) ,(A-5)
{− Z } {− Z K } − Z K AF I D K
0 0 (cid:8) 0 (cid:0) (cid:1) (cid:9)
6
whereU (τ,0)=Texp i τdt′H (t′) . Theproofforthisexactfactorizationcanbefoundin[27]. Theinteraction
K {− 0 K }
Hamiltonian in the second tRime-ordered exponential of the above becomes
H (τ)=U†(τ,0)H (τ)U (τ,0)
I K I K
2
1
= dz c Φˆ† (z)+c Φˆ† (z)+c Ψˆ†(zτ,l) c Φˆ (z)+c Φˆ (z)+c Ψˆ (zτ,l) Vˆ(z,τ), (A-6)
2 Z 1 +,l 2 −,l 3 l 1 +,l 2 −,l 3 l l
Xl=1 (cid:0) (cid:1)(cid:0) (cid:1)
where
Vˆ(z,τ) = dζ∆(z ζ) c Φˆ† (ζ)+c Φˆ† (ζ)+c Ψˆ† (ζτ,3−l) c Φˆ (ζ)+c Φˆ (ζ)+c Ψˆ (ζτ,3−l)
l Z − 1 +,3−l 2 −,3−l 3 3−l 1 +,3−l 2 −,3−l 3 3−l
(cid:0) (cid:1)(cid:0) (cid:1)
(A-7)
with zτ,l = z τ v (t′)dt′. The effect of the above transformation is the displacement of the coordinates for the
− 0 g,l
DSP field operatRors. The other terms in the second time-ordered exponential of (A-5) are not changed.
Secondly, the system-reservoircoupling processin the secondtime-orderedexponentialof (A-5) is separatedoutto
the right side as follows:
t
Texp i dτ H +H (τ)+H
AF I D
{− Z }
0 (cid:0) (cid:1)
t t
= Texp i dτU (t,τ) H +H (τ) U†(t,τ) Texp i dτH (τ) , (A-8)
{− Z D AF I D } − Z D
0 (cid:0) (cid:1) (cid:8) 0 (cid:9)
where U (t,τ) = Texp i tdt′Hˆ (t′) . In the first time-ordered exponential of the above, the BSP fields will be
D {− τ D }
transformed to R
U (t,τ)Φˆ (z)U†(t,τ)
D +,l D
t
= e−cos2φγ(t−τ)/2Φˆ (z)+cosφ√γ dt′e−cos2φγ(t′−τ)/2ξˆ(z,t′) Ξˆ (z,τ); (A-9)
+,l l +,l
Z ≡
τ
nˆ+,l(z,τ)
| {z }
U (t,τ)Φˆ (z)U†(t,τ)
D −,l D
t
= e−sin2φγ(t−τ)/2Φˆ (z) sinφ√γ dt′e−sin2φγ(t′−τ)/2ξˆ(z,t′) Ξˆ (z,τ). (A-10)
−,l l −,l
− Z ≡
τ
nˆ−,l(z,τ)
| {z }
The transformed BSP operators therefore satisfy the following commutation relations:
[Ξˆ (z,τ ),Ξˆ† (z′,τ )]=e−cos2φγ|τ1−τ2|/2δ(z z′) g (τ ,τ )δ(z z′);
+,l 1 +,l 2 − ≡ 1 1 2 −
[Ξˆ (z,τ ),Ξˆ† (z′,τ )]=e−sin2φγ|τ1−τ2|/2δ(z z′) g (τ ,τ )δ(z z′). (A-11)
−,l 1 −,l 2 − ≡ 2 1 2 −
Then the first time-ordered exponential in (A-8) takes the form
Te−i21P2l=1R0tdτRdzΠˆ†l(z,τ)Πˆl(z,τ)Vˆl(z,τ)+iPlR0tdτRdz{ω+Ξˆ†+,lΞˆ+,l(z,τ)+ω−Ξˆ†−,lΞˆ−,j(z,τ)} (A-12)
with Πˆ (z,τ)=c Ξˆ (z)+c Ξˆ (z)+c Ψˆ (zτ,l).
l 1 +,l 2 −,l 3 l
The actionof the secondterminside the time-orderedexponential in (A-12) canbe further separatedout as in Eq.
(A-5), and the accompanying effect is to transform the BSP operators inside the other time-ordered exponential as
follows:
Te−iR0τdt′PlRdζ{ω+Ξˆ†+,lΞˆ+,l(ζ)+ω−Ξˆ†−,jΞˆ−,l(ζ)} Ξˆ+,l(z,τ) TeiR0τdt′PlRdζ{ω+Ξˆ†+,lΞˆ+,l(ζ)+ω−Ξˆ†−,lΞˆ−,l(ζ)}
τ ( 1) τ τ
= Ξˆ (z,τ)+i dt ω+g (t ,0)Ξˆ (z,t )+ − dt dt (ω+)2g (t ,0)g (t ,0)Ξˆ (z,t )+
+,l 1 1 1 +,l 1 2 1 1 2 1 1 +,l 2
Z 2! Z Z ···
0 0 0
Ξˆ′ (z,τ);
≡ +,l
Te−iR0τdt′PlRdζ{ω+Ξˆ†+,lΞˆ+,l(ζ)+ω−Ξˆ†−,lΞˆ−,l(ζ)} Ξˆ+,l(z) TeiR0τdt′PlRdζ{ω+Ξˆ†+,lΞˆ+,l(ζ)+ω−Ξˆ†−,lΞˆ−,l(ζ)}
τ ( 1) τ τ
= Ξˆ (z,τ)+i dt ω−g (t ,0)Ξˆ (z,t )+ − dt dt (ω−)2g (t ,0)g (t ,0)Ξˆ (z,t )+
−,l 1 2 1 −,l 1 2 1 2 2 2 1 −,l 2
Z 2! Z Z ···
0 0 0
Ξˆ′ (z,τ). (A-13)
≡ −,l
7
HerewehaveneglectedthemixingofthetwotransformedBSPfieldsduetotheircouplingtothesamereservoir. The
pulse interaction process after the factorization now takes the form
1 2 t
U (t,0) = Texp i dτ dzΘˆ†(z,τ)Θˆ (z,τ)Wˆ (z,τ) , (A-14)
I − 2 Z Z l l l
(cid:8) Xl=1 0 (cid:9)
with Wˆ (z,τ) = dz′∆(z z′)Θˆ† (z′,τ)Θˆ (z′,τ) and Θˆ (z,τ) = c Ξˆ′ (z,τ)+c Ξˆ′ (z,τ)+c Ψˆ (zτ,l). For a
l − 3−l 3−l l 1 +,l 2 −,l 3 l
sufficiently large dRamping rate γ, the commutators in (A-11) can be regarded as vanishing for τ1 =τ2, and then the
BSPfield operatorsinthe aboveequationcanbe approximatedasΞˆ with g (t ,t ) 0 foran6y pairoft andt .
±,l 1(2) 1 2 1 2
≈
So far the joint evolution operator has been decomposed as
Te−iR0tdτH(τ) =Te−iR0tdτHK(τ) TeiR0tdt′PlRdζ{ω+Ξˆ†+,lΞˆ+,l(ζ)+ω−Ξˆ†−,lΞˆ−,l(ζ)}
Te−i12P2l=1R0tdτRdzΘˆ†l(z,τ)Θˆl(z,τ)Wˆl(z,τ) Te−iR0tdτHD(τ)
×
U (t,0)U (t,0)U (t,0)U (t,0). (A-15)
K AF I D
≡
B. Evolution of Pulse Quantum State
Now we study the evolution of the joint state
ψ = dz f(z )Ψˆ†(z ) dz f(z )Ψˆ†(z )0 0 , (B-1)
| ini Z 1 1 1 1 Z 2 2 2 2 | i⊗| ic
for two identical pulses, where 0 is the reservoir vacuum state, under U(t,0). It is equivalent to finding the
c
| i
transformation U(t,0)Ψˆ†(z )U†(t,0) (or U(t,0)Ψˆ (z )U†(t,0)) because U(t,0)0 0 = 0 0 0 , i.e.
l l l l | i⊗| ic | i⊗| ic ≡| it
U(t,0)ψ = U(t,0) dz f(z )Ψˆ†(z ) dz f(z )Ψˆ†(z )0
| ini Z 1 1 1 1 Z 2 2 2 2 | it
= dz f(z )U(t,0)Ψˆ†(z )U†(t,0) dz f(z )U(t,0)Ψˆ†(z )U†(t,0)0 . (B-2)
Z 1 1 1 1 Z 2 2 2 2 | it
We will apply the decomposed form in (A-15) for the purpose.
The operation U (t,0) does not change ψ . The transformation by Uˆ (t,0) is found through
D in I
| i
UI(t,0)Ψˆl(zl)UI†(t,0)= Te−ic23R0tdτ{UI(t,τ)Wˆl(zl−τ,τ)UI†(t,τ)} †Ψˆl(zl)
t (cid:0) (cid:1)
+ ic3 dτ Te−ic23R0τdt′{UI(t,t′)Wˆl(zl−t′,t′)UI†(t,t′)} †
Z
0 (cid:0) (cid:1)
U (t,τ) c Ξˆ′ (z−τ)+c Ξˆ′ (z−τ) Wˆ (z−τ,τ)U†(t,τ), (B-3)
× I 1 +,l l 2 −,l l l l I
(cid:0) (cid:1)
where z−τ = z + τv (t′)dt′. This is an exact form obtained by expressing Uˆ (t,0) as an infinite product of the
l l 0 g,l I
small elements arouRnd each moment, which transform the DSP operator as follows:
t
U (t+dt,t)Ψˆ (z )U†(t+dt,t)= Iˆ+ic2Wˆ (z + dt′v (t′),t)dt Ψˆ (z )
I l l I 3 l l Z g,l l l
(cid:0) 0 (cid:1)
t
+ ic c Ξˆ′ (z−t)+c Ξˆ′ (z−t) Wˆ (z + dτ′v (τ′),t)dt. (B-4)
3 1 +,l l 2 −,l l l l Z g,l
(cid:0) (cid:1) 0
TheoperationbyUˆ (t,τ)insidethetime-orderedexponentialandintegralof(B-3)canbefurtherperformedtoobtain
I
a form of this exact transform in terms of an infinite series. There is the following commutator
[Wˆ (z ,τ),Wˆ (z ,τ′)]=[ dz′∆(z z′)Θˆ† (z′,τ)Θˆ (z′,τ), dζ′∆(z ζ′)Θˆ† (ζ′,τ′)Θˆ (ζ′,τ′)]
l 1 l 2 Z 1− 3−l 3−l Z 2− 3−l 3−l
ic2e−γsin2φ|τ−τ′|/2sing (τ,τ′)+ic2e−γcos2φ|τ−τ′|/2sing (τ,τ′) dz′∆(z z′)∆(z z′)Θˆ† (z′,τ)Θˆ (z′,τ′)
≈ { 1 1 2 2 }Z 1− 2− 3−l 3−l
(B-5)
8
for Wˆ (z,τ). Together with the fact e−γsin2φ|τ−τ′|/2 1, e−γcos2φ|τ−τ′|/2 1 for a sufficiently large damping
l
≪ ≪
rate γ (this means a negligible correlation time window for the colored noises nˆ introduced in Eqs. (A-9) and
±,l
(A-10)), the above commutator can be approximated as vanishing for τ = τ′ in a slow light regime with c 1
1
6 | | ≪
and c 1, which is created for the input photons under the EIT condition. Meanwhile one has sing (τ,τ′) =
2 1(2)
| | ≪
0 for τ = τ′. Then there is the relation Uˆ (t,τ)Wˆ (z−τ,τ)Uˆ†(t,τ) Wˆ (z−τ,τ) from the approximation of the
I l l I ≈ l l
vanishing commutator in (B-5), and the non-Abelian phases in (B-3) can be reduced to the Abelian ones due to such
approximate commutativity of Wˆ (z−t,t) at the different time. Moreover, in a slow light regime where the BSPs
l l
containing negligible Rydberg excitation quickly decouple from the system through decaying to the environment
and escaping from the medium, the DSP components transformed back from the BSP components through the
transformationU (t,τ) c Ξˆ′ (z )+c Ξˆ′ (z ) U†(t,τ) in the second term of (B-3) is negligible. Therefore, the DSP
I 1 +,l l 2 −,l l I
operator transformation(cid:0) in (B-3) can be finally(cid:1)approximated as
t
UI(t,0)Ψˆl(zl)UI†(t,0) = eic23R0tdτWˆl(zl−τ)Ψˆl(zl)+ic3Z dτeic23R0τdt′Wˆl(zl−t′)
0
c Ξˆ′ (z−τ)+c Ξˆ′ (z−τ) Wˆ z−τ (B-6)
× 1 +,l l 2 −,l l l l
(cid:0) (cid:1) (cid:0) (cid:1)
in the regime considered in the main text. In Eq. (8) of the main text we express this DSP operator evolution with
a further approximatedform, consideringthe vanishing commutatorsin (A-11) for different time due to a sufficiently
large damping rate γ.
To find the evolution for the state ψ , one needs the following operation
in
| i
t
UI(t,0)Ψˆ†1(z1)Ψˆ†2(z2)|0it = e−ic23R0tdτWˆ1(z1−τ)Ψˆ†1(z1)−ic3Z dτe−ic23R0τdt′Wˆ1(z1−t′)
(cid:8) 0
c Ξˆ′† (z−τ)+c Ξˆ′† (z−τ) Wˆ z−τ Ψˆ†(z )0 (B-7)
× 1 +,1 1 2 −,1 1 1 1 2 2 | it
(cid:0) (cid:1) (cid:0) (cid:1)(cid:9)
based on (B-6), where the relation Wˆ z−τ 0 = 0 has been considered. Similar to Eq. (B-6), the phase operator
2 2 | it
e−ic23R0tdτWˆ1(z1−τ) in the first term acts(cid:0)on th(cid:1)e second DSP field operator Ψˆ†2(z2) via the transformation
e−ic23R0tdτWˆ1(z1+R0τdt′vg,1(t′))Ψˆ†2(z2)eic23R0tdτWˆ1(z1+R0τdt′vg,1(t′)) =e−ic43R0tdτ∆ z1−z2+R0τ{vg,1(t′′)−vg,2(t′′)}dt′′ Ψˆ†2(z2)
(cid:0) (cid:1)
− ic33Z0tdτe−ic43R0τdt′∆(cid:0)z1−z2+R0t′{vg,1(t′′)−vg,2(t′′)}dt′′(cid:1)∆(cid:0)z1−z2+Z0τ{vg,1(t′′)−vg,2(t′′)}dt′′)
c Ξˆ′† (z−τ)+c Ξˆ′† (z−τ) . (B-8)
× 1 +,2 2 2 −,2 2
(cid:0) (cid:1)
The second term’s action on Ψˆ†(z ) is found through the commutator
2 2
τ
[e−ic23R0τdt′Wˆ1(z1+R0t′dτ′vg,1(τ′))Wˆ1(z1+Z dτ′vg,1(τ′)),Ψˆ†2(z2)]
0
= c23e−ic43R0τdτ′∆(cid:0)z1−z2+R0τ′{vg,1(t′′)−vg,2(t′′)}dt′′(cid:1)∆(cid:0)z1−z2+Z0τ{vg,1(t′′)−vg,2(t′′)}Ψˆ†2(z2)
+ c3e−ic43R0τdt′∆ z1−z2+R0t′{vg,1(t′′)−vg,2(t′′)}dt′′ ∆ z1 z2+ τ vg,1(t′′) vg,2(t′′) dt′′
(cid:0) (cid:1) (cid:0) − Z0 { − } (cid:1)
c Ξˆ′† (z−τ)+c Ξˆ′† (z−τ) . (B-9)
× 1 +,2 2 2 −,2 2
(cid:0) (cid:1)
Putting all these together one will obtain the entangled state (unnormalized and to the first order of c )
1(2)
U (t,0) dz f(z )Ψˆ†(z ) dz f(z )Ψˆ†(z )0
I Z 1 1 1 1 Z 2 2 2 2 | it
= Z dz1Z dz2f(z1)f(z2)e−ic43R0tdτ∆(cid:0)z1−z2+R0τ{vg,1(t′′)−vg,2(t′′)}dt′′(cid:1)Ψˆ†1(z1)Ψˆ†2(z2)|0it
2 t
− ic33 Z dz1Z dz2f(z1)f(z2)Z dτe−ic43R0τdt′∆(z1−z2+R0t′{vg,1(t′′)−vg,2(t′′)})
Xl=1 0
τ
∆ z z + v (t′′) v (t′′) dt′′ c Ξˆ′† (z−τ)+c Ξˆ′† (z−τ) Ψˆ†(z )0
× 1− 2 Z { g,1 − g,2 } 1 +,3−l 2 2 −,3−l 2 l l | it
(cid:0) 0 (cid:1)(cid:0) (cid:1)
(B-10)
9
due to the evolution under U (t,0). After finding the above U (t,0)ψ , it will be straightforwardto do the further
I I in
| i
transformationsunderU (t,0),whichtransformstheBSPcomponentsasin(A-13),andbyU (t,0),whichdisplaces
AF K
theDSPcoordinates. TracingoutthereservoirdegreesoffreedommakesnodifferencetotheDSPpartfortheoutput
state of the system.
Asweexplaininthemaintext,thecrossphasefortheoutputstate ψ =U(t,0)ψ anditsfidelitywithanideal
out in
| i | i
outputstatefromXPMarefoundthroughitsoverlapwiththereferencestate ψ0 =U (t,0)U (t,0)U (t,0)ψ .
| outi K AF D | ini
In the absence of the pulse interaction process, there is no BSP components in the reference state ψ0 . The
| outi
approximationswemakeforthesimplificationoftheevolvedstateinthemaintext,therefore,donotaffectthevalues
of the cross phase and fidelity in Fig. 4 of the main text.
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