Table Of ContentTwo-way interconversion of millimeter-wave and optical fields in Rydberg gases
Martin Kiffner1,2, Amir Feizpour2, Krzysztof T. Kaczmarek2, Dieter Jaksch2,1, and Joshua Nunn2
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 1175431 and
Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom2
We show that cold Rydberg gases enable an efficient six-wave mixing process where terahertz or
microwave fields are coherently converted into optical fields and vice versa. This process is made
possiblebythelonglifetimeofRydbergstates,thestrongcouplingofmillimeterwavestoRydberg
transitionsandbyaquantuminterferenceeffectrelatedtoelectromagneticallyinducedtransparency
(EIT).Ourfrequencyconversionschemeappliestoabroadspectrumofmillimeterwavesduetothe
abundanceoftransitionswithintheRydbergmanifold,andwediscusstwopossibleimplementations
basedonfocussedterahertzbeamsandmillimeterwavefieldsconfinedbyawaveguide,respectively.
We analyse a realistic example for the interconversion of terahertz and optical fields in rubidium
atoms and find that the conversion efficiency can in principle exceed 90%.
I. INTRODUCTION crowaves [27] and hybrid atom-photon gates [28]. Here
we show how to achieve efficient and coherent MMOC
6 withouttheneedforcavities,microfabricationorcooling.
1 Two-way conversion between optical fields and tera-
Our MMOC scheme is made possible by an EIT-related
0 hertz/microwave radiation is a highly desirable capa-
quantum interference effect and the long lifetime of the
2 bility with applications in classical and quantum tech-
p nologies, including the metrological transfer of atomic Rydbergstates. Incontrasttopreviousfrequencymixing
schemesinEITmedia[29–31],thisquantuminterference
e frequency standards [1], novel astronomical surveys [2],
effectimplementsacoherentbeamsplitterinteractionbe-
S long-distance transmission of electronic data via pho-
tween the millimeter and optical fields which underpins
6 tonic carriers [3], and signal processing for applications the conversion effect. Our main result is a theoretical
1 in radar and avionics [4]. Efficient conversion of tera-
model establishing the principle of operation of the pro-
hertzradiationintovisiblelightwouldfacilitatethegen-
posed device, which it is shown could be implemented in
] eration, detection and imaging of terahertz fields [5, 6]
h an ensemble of cold trapped Rb atoms.
for stand-off detection, biomedical diagnostics and spec-
p
- troscopy. In the quantum domain, coherent microwave- Thepaperisorganisedasfollows. Weintroduceourthe-
nt optical conversion could enable quantum computing via oretical model based on the standard framework of cou-
a optically-mediated entanglement swapping [7–9] in solid pled Maxwell-Bloch equations in Sec. II, where we also
u state systems such as spins in silicon [10] or supercon- describe how to include interactions between Rydberg
q ducting qubits [11], which lack optical transitions but atoms. In Sec. III we discuss the principle of operation
[ couple strongly to microwaves. Moreover, Josephson of our scheme and show that both time-independent and
3 junctionscanmediatemicrowavephotonicnon-linearities pulsed input fields of arbitrary (band-limited) shape can
v thatcannoteasilybereplicatedforopticalphotons[12]so be efficiently converted. We go on to consider the si-
0 thatcoherentmicrowave-opticalconversionalsoprovides multaneous spatial confinement of mm-wave and optical
0 a route to freely-scalable all-photonic quantum comput- fields, and we show that high conversion efficiencies are
1 ing. predicted for a realistic implementation in trapped Rb
7
vapour. A brief summary of our work is presented in
0 Recent proposals for conversion between the optical and
Sec. IV.
. mm-wave domains have been based on optomechani-
1
cal transduction [13–15], or frequency mixing in Λ-type
0
6 atomic ensembles [16–20]. Both approaches require high
1 quality frequency-selective cavities limiting the conver-
: sion bandwidth, as well as aggressive cooling or optical II. MODEL
v
i pumpingtobringtheconversiondevicesintotheirquan-
X
tum ground states. We consider an ensemble of cold trapped atoms inter-
r
Inthispaper,weproposeinsteadtousefrequencymixing acting with laser fields and mm-waves and model these
a
inRydberggases[21–23]fortheconversionofmillimeter interactions using the standard framework of coupled
waves to optical fields (MMOC) (see Fig. 1). We use the Maxwell-Blochequations. Asummaryofthegeneralap-
terminology ‘mm-wave’ broadly to refer to fields with proachispresentedinSec.IIA,andadetailedderivation
carrier frequencies between 10 and 10,000 GHz, corre- canbefoundintheSupplementaryInformation. Thean-
sponding to resonant transitions between highly excited alytical solution of the Maxwell-Bloch equations is out-
Rydberg states in an atomic vapour. Our scheme ben- lined in Sec. IIB and complemented by Appendix A. In
efits from the strong coupling between Rydberg atoms Sec. IIC we include Rydberg-Rydberg interactions into
and millimeter waves which has previously been used our model. This allows us to identify parameter regimes
for detection and magnetometry [24–26], storage of mi- in Sec. III where these interactions are negligible.
2
(a) mm-wave lens (c)
mm-wave fields
optical fields
(b)
cold atoms
mm-wave guide
FIG.1. (a)and(b)depicttwopossibleimplementationsoftheconsideredMMOCdevicewhereamm-wavefield(red)andan
optical field (blue) are coherently interconverted through their interaction with an ensemble of cold atoms (green). The other
fields needed to drive the interaction are not shown. (a) The mm-wave fields are coupled into and out of the atomic ensemble
with dielectric lenses; the optical fields are directed through small, sub-mm holes. (b) Tight confinement of microwaves with
longer wavelengths could be achieved by trapping the atoms inside the core of a hollow waveguide. (c) Atomic level scheme.
Transition frequencies and detunings are not to scale. Ω and Ω are the Rabi frequencies associated with the mm-wave and
M L
the optical fields, respectively. Ω , Ω , Ω and Ω are Rabi frequencies of the auxiliary fields, and ∆ is the detuning of the
P R C A k
fieldswithstate|k(cid:105)(k∈{4,5,6}). Levels|3(cid:105), |4(cid:105)and|5(cid:105)areRydbergstateswithdecayrateΓ(cid:28)γ, whereγ isthedecayrate
of states |2(cid:105) and |6(cid:105).
A. Maxwell-Bloch equations and A = i j are atomic transition operators. The
ij
| (cid:105)(cid:104) |
detuning ∆ in Eq. (2) is defined as
k
In a first step we neglect atom-atom interactions and ∆ =ω +ω ω , (3a)
3 P R 3
−
consider the Bloch equations for a single atom with level
∆ =ω +ω +ω ω , (3b)
4 P R M 4
scheme as shown in Fig. 1. The millimeter wave Ω of −
M ∆ =ω +ω +ω ω ω , (3c)
interest couples to the transition 3 4 , where 3 5 P R M− C− 5
and 4 areRydbergstateswithprin|c(cid:105)ip↔alq|u(cid:105)antumnum| (cid:105)- ∆6 =ωL ω6, (3d)
ber |n(cid:105)(cid:38) 20. The optical field Ω of interest couples −
L where (cid:126)ω denotes the energy of state k with respect
to the 1 6 transition, and the conversion between k | (cid:105)
ΩM and| (cid:105)Ω↔L i|s (cid:105)facilitated by four auxiliary fields. The tXowthiteheRnearbgiyfroefqlueevnecly|1Ω(cid:105) an(dXωX iPs,tRhe,Cfr,eAqu,Men,cLy o)f. fiTehlde
resonant fields ΩP and ΩR create a coherence on the term (cid:37) in Eq. (1) accoXunts∈fo{r spontaneous e}mission
1 3 transition through coherent population trap- Lγ
| (cid:105) ↔ | (cid:105) from the excited states. These processes are described
ping [32]. The two other auxiliary fields Ω and Ω are
C A by standard Lindblad decay terms. The full decay rate
in general off-resonant and establish a coherent connec-
of the states 2 and 6 is γ, and the long-lived Rydberg
tion between the 3 4 and 1 6 transitions. We | (cid:105) | (cid:105)
| (cid:105)↔| (cid:105) | (cid:105)↔| (cid:105) states decay with Γ γ. The six fields drive a resonant
model the time evolution of the atomic density operator (cid:28)
loop,
by a Markovian master equation
ω +ω +ω ω ω ω =0, (4)
P R M C A L
− − −
i
and we impose the phase matching condition
∂ (cid:37)= [H,(cid:37)]+ (cid:37). (1)
t −(cid:126) Lγ
k +k +k k k =k . (5)
P R M C A L
− −
In the electric-dipole and rotating-wave approximations, In the following, we assume that Ω and Ω are co-
M L
the Hamiltonian H in Eq. (1) is given by propagating, while the directions of the auxiliary fields
are chosen such that Eq. (5) holds. Note that this phase
matching condition is automatically fulfilled by virtue of
6
(cid:88) Eq. (4) if all fields are co-propagating.
H = (cid:126) ∆ A (cid:126)(Ω A +Ω A +Ω A
− k kk− P 21 R 32 M 43 The strong auxiliary fields are not significantly affected
k=3 by their interaction with the mm-wave and optical sig-
+Ω A +Ω A +Ω A + H.c.) , (2)
C 45 A 56 L 61 nals. We therefore consider only these signal fields and
3
the atomic coherences as dynamical variables. In the for the mm-wave and optical fields can then be written
paraxial approximation we find as
(cid:18) (cid:19)
i 1 ∂ Ω=i Ω, (10)
∆⊥+ ∂t+∂z ΩM =iηM(cid:37)43, (6a) z M
−2k c
M
where
(cid:18) (cid:19)
i 1
−2kL∆⊥+ c∂t+∂z ΩL =iηL(cid:37)61, (6b) =η (cid:32)b2χM43 b2χL43 (cid:33), Ω=(cid:18)ΩM (cid:19) . (11)
M L χM χL ΩL
where k (k ) is the wavenumber of the mm-wave (op- 61 61
M L
tical) field and ∆ = ∂2 +∂2 is the transverse Laplace
⊥ x y When the auxiliary fields are time-independent and spa-
operator. The coupling constants η and η are given
M L tially uniform, the solution to Eq. (10) is
by
Ω=exp(i z)Ω , (12)
d 2 M 0
43
η = | | ω , (7a)
M N 2(cid:126)(cid:15) c M where Ω is the initial condition Ω evaluated at z = 0.
0 0
d 2 The matrix exponential in Eq. (12) can be expressed in
61
ηL =N|2(cid:126)(cid:15)0|cωL, (7b) termsofthe2×2identitymatrix1andthePaulimatrices
σ [33],
k
where d = k dˆl is the matrix element of the electric
dipole mkolme(cid:104)nt|o|p(cid:105)erator dˆon the transition transition exp(iMz)=exp(ia0z)
(cid:20) (cid:16) (cid:17) a σ (cid:16) (cid:17)(cid:21)
|k(cid:105) ↔ |l(cid:105), c is the speed of light and N is the density cos √a2z 1+i · sin √a2z ,
of atoms. In the following the ratio of the coupling con- √a2
stants is denoted by (13)
d 2ω where
b2 =η /η = | 43| M. (8)
M L d61 2 ωL 1 1
| | a = Tr( ), a= Tr( σ). (14)
0
2 M 2 M
Numerical solutions of Eqs. (1), (6) are presented in
Sec. IIIC, but first it is instructive to derive analytic The solution presented here treats the signal fields Ω ,
M
solutions in the limit that diffraction over the length of Ω as c-numbers. However, the generalisation to quan-
L
the atomic ensemble can be neglected. tum fields is straightforward since the coherences in
Eq. (9) are linear in the signal fields. Apart from quan-
tumnoiseoperators, ourcalculationsarethusequivalent
B. Analytical solution toaHeisenberg-LangevinapproachwherethesignalRabi
frequencies Ω , Ω are replaced by quantum fields [34–
M L
The first-order solution of Eq. (1) with respect to 36]. Since the Langevin noise operators represent only
the Rabi frequencies Ω , Ω takes the form (see Ap- vacuum noise, they do not contribute to normally or-
M L
pendix A) deredexpectationvalues,whichdeterminetheconversion
efficiency.
(cid:37) χMΩ +χL Ω , (9a)
43 ≈ 43 M 43 L
(cid:37) χMΩ +χL Ω . (9b)
61 ≈ 61 M 61 L C. Interaction-induced imperfections
Theresponseoftheatomicsystemonthe 3 4 tran-
| (cid:105)↔| (cid:105)
sition induced by the mm-wave field is described by χM, Next we consider the effects of dipole-mediated interac-
43
and χL accounts for the atomic response on the tran- tions between atoms excited into their Rydberg man-
61
sition 1 6 due to the optical field. In addition, ifolds. In general, Rydberg interactions will prevent
| (cid:105) ↔ | (cid:105)
the mm-wave field can induce a coherence proportional some fraction of atoms from participating in the conver-
to χM on the optical transition 1 6 , and the op- sion process and lead to absorption of the signal fields,
61 | (cid:105) ↔ | (cid:105)
tical field can create a coherence proportional to χL on and therefore will reduce the conversion efficiency. The
43
the transition 3 4 . The cross-terms proportional atomic level scheme in Fig. 1(b) contains three Rydberg
| (cid:105) ↔ | (cid:105)
to χL and χM in Eq. (9) originate from the closed-loop states, and the population in state 3 is continuously
43 61 | (cid:105)
character of the atomic level scheme. kept at ρ Ω /Ω 2 via coherent population trap-
33 P R
≈ | |
NextwecombineEq.(9)withEq.(6)andmakethesim- ping. On the other hand, the population in the other
plifying assumption that diffraction over the ensemble Rydberg states 4 and 5 is negligibly small for weak
| (cid:105) | (cid:105)
lengthcanbeneglected,sothatthetransverseLaplacians fieldsΩ andΩ . Thedominantperturbationtothecon-
M L
can be dropped. Making a coordinate transformation version mechanism will thus stem from nearby Rydberg
fromthelaboratoryframe(t,z)toaframe(τ =t z/c,z) atoms in state 3 . In order to model this, we consider
− | (cid:105)
co-moving with the signal fields, the evolution equation a system of two atoms where atom A is located at the
4
(a) (b)
)
2 2
2 γ 2 b
γ / γ 2
2/M| 2ΩL| 2/L| 2/(γ
Ω | Ω M|
| 2b | Ω
|
z/labs z/labs
FIG. 2. Frequency conversion of stationary fields. Intensities of the millimeter wave (red) and optical field (blue) inside the
medium. DotsindicatetheresultsfromanumericalintegrationofMaxwell-Blochequations. (a)ACWmillimeterwaveenters
themediumatz=0. (b)ACWopticalfieldentersthemediumatz=0. In(a)and(b),thedashedlineisproportionaltothe
√
envelopee−2κz andwesetΓ/γ =1/285,Ω =2γ,Ω =2γ,Ω =2γ,Ω =0.3γ,∆ =2γ,∆ =2γ,∆ =2γ andb= 0.72.
A C R P 4 5 6
These parameters correspond to the realisation of the level scheme with Rubidium atoms in Sec. IIIB.
coordinate origin. The conversion process in atom A is duce the effective detuning parameters
disturbed by Rydberg-Rydberg interactions with atom
B, which is prepared in state 3 and positioned at R. ∆(cid:101)3 =∆3 ∆vdW, (17a)
| (cid:105) −
Next we discuss the two dominant effects caused by the
∆(cid:101)4 =∆4 ∆DD, (17b)
presence of atom B. First, atom B gives rise to a van der −
Waals shift of state 3 in atom A [37], and replace ∆ and ∆ in the general expression for the
| (cid:105) 3 4
matrix in Eq. (11) by ∆(cid:101)3 and ∆(cid:101)4. Since ∆vdW and
M
C ∆ depend on the relative position R, we average
(cid:126)∆vdW =−R66, (15) ovDerDR, M
(cid:90)
where the coefficient C6 depends on the quantum num- (cid:102)= d3R (R)w(R), (18)
M−→M M
bers of state 3 . If R is smaller than the blockade
| (cid:105)
radius R , atom A cannot be excited to the Rydberg
b where the distribution of nearest neighbours in a ran-
state and thus does not participate in the conversion.
dom sample of Rydberg atoms follows the probability
The blockade radius is determined by the single-atom
density [40],
EIT linewidth γ = Ω 2/γ and given by R =
EIT R b
[2C /((cid:126)γ )]1/6 [38]. S|econ|d, atom B gives rise to a (cid:18) (cid:19)2 (cid:34) (cid:18) (cid:19)3(cid:35)
6 EIT 1 3 R R
fre|qu|ency shift of state 4 in atom A via the resonant w(R)= exp , (19)
dipole-dipole interaction|[3(cid:105)9], 4πrws rws − rws
with the parameter
1 d 2 3d R(cid:126) 2
(cid:126)∆DD = 4πε | 43| −R|3 43· | , (16) (cid:20) 3 (cid:21)1/3
0 r = (20)
ws
4π
Ry
N
where R(cid:126) =R/R. In contrast to the van der Waals shift the Wigner-Seitz radius for a given density of Rydberg
in Eq. (15), ∆DD depends on the relative orientation of atoms Ry.
the two atoms. In principle, state 5 in atom A can ex- This aNccount of Rydberg-Rydberg interactions is ex-
| (cid:105)
perience a similar shift ∆DD if the dipole moment d53 is pectedtoworkwellforweakopticalandmm-wavefields.
different from zero. Here we assume that states 5 and IftheintensitiesofΩ andΩ areincreasedsuchthatthe
| (cid:105) M L
3 have the same parity so that d53 =0, consistent with population in 4 and 5 is not negligible, other dipole-
|th(cid:105)eexampleimplementationinRbthatweintroducebe- dipole interact|io(cid:105)ns can| o(cid:105)ccur that are not captured by
low in Sec. IIIB. our model. Furthermore, our model neglects cooperative
The preceding discussion shows that Rydberg-Rydberg effectslikesuperradiance[41]andfrequencyshiftsdueto
interactions change the energies of states 3 and 4 . a ground state atom within the electron orbit of a Ryd-
| (cid:105) | (cid:105)
In order to incorporate these frequency shifts into our bergstate[42]. However,experimentalresults[43–45]for
model, we find the general first-order solution of the EITinvolvingaRydbergstateshowthattheseeffectscan
atomic coherences in Eq. (9) for arbitrary detunings and be negligible for low principal quantum numbers n(cid:46)40,
Rabi frequencies of the auxiliary fields. We then intro- for weak probe fields and low atomic densities.
5
III. RESULTS
In a first step we analyse the simplified analytical model
of Sec. IIB in order to explain the principle of the con- %]
versionmechanism. ThisispresentedinSec.IIIAwhere [
F
we also investigate the maximally achievable conversion
efficiencies. We then introduce one possible implemen-
tation of our scheme in rubidium vapour in Sec. IIIB
andfindasetofparametersforwhichRydberg-Rydberg
interactions are negligibly small. Finally, we present nu-
mericalresultsforMMOCinthephysicalsystemsshown
D
c
in Figs. 1 (a) and (b) in Sec. IIIC.
FIG. 3. Conversion efficiency as a function of optical depth
A. Conversion mechanism D . WesetΓ/γ =3.9×10−3 (blackdottedline),Γ/γ =10−3
c
(redsolidline)andΓ/γ =3.8×10−4(bluedashedline). These
The conversion efficiency between mm-wave and optical parameters correspond to rubidium Rydberg states at zero
fields according to Eq. (12) will be small for a generic temperature with n ≈ 20, n ≈ 30 and n ≈ 40, respectively.
CommonparametersinallthreecurvesareΩ =2γ,Ω =2γ
matrix , but complete conversion can be achieved if A C
M and ∆ =2γ.
the atomic ensemble realises a beam splitter interaction 4
(cid:16) (cid:17)
V Ωˆ Ωˆ† +Ωˆ† Ωˆ , (21)
BS ∝ M L M L by Ω , Ω and Ω , respectively. However, these three
P M L
systems are coupled and hence the normal two-photon
where the ‘hat’ notation emphasises the operator nature
resonance condition for transparency of the Ω and Ω
of the fields. Formally, such an interaction corresponds M L
fields is changed. The conditions in Eq. (22) approxi-
to the case where the diagonal elements of vanish.
We find that this condition, such that χM MχL 0, matelyrestoretransparencyforthefieldΩM (ΩL)onthe
43 ≈ 61 ≈ transition 3 4 (1 6 )andinthepresenceofthe
can indeed be met if the intensities and detunings of the | (cid:105)↔| (cid:105) | (cid:105)↔| (cid:105)
otherlevelsandfieldssuchthatχM 0(χL 0). How-
auxiliary fields satisfy 43 ≈ 61 ≈
ever,Ω stillcreatesacoherenceontheopticaltransition
M
Ω 2 Ω 2 and ΩL induces a coherence on the Rydberg transition
C A
ΩR ΩP , ∆5 = | | , ∆6 = | | . (22) such that the fields are interconverted as they propagate
| |(cid:29)| | ∆ ∆
4 5 along the medium. With Eqs. (23) and (24) the gen-
To first order in Γ/γ the susceptibilities in Eq. (9) are eral solution for the spatial distribution of the fields in
then given by Eq. (12) is given by
χM i 8 ε2, χL α, (23a) Ω(z) e−κz(cid:18) cos(kz) ibsin(kz)(cid:19)Ω(0), (25)
43 ≈ b2γ 43 ≈ ≈ i sin(kz) cos(kz)
b
8
χM α∗, χL i ε , (23b)
61 ≈ 61 ≈ γ Γ whereκ=(ε2+ε )/l andk =ε/l determinetheloss
Γ abs abs
and the spatial oscillation period of the interconversion,
where
respectively, l = γ/(4η ) is the resonant absorption
abs L
Ω Ω∗ lengthonthe 6 1 transitionandweassumedε 1
α= C P , (24a) and ε /ε 1.| (cid:105)↔| (cid:105) (cid:28)
− ∆4Ω∗AΩR The sΓpatia(cid:28)l oscillations of optical and mm-wave intensi-
b γ Ω Ω
ε= | C|| P| , (24b) tiesaccordingtoEq.(25)areshowninFig.2. Ourmodel
4 ∆4 ΩA ΩR is in excellent agreement with a full numerical solution
| || || |
Γγ (cid:18) Ω 2(cid:19) of the Maxwell-Bloch equations. Note that the small de-
C
ε = 1+2| | . (24c)
Γ 16Ω 2 ∆2 viations for large z vanish if the approximations leading
| A| 4 to Eq. (25) are omitted. Complete MMOC occurs af-
εandε aredimensionlessparametersthataregenerally ter a length L = π/(2k), and thus requires an optical
Γ c
smaller than unity. Since ε Γ, ε is typically of the depth D = L /l that is inversely proportional to ε
Γ Γ c c abs
∝
order of ε2. On the other hand, α ε and hence the in Eq. (24b), D = π/(2ε). Since the value of ε can be
c
| | ∝
off-diagonal elements of the matrix are indeed much adjusted through the intensities and frequencies of the
M
larger than the diagonal elements. auxiliary fields, the condition for complete MMOC can
This result can be understood as follows. The level be met for various densities and sizes of atomic gases.
scheme in Fig. 1 can be regarded as three consecutive In the example in Fig. 2, we find L = 100l . The
c abs
EITsystemswheretheweakprobefieldsarerepresented efficiency F =e−2κLc for complete conversion can be ex-
6
(a) |ΩγM2|2 (b) b2|γΩ2L|2
tγ tγ
z/l z/l
abs abs
FIG. 4. Frequency conversion of pulsed fields. (a) Density plot of the incoming millimeter wave pulse with a Gaussian
envelope. (b) Density plot of the outgoing optical pulse. The parameters in (a) and (b) are the same as in Fig. 2.
pressed in terms of the optical depth D , to demonstrate this, we present numerical solutions of
c
the Maxwell-Bloch equations for a mm-wave input pulse
(cid:20) π2 (cid:21) as shown in Fig. 4. The intensity of a mm-wave in-
F(D )=exp exp[ 2ε D ] , (26)
c −2D − Γ c put pulse with Gaussian envelope is shown in Fig. 4(a),
c
and the corresponding optical output field is shown in
and F(Dc) is shown in Fig. 3 for three different values Fig. 4(b). The input pulse has a bandwidth on the order
of εΓ. The maximum efficiency Fmax = exp( 2π√εΓ) of∆ν 2π 80kHzandisconvertedwithoutdistortion
is attained at an optical depth Dcmax = π/(2√−εΓ) and of its ≈shape×. We thus find that the bandwidth of our
tendstounityforεΓ 0. SinceεΓ Γ,efficienciesclose conversionschemeisatleast 80kHzforthechosenpa-
to unity are only pos→sible because ∝of the slow radiative rameters. This bandwidth ca∼n be significantly increased
decay rate Γ of the Rydberg levels 3 , 4 and 5 . Γ by increasing the detunings and Rabi frequencies of the
decreases with increasing n as Γ n|−(cid:105)3 [4|6(cid:105)] and i|s(cid:105)thus auxiliary fields. Finally, we note that the conversion of
∝
typically several orders of magnitude smaller than the optical pulses to mm-waves works equally well.
decay rate γ of the low-lying states 2 and 6 . The effi-
| (cid:105) | (cid:105)
ciency for complete MMOC for the parameters in Fig. 2
is F 92.1%. B. Rubidium parameters
≈
Note that our definition of the efficiency is based on
photon fluxes and not intensities as required for a co- Here we discuss one possible realisation of our scheme
herent conversion scheme that conserves the total pho- based on an ensemble of 87Rb atoms. The atomic level
ton flux. In order to see this, we consider the perfectly scheme is shown in Fig. 5, where the optical field L cou-
coherent conversion of an optical field to a millimeter ples to the D line, and the auxiliary P field couples to
2
wave with F(Dc) = 1. According to Eq. (25), we ob- the D1 line. The transition dipole matrix elements for
tain ΩM(Lc)2 = b2 ΩL(0)2. With the definition of b in theopticaltransitionscanbefoundin[47], andfortran-
| | | |
Eq. (8) and the definition of the Rabi frequencies we get sitions between Rydberg states we follow the approach
described in [48]. The intensities of the auxiliary fields
ω
M(Lc)2 = M L(0)2, (27) arechosensuchthattheycorrespondtotheRabifrequen-
|E | ω |E |
L cies in Fig. 2, and the values of the detuning parameters
where and are the electric field amplitudes of the in Figs. 5 and 2 are also equivalent.
M M
mm-waEve andEoptical fields, respectively. The ratio of Next we show that Rydberg-Rydberg interactions are
the intensities (I 2) is thus Iout/Iin = ω /ω , as negligibleforthelevelschemeinFig.5andforanatomic
it should be. Simi∝lar|lEy,|we obtain IMout/ILin = ωM/ωL for densityof =2 1017m−3. FirstwenotethattheRyd-
L M L M N ×
the conversion of mm-waves into optical fields. berg blockade radius is Rb 0.63µm for the parameters
≈
Next we consider the conversion of pulsed fields. The of Fig. 5. This is significantly smaller than the mean
derivationofEq.(25)showsthatourschemeisnotmode- distance between atoms, and hence the density of Ryd-
selective and works for broadband pulses. The only re- berg atoms is simply given by Ry ρ33 [49, 50]. By
N ≈ N
quirement is that the atomic dynamics remains in the carrying out the average in Eq. (18), we find that the
adiabatic regime, which holds if the bandwidth δν of matrix (cid:102)leads to the same conversion efficiency as ,
M M
the input pulse is smaller than all detunings ∆ and the i.e., there is no notable difference between the curves in
k
Rabi frequencies Ω , Ω and Ω (see Sec. A). In order Fig.2generatedby andthecorrespondingcurvespro-
R C A
M
7
M
C
A
R
L
P
FIG.5. RealisationofthelevelschemeinFig.1basedontransitionsin87Rb. Allquantumnumbersoftheemployedstatesas
well as intensities, polarisations and detunings are indicated. Note that energy spacings are not to scale. The intensities and
detunings correspond to the parameters in Fig. 2. The decay rate γ = 2π×6.1MHz corresponds to the D line. We set the
2
decay rate Γ of all Rydberg states equal to the decay rate of the |23S (cid:105) state at T = 300K, which is faster than the decay
1/2 √
rate of the |24P (cid:105) state. We find [46] Γ/γ =1/285 and the ratio of the coupling constants is b= 0.72.
1/2
duced with (cid:102). On the other hand, if we choose 3 = hand,∆DD increasesbyafactorof100bychoosing 3 =
M | (cid:105) | (cid:105)
24S ,m = 1/2 instead of 3 = 23S ,m = 1/2 , 24S ,m = 1/2 instead of 3 = 23S ,m = 1/2 .
1/2 J 1/2 J 1/2 J 1/2 J
| (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:105)
the conversion efficiency drops to 61%. This explains why the conversion efficiency drops signifi-
In order to obtain more insight into these results, we cantlybyusingthestrong ns np transitioninstead
| (cid:105)↔| (cid:105)
considerthedistanceR where90%ofallRydbergatom of (n 1)s np as in Fig. 5.
90
| − (cid:105)↔| (cid:105)
pairs will have a larger separation than R , The absorption length for the field L is l = 5.1
90 abs
(cid:90) 10−2mm for our parameters. Since full conversion re×-
d3R w(R)=0.9. (28) quiresanopticaldepthof 100,thelengthofthemedium
∼
R>R90 needs to be Lc 5.1mm. These parameters are exper-
≈
Our parameters give 4.4 1015m−3 and thus imentally achievable. For example, much higher opti-
Ry
N ≈ × cal depths 1000 have been reported [51, 52], and the
R 1.79µm. The van der Waals shift between two
90 ∼
≈ atomic cloud size considered here is similar to the di-
atoms in state 3 and separated by R is [37]
90
| (cid:105) mensionsoftheexperimentin[52], wherecoldRbatoms
C6 were trapped in a cylindrical geometry of length 4.6mm
∆ = 2π 24.5kHz. (29)
vdW −R6 ≈ × and width 0.45mm.
90
This is much smaller than all detuning parameters and
Rabi frequencies entering the matrix . Since the fre-
C. Physical implementation
M
quency shifts for 90% of all atoms are even smaller, av-
eraging over all nearest neighbour distances does not
We first consider the setup in Fig. 1(a) where the mm-
change the matrix . Similarly, the dipole-dipole shifts
M wavefieldisfocussedintotheatomicensemblebylenses.
in Eq. (16) with 3 = 24S ,m = 1/2 are on the
| (cid:105) | 1/2 J (cid:105) We assume that the focal spot is at z = 0 such that the
order of
mm-wave beam profile is
1 3dˆ4 2
∆DD ∝ 4πε (cid:126)|(cid:104) |R3| (cid:105)| ≈2π×62.6kHz, (30) ΩM(z =0,r)=Ω(M0)e−r2/σM2 , (31)
0 90
(cid:112)
whichisalsosmallcomparedtothedetuningparameters where r = x2+y2 is the radial coordinate in the x-y
andRabifrequenciesoftheauxiliaryfields. Ontheother plane, σ is the beam waist and Ω(0) is the peak Rabi
M M
8
(a) (b)
22/γM| 22Ω/γL|
Ω| x/labs 2b| x/labs
z/l z/l
abs abs
FIG. 6. (a) Spatial intensity profile |Ω |2 of an incident mm-wave field in the x-z plane. The beam profile of the mm-wave
M
at z = 0 is given by Eq. (31). (b) Spatial intensity profile |Ω |2 of the resulting generated optical field in the x-z plane.
L
The fields in (a) and (b) are cylindrically symmetric, and the parameters are σ = 8l ≈ 413µm, σ = 1.9λ ≈ 509µm,
c abs M M
N(0) =2×1017m−3 and Ω(0) =10−4γ. All other parameters are specified in Sec. IIIB.
M
frequencyatthecenterofthebeam. Wemodelthetrans- which is consistent with our definition of the conversion
versedensityprofileoftheatomcloudbyaGaussianwith efficiencyinSec.IIIA.WefindF 26%fortheparame-
≈
peak density (0) and width σ , ters in Fig. 6, and this value can be further increased by
c
N
increasing the transverse size of the atomic cloud. For
(r)= (0)e−2r2/σc2, (32) example, for an atomic ensemble with transverse size
N N
σ 1mm we obtain F 61%. In addition, the con-
c
Inordertocalculatetheconversionefficiency,wefindthe vers≈ion of optical fields to≈mm-waves works equally well.
stationary solution of Eq. (6) with the boundary con- For a Gaussian optical beam of width σ 509µm and
L
dition in Eq. (31), the density profile in Eq. (32) and all other parameters as in Fig. 6, we find F≈ 24%. This
with the analytical expression for the atomic coherences value increases toF 72% ifthe atomicclo≈ud sizeis in-
in Eq. (23). The result for the parameters specified in creasedtoσ 1mm≈. However,increasingthetransverse
c
Sec. IIIB is shown in Fig. 6, where we consider a beam size of the ato≈mic ensemble requires auxiliary fields with
waist of σM = 1.9λM and an atomic cloud with trans- higher power in order to maintain the intensities shown
verse size σc 413µm. The intensity of the millimeter in Fig. 5.
≈
wave is shown in Fig. 6(a) and decreases due to the con-
Next we discuss the implementation shown in Fig. 1(b),
versionmechanism. Inaddition,itbroadensslightlywith
wherethemm-wavesareconfinedbyawaveguideandan
increasingz whichcanbeunderstoodasfollows. Forthe
elongated atomic cloud is trapped inside the waveguide
given parameters the Rayleigh length z = πσ2 /λ of
M M M core. This setting can be approximately described by
the mm-wave is z 11.3λ , which is about half the
M M the one-dimensional model in Eq. (10) if the ratio of the
≈
length of the medium. The broadening is thus caused
coupling constants in Eq. (8) is replaced by
by the strong focussing of the beam before it enters the
atomic ensemble. Note that the Rayleigh length is much A
larger than the wavelength λM, and hence the paraxial b2wg = ALb2, (35)
approximation is justified. The intensity of the optical M
wave is shown in Fig. 6(a) and increases with increasing
where A is the effective area of the mm-wave guided
M
z. In order to quantify the conversion efficiency, we con-
mode, and A is the transverse size of the optical beam
L
sider the total power of the incoming mm-wave and of
which is assumed to match the transverse density profile
the outgoing optical field,
of the atoms [35, 53]. In principle, the setup in Fig. 1(b)
can thus be employed to interconvert mm-waves with
∞
Pin = πε0c(cid:126)2 (cid:90) Ω (z =0,r)2rdr, (33a) longer wavelengths that cannot be focussed down to re-
M d 2 | M | alistic dimensions of cold atom clouds. However, since
34
| |
0 A /A 1, this results in smaller values of the param-
L M
∞ (cid:28)
πε c(cid:126)2 (cid:90) eter ε bwg definedinEq. (24)andthus inlargervalues
PLout = d0 2 |ΩL(z =Lc,r)|2rdr, (33b) of the∝optical depth required for complete conversion,
| 16| 0 Dc = π/(2ε). In order to achieve the required optical
depths, the atoms could be confined inside hollow core
where ε0 is the dielectric constant. We then define the fibres where extremely large optical depths have been
conversion efficiency by observed [54, 55]. In addition, mm-waves can similarly
beguidedbyphotoniccrystalfibres[56]. Thestrongcou-
ω Pout
F = M L , (34) pling of atoms with mm-wave and optical fields required
ωL PMin for efficient conversion might then be achievable by em-
9
bedding a small hollow-core photonic crystal fibre into a Appendix A: Atomic coherences
larger mm-wave photonic crystal fibre.
Here we derive the adiabatic solutions for the atomic co-
herences (cid:37) and (cid:37) in Eq. (9). To this end, we assume
43 61
IV. SUMMARY that the fields ΩM and ΩL are sufficiently weak and ex-
pand the atomic density operator as follows [57, 58],
We have shown that frequency mixing in Rydberg gases ∞
(cid:88)
enables the coherent conversion between mm-wave and (cid:37)= (cid:37)(k), (A1)
optical fields. Due to the numerous possibilities for k=0
choosing the 3 4 transition within the Rydberg
| (cid:105) ↔ | (cid:105) where (cid:37)(k) denotes the contribution to (cid:37) in kth order in
manifold, our proposed MMOC scheme enables the con-
the Hamiltonian
version of various frequencies ranging from terahertz ra-
diationtothemicrowavespectrum,thatisforfrequencies H = (cid:126)(Ω A +Ω A )+H.c.. (A2)
1 M 43 L 61
−
in the range 10 10,000 GHz. The degree of conver-
sion can be adjus−ted through the atomic density and the Thesolutions(cid:37)(k) canbeobtainedbyre-writingthemas-
ancillary drive field intensities and frequencies. Conver- ter equation (1) as
sion efficiencies are limited by the lifetime of the Ryd-
i
berg levels and dipole-dipole interactions between Ryd- (cid:37)= (cid:37) [H ,(cid:37)], (A3)
L L0 − (cid:126) 1
berg atoms. Imperfections due to Rydberg interactions
canbeminimisedinensembleswithlowatomicdensities wherethelinearsuper-operator 0 isindependentofΩM
L
andbythechoiceoftheatomicstatesandparametersof andΩL. Insertingtheexpansion(A1)intoEq.(A3)leads
the auxiliary fields. We have analysed a realistic imple- to the following set of coupled differential equations
mentationfortheinterconversionofterahertzandoptical
(cid:37)˙(0) = (cid:37)(0), (A4)
fields with an ensemble of trapped rubidium atoms, and L0
find that the conversion efficiency can exceed 90%. (cid:37)˙(k) = (cid:37)(k) i[H ,(cid:37)(k−1)], k >0. (A5)
Efficient conversion requires a large spatial overlap be- L0 − (cid:126) 1
tween the mm-wave and optical fields, and we have Equation(A4)describestheinteractionoftheatomwith
discussed two possible scenarios how to achieve this. the fields Ω , Ω , Ω and Ω to all orders and in the
P R C A
First, we have considered focussed terahertz beams and absence of H . Higher-order contributions to (cid:37) can be
1
found that high conversion efficiencies are possible if the obtainedifEq.(A5)issolvediteratively. Equations(A4)
Rayleighlengthofthebeamsiscomparabletothelength and(A5)mustbesolvedundertheconstraintsTr((cid:37)(0))=
of the atomic cloud. Second, we investigated a setup 1 and Tr((cid:37)(k))=0 (k >0).
wherethemm-wavefieldsaretransversallyconfinedbya The zeroth-order solution (cid:37)(0) is the EIT dark state of
waveguide and the atoms are trapped inside the waveg- the three-level ladder system 1 , 2 and 3 . For the
uide core. The optical depth required for complete con- specialcase∆ =0andifthes|m(cid:105)all|d(cid:105)ecayra|te(cid:105)Γofstate
(cid:112) 3
versionincreasesby AM/ALcomparedtothefree-space 3 is neglected, we find
implementation, where A is the effective area of the | (cid:105)
M
mm-wave guided mode and AL is the transverse size of (cid:37)(0) = |ΩR|2 , (A6a)
the atomic cloud. This waveguide setting enables high 11 Ω 2+ Ω 2
P R
| | | |
conversion efficiencies close to the theoretical limit set Ω 2
by the lifetime of the Rydberg states and Rydberg inter- (cid:37)(0) = | P| , (A6b)
33 Ω 2+ Ω 2
actions. | P| | R|
Ω∗Ω∗
(cid:37)(0) = P R . (A6c)
13 − Ω 2+ Ω 2
P R
| | | |
ACKNOWLEDGMENTS For Ω Ω thesteadystateisreachedwithinseveral
P R
| |(cid:28)| |
inverse decay times 1/γ.
In general, we obtain the zeroth-order solution (cid:37)(0) for
We thank the National Research Foundation and the
∆ =0 and substitute it in the first-order equation (A5)
Ministry of Education of Singapore for support. The 3
(cid:54)
with k =1. The formal solution of this differential equa-
research leading to these results has received funding
tion is given by
fromtheEuropeanResearchCouncilundertheEuropean
Union’s Seventh Framework Programme (FP7/2007- i
2013)/ERC Grant Agreement no. 319286 Q-MAC, (cid:37)(1)(t)=(cid:126)L−01[H1(t),(cid:37)(0)]
and from the UK EPSRC through the standard grant
t
EanPd/Jt0h0r0o0u5g1h/1t,hethHeubprofogrramNemtweogrkraedntQEuPan/Ktu0m344In8f0o/r1- − (cid:126)iL−01(cid:90) dt(cid:48)eL0(t−t(cid:48))∂t(cid:48)(cid:16)[H1(t(cid:48)),(cid:37)(0)](cid:17),
mation Technologies (NQIT). JN acknowledges support 0
(A7)
from a Royal Society University Research Fellowship.
10
where we assumed H (0)=0. If H (t) varies sufficiently find that this condition is satisfied if all detunings ∆
1 1 k
slowlywithtime, thesecondterm ontheright-handside (k 4,5,6 ) and the Rabi frequencies Ω , Ω and Ω
R C A
∈ { }
in Eq. (A7) involving the time derivative of H can be are large as compared to the bandwidth δ . In general,
1 ν
neglected. More precisely, this approximation is justi- the analytical expression for the first-order density oper-
fied if the bandwidth δ of the pulses Ω and Ω is ator (cid:37) is too bulky to display here. A special solution if
ν M L
small as compared to the relevant differences between the conditions in Eq. (22) are met is given in Eq. (23).
eigenfrequencies of H . Through a numerical study we
0
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