Production of entanglement in Raman three-level systems using feedback

0.87 MB·English

by  R. N. Stevenson

Production of entanglement in Raman three-level systems using feedback
R. N. Stevenson and A. R. R. Carvalho
Department of Quantum Science, Research School of Physics and Engineering,
The Australian National University, ACT 0200, Australia
J. J. Hope
Australian Centre for Quantum-Atom Optics, Department of Quantum Science,
Research School of Physics and Engineering, The Australian National University, ACT 0200, Australia
(Dated: January 12, 2010)
We examine the theoretical limits of the generation of entanglement in a damped coupled ion-
cavity system using jump-based feedback. Using Raman transitions to produce entanglement be-
tween ground states reduces the necessary feedback bandwidth, but does not improve the overall
effectofthespontaneousemissiononthefinalentanglement. Wefindthatthefidelityoftheresult-
0 ing entanglement will be limited by the asymmetries produced by vibrations in the trap, but that
1 the concurrence remains above 0.88 for realistic ion trap sizes.
0
2 PACSnumbers: 03.67.Mn,42.50.Lc,03.65.Yz

n
a
J I. INTRODUCTION be achieved much quicker, with less time spent in the
light state as the transition to the dark state is forced
0
1 The production and manipulation of entangled states by the feedback, rather than switched through sponta-
neousemission. Wewillalsoshowthat,althoughtheRa-
has been a major feature of quantum information re-
] man scheme produces entanglement between metastable
h search in the last several years. The efforts in this di-
levels, which have orders of magnitude smaller decay
p rection were rewarded with extraordinary experimental
- advances that led to the realisation of entangled states ratesthanopticaltransitions,thisdoesnottranslateinto
t higher or more stable entanglement in our scheme while
n in a variety of physical systems [1–5], including entan-
a glementinvolvingmultipleparticles[6,7],andlong-lived feedback is on. In fact, the effect of spontaneous emis-
u sion from the upper level used for the Raman transition
entangled states [8, 9]. The latter are achieved either by
q can be detrimental enough to put optical and Raman
reducingtheexperimentalimperfectionsandundesirable
[
schemes at the same level in terms of entanglement pro-
interactions with the environment that are responsible

1 for entanglement deterioration, or by encoding entangle- duction. However, the Raman scheme will be shown to
v significantly reduce the necessary feedback bandwidth.
ment in a decoherence-free subspace [10, 11]. A promis-
7 The paper is organised as follows: In Section II we de-
ing approach to deal with the decoherence problem is
6
scribe the Raman model and derive the equivalent two-
the use of active quantum feedback control [12–15]. In
5
level description after the elimination of the upper level.
1 fact, quantum feedback has been recently proposed and
Weestablishtheequivalencebetweenthismodelandthe
. used to improve entanglement production and stability
1 one in [19] in the absence of spontaneous emission, and
in both continuous [16, 17] and discrete [18–22] variable
0
presenttheimportanteffectiveparametersfortheentan-
0 systems.
glement dynamics. In Section III we analyse the effect
1 Recently, two of us have shown that direct feedback
: under the appropriate detection strategy leads to the ro- of spontaneous emission, while in Section IV we include
v important experimental limitations as the delocalisation
bust production of highly entangled states of two atoms
i
X or ions in a cavity [19, 20]. Motivated by the perspec- oftheionswithinthestandingwaveofthecavityandde-
tection inefficiency. The latter seems to be an important
r tiveofexperimentalimplementations,andthepossibility
a limiting factor for current experimental parameters.
of improving even further the proposed scheme, in this
paper we analyse the use of Raman transitions in place
of the optical dipole ones considered in [19, 20]. A Ra-
man scheme has also been considered by Metz et al. in II. MODEL
a different approach to generate entanglement through
thedetectionofmacroscopicquantumjumps[23]without The systems consists of a pair of atoms (or ions) with
feedback. Theentanglementisconditionedonthedetec- the Raman level configuration shown in Fig. 1. The
tion of certain spontaneous emission events that transi- two target levels are labelled 0 and 1 , and have en-
tionthesystemfromalighttoadarkstate. Moreover,as ergies (cid:126)ω and (cid:126)ω respective|ly(cid:105), with|th(cid:105)e intermediate
0 1
in[19,20],entanglementisheraldedasprolongedperiods upper level labelled 2 , with energy (cid:126)ω . The parti-
2
| (cid:105)
ofnocavityphotondetectionindicatethattheentangled cles are coupled to a single mode of an optical cavity
state has been prepared. which is detuned from the 0 2 transition by ∆, i.e.
| (cid:105)→| (cid:105)
In this paper we will show that the use of feedback ω =(ω ω ) ∆. Thetransitionfromlevel 1 tolevel
C 2 0
− − | (cid:105)
in the Metz et al. setup will allow the desired state to 2 is pumped using light detuned by ∆ δ from reso-
| (cid:105) −
2
nance, i.e. of frequency ω =(ω ω ) (∆ δ), where ω
L 2− 1 − − M
δ << ∆, and with a coupling strength V . This laser
L
pumps the internal levels directly, and does not pump
the cavity mode. The transition from level 0 to level
| (cid:105) 1 2 D
1 is pumped using microwaves detuned from resonance
| (cid:105)
by δ, with coupling strength V , or equivalently using a
M
differentRamantransition,asin[24]. Weneglectsponta-
neous emission for now, but will include it further down. ω U
L fb
Here the particles are assumed to have constant and
equal cavity coupling strengths g. In section IVA, the
position of the particles will be allowed to vary in time
FIG. 2: A diagram representing the model. The
and therefore the coupling strengths with the cavity
particles are pumped from the side of the cavity.
mode, g (t) and g (t), will be time-dependent and dif-
a b Light leaks from the cavity at a rate κ and is de-
ferent for each particle.
tected using a photodetector. When a photon is
detected a finite evolution represented by the uni-
2 tary matrix Uˆ is applied.
fb
| !
∆
tors of the cavity mode, respectively. is the decoher-
D
ence superoperator
ω ω (cid:2)cˆ(cid:3)ρˆ=cˆρˆcˆ† 1(cid:2)cˆ†cˆρˆ+ρˆcˆ†cˆ(cid:3), (3)
C L D − 2
and Uˆ is a unitary matrix describing the effect of the
control Hamiltonian Fˆ applied for a finite time δt:
ω δ
M
(cid:34) (cid:35)
1 iFˆδt
0 | ! Uˆ =Exp (cid:126) . (4)
| !
FIG.1: ARamanlevelscheme. Electronsareexcited Following the results in [19], we use an asymmetric form
fromlevel|0(cid:105)tolevel|1(cid:105)byatwophotontransitionvia of control, acting on either of the particles. We choose
level |2(cid:105). The transition with frequency ωL is excited Fˆ =α(|1(cid:105)1(cid:104)2|+|2(cid:105)1(cid:104)1|),i.e. Fˆ =α(σ1−+σ1+)onthefirst
byalaser,thetransitionwithfrequencyω isexcited particle, where α represents the strength of the control.
C
by the cavity mode and a direct transition between In the regime of large detuning, i.e.
level |0(cid:105) and |1(cid:105) transition with frequency ω could
M
be directly excited by microwaves. ∆>>V ,g,κ>V , (5)
L M
Light that escapes through one of the cavity mirrors wecanadiabaticallyeliminatetheupperlevelandobtain
is monitored using a photodetector. When a photon is aneffectivetwo-leveldynamicsgivenbytheHamiltonian
dsyestteecmtedu,sainfignciotentarmololuansetrosfoervomluictrioownaivseism. pAossecdheomnatthiec Hˆ2 = (cid:126)VLg(cid:0)Aˆ0,1aˆ†+Aˆ1,0aˆ(cid:1)
− 2∆
of this feedback setup is shown in Fig. 2.
Themasterequationforthissystem,intheinteraction (cid:126)g2aˆ†aˆAˆ + (cid:126)VM(cid:0)Aˆ +Aˆ (cid:1)
0,0 0,1 1,0
picture, using the rotating wave approximation, is − ∆ 2
ρˆ˙ = i (cid:104)Hˆ,ρˆ(cid:105)+κ (cid:2)Uˆaˆ(cid:3)ρˆ, (1) + (cid:126)4V∆L2(cid:0)Aˆ1,1−Aˆ0,0(cid:1)−(cid:126)δ(cid:0)Aˆ1,1−Aˆ0,0(cid:1). (6)
−(cid:126) D
Notethatbychoosingtheseconddetuningappropriately,
where Hˆ is the Hamiltonian δ = VL2, the last two terms cancel. Now all the terms in
4∆
Hˆ = (cid:126)δAˆ1,1+(cid:126)∆Aˆ2,2 the Hamiltonian are on the order of ∆1 or smaller, so the
−+ (cid:126)VL(cid:0)Aˆ +Aˆ (cid:1) coafvtihtye sdyesctaeymr,aatnedκthisetchaeviftaystmesotdreatsepeinndtshmeodsytnoafmtihces
1,2 2,1
2
timeinthezerooccupationstate. Thismeansthecavity
+ (cid:126)VM(cid:0)Aˆ +Aˆ (cid:1) mode can also be adiabatically eliminated in a similar
1,0 0,1
2 way to [18]. The resulting master equation after elimi-
+(cid:126)g(cid:0)Aˆ0,2aˆ†+Aˆ2,0aˆ(cid:1), (2) nating the cavity is
wanitdhaˆthaendopaˆe†rabteoinrsgAtˆhedeafinnneidhilaastiAoˆni,jan=d|ci(cid:105)r1e(cid:104)ajt|io+n|oip(cid:105)2e(cid:104)rja|-, ρ˙ = −i2VM (cid:104)(cid:16)Aˆ0,1+Aˆ1,0(cid:17),ρ(cid:105)+ V∆L22gκ2D[UˆAˆ0,1]ρ. (7)
3
This is completely equivalent to the feedback equation different choices of jump operators can be interpreted as
for a two-level system derived in [19], but with an effec- differentwaystomonitorthesystemintheframeworkof
tive cavity coupling constant of G = VLg. The system quantum trajectories [25, 26]. In Eq.(9), we followed ref-
eff 2∆
will therefore evolve towards the maximally entangled erence [24] and broke up the spontaneous emission into
Bell state a01 = √1 (01 10 ), as shown previously four jump terms Ri,j
| (cid:105) 2 | (cid:105)−| (cid:105)
in [19, 20]. This jump-based feedback has been shown
(cid:114)
to be fairly robust against spontaneous emission effects R = γ0VL2 (00 s + s 11)
as well as to detection inneficiencies. In the next sec- 0,s 4∆2 | (cid:105)(cid:104) 01| | 01(cid:105)(cid:104) |
tions we will investigate the impact of these phemomena (cid:114)γ V2
on the Raman system, as well as looking at the effect R0,a = 40∆L2 (|00(cid:105)(cid:104)a01|−|a01(cid:105)(cid:104)11|)
of imperfect trapping of the particles within the cavity
(cid:114)
mode. R = γ1VL2 (a a + s s +211 11)
1,s 8∆2 | 01(cid:105)(cid:104) 01| | 01(cid:105)(cid:104) 01| | (cid:105)(cid:104) |
(cid:114)
γ V2
III. SPONTANEOUS EMISSION EFFECTS R = 1 L (s a a s ). (11)
1,a 8∆2 | 01(cid:105)(cid:104) 01|−| 01(cid:105)(cid:104) 01|
Spontaneous emission in a Raman system is signifi- This choice reflects the cooperative effects in the fluores-
cantlydifferentfromspontaneousemissioninatwo-level cence[27]ifonecouldmonitorthequantumjumpsusing
system. Therateofemissionbetweenthetwolowerlevels an electron shelving technique [28]. In [23], Metz et al.
0 and 1 is negligible, as the energy difference between consideredtheevolutionofthesystemconditionedonthe
| (cid:105) | (cid:105)
them is small. The only significant spontaneous emis- observation of cavity photons and the spontaneous emis-
sion is from the upper level 2 to the other two levels. sion jumps in Eq.(11) to show that entanglement can be
| (cid:105)
The rates of spontaneous emission are γ0 from 2 to 0 , producedinthissystem,butwithnofeedback. Theirre-
| (cid:105) | (cid:105)
and γ1 from 2 to 1 , with γ0 +γ1 = γ. We choose sults show that some spontaneous emission events bring
| (cid:105) | (cid:105)
the parameter regime given in Eq.(5), so that the rate of thesystemtoan entangleddark state, namelythe maxi-
spontaneous emission is on the order of the other main mally entangled antisymmetric Bell state a , and that
01
| (cid:105)
rates in the system. thesystemchangesfromdark(entangled)tolight(unen-
The full master equation including the spontaneous tangled)periodsaccordingtothemeasurementevents. It
emission is wasshownthatthesystemspendsapproximatelyaquar-
ρˆ˙ = i (cid:104)Hˆ,ρˆ(cid:105)+κ (cid:2)Uˆaˆ(cid:3)ρˆ+ (cid:88) (cid:88) γ (cid:18) (cid:2)j 2(cid:3)ρˆ(cid:19), ter of the time in the dark state.
−(cid:126) D j D | (cid:105)i(cid:104) | We have investigated the effect of feedback in this sit-
i=1,2j=0,1 uation and the results are shown in Fig. 3. The entan-
(8)
glement (as measured by concurrence [29]) and jumps
with the Hamiltonian Hˆ given by Eq.(2). One can per- areshownforthecaseswith(bottom)andwithout(top)
form the adiabatic elimination of the upper level and feedback. In the simulations without feedback there is
the cavity mode in the same way it was done in Section a clear distinction between the dark state, where there
II. In these calculations it is useful to consider a differ- arenocavityemissionsandtheconcurrenceis1,andthe
ent set of basis states: The symmetric states 00 , 11 light state, where there are many cavity emissions and
and s01 = √1 (01 + 10 ), and the antisymme|tric(cid:105)st|ate(cid:105) the concurrence oscillates rapidly, much faster than the
| (cid:105) 2 | (cid:105) | (cid:105)
a01 = √1 (01 10 ). These states are chosen because samplingrate. Whenfeedbackisturnedon,theduration
| (cid:105) 2 | (cid:105)−| (cid:105) ofthe darkperiodsstaysthe samebut thelengths ofthe
the Hamiltonian and cavity emission terms are symmet-
light periods are vastly reduced, and the system spends
ric in the two ions, so they don’t couple between the
a greater portion of the time in the desired state.
symmetric and antisymmetric subspaces. We can write
the final master equation as With the feedback on, the entanglement is robust
enough that the system doesn’t need to be conditioned
ρˆ˙ = i (cid:104)Hˆ ,ρˆ(cid:105)+ VL2g2 (cid:2)UˆAˆ (cid:3)ρˆ on the monitoring of spontaneous emission events. Mon-
−(cid:126) red ∆2κD 0,1 itoring particular kinds of spontaneous emission events
(cid:18) (cid:19)
(cid:88) (cid:88) (cid:2) (cid:3) would have additional experimental complications and
+ R ρˆ , (9)
D i,j a finite quantum efficiency, which that could degrade
i=s,aj=0,1 the conditional entanglement, so achieving high entan-
where Hˆ is the adiabatically reduced Hamiltonian as glement without this process is a useful design feature.
red
in Eq.(7): Figure 4 shows simulations for the system when the evo-
lution is conditioned only on the cavity emissions, and
(cid:126)V (cid:16) (cid:17)
Hˆ = M Aˆ +Aˆ . (10) the spontaneous emission is averaged out. We used the
red 0,1 1,0
2 experimental parameters for single atoms in a cavity
Notethatthedecoherencetermcanbewritteninmany from [30] ( g,κ,γ = 2π 1.61,0.054,11.1 MHz) (top
{ } ×{ }
different ways that are equivalent when it comes to solv- plot), and from [31] ( g,κ,γ = 2π 10,0.4,2.6 MHz)
{ } ×{ }
ing the full master equation. Note also, however, that (bottom plot). The detuning ∆ was chosen to be large
4
SE a1 1
SE a0
mppmSE s1 0.8
Juu e
JSE s0 c
n
Cavity e0.6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 r
Time(inunitsof1/2πg) x 105 ur
c
n0.4
1 o
e C
c
n
urre0.5 0.2
c
n
o
C 0
0 0 0.2 0.4 0.6 0.8 1
0 0.5 1 1T.5ime(2inun2i.t5sof31/2πg3.)5 4 4.5 5x 105 Time (seconds) x 10−3
1
SE a1
SE a0
JumppmuJSSEE ss01 ence00..68
r
Cavity0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ur
Time(inunitsof1/2πg) x 105 nc0.4
o
e 1 C
c
n 0.2
e
urr0.5
c
on 0
C 0 0.2 0.4 0.6 0.8 1
00 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (seconds) x 10−4
Time(inunitsof1/2πg) x 105
FIG. 4: (Color online) Simulations of the system with the
FIG.3: (Coloronline)Singlerunsofsinglepathsimulations. evolutionconditionedonthecavityemissiononly. Thesesim-
The top pairs of panels show a run with no feedback, while ulationsuseparameterstakenfromRusso[30](top)andKhu-
the bottom pair of panels show the same with feedback. For daverdyan[31](bottom),withadetuningbigenoughtobein
each of these, the bottom panel shows the concurrence while the adiabatic regime.
the top panel shows the occurrence of jumps. The bottom
line of jumps is the cavity emission jumps and the rest are
spontaneous emission events, with the label corresponting to
lations of the other two levels 0 and 1 as part of the
the jumps in Equations 11. These simulations use the fol- | (cid:105) | (cid:105)
adiabatic elimination process. This leads to an effective
lowing parameters: g = κ = V , γ = 0.1g, V = 0.05 and
L M spontaneous emission rate of
∆=50g, with γ =γ = g/20. The feedback Hamiltonian is
1 2
Fˆ˜ = Fˆ(cid:126)δt = π2σˆ1 γ = VLγ (12)
eff 4∆2
thatcanbeadjustedbychangingthedetunigofthepump
enough that the parameters were within the two adia-
laser and cavity mode.
batic regimes described above, with ∆ = 6GHz (top)
Toseewhetherthereducedeffectivespontaneousemis-
and ∆ = 0.5GHz (bottom). These figures are similar
sion rate due to the Raman transition is useful, we com-
to those conditioned on both the cavity emissions and
pare this ratio to the equivalent ratio in a two-level sys-
the spontaneous emissions, however the steady state is
tem [19, 20]. The rate of feedback in the two-level sys-
no longer maximally entangled. If one averages out also
the cavity decay, one recovers the solution of the master tem is proportional to g2 so the entanglement is max-
κ
equation (9) as shown in Fig. 5. imised when g2, which is the single atom cooperativity,
γκ
ismaximised. TakingtheequivalentratiofortheRaman
scheme, one has, as the new figure of merit,
A. Raman versus Optical scheme
G2 (VLg)2 g2
eff = 2∆ = . (13)
As can be seen from the results of the previous sec- κγeff VL2γκ γκ
4∆2
tion, the important factor that determines the amount
ofentanglementistheratiobetweenthetimethesystem Unfortunatelythisshowsthat,althoughtherateofspon-
spendsinthemaximallyentangledstatebeforeitdecays taneous emission is decreased as the detuning increases,
through spontaneous emission to the time the feedback therateoffeedbackdecreasesbythesamefactor,sothat
takes to restore the entangled state after it has decayed. the Raman scheme does not offer advantages over the
The rate at which spontaneous emission occurs is pro- optical two-level scheme in terms of maximum entangle-
portional to the population of level 2 . The population ment produced. It can, however, present some practical
| (cid:105)
of this level can be calculated as a function of the popu- advantages. In the optical scheme, the rate at which
5
1 usually being on the order of micrometres. Due to their
strong interaction with electric fields, ions are a prime
0.8 candidate for tight trapping, and current ion trapping
e
nc techniques are able to trap an ion to a little less than an
re0.6 optical wavelength. In [32], 40Ca+ ions are trapped to
r
u within 70nm. This is not a lot smaller than the distance
c
n0.4
o between two nodes of the standing wave, so the change
C
in coupling strength as the ion moves in the trap will
0.2
be significant. A similar problem would be present in
experiments with neutral atoms.
0
0 0.2 0.4 0.6 0.8 1 The reduced master equation when the coupling con-
Time (seconds) x 10−3 stants of the two atoms are allowed to vary is (without
1 spontaneous emission for now),
0.8 (cid:126)V (cid:104)(cid:16) (cid:17) (cid:105)
nce ρˆ˙ = 2M Aˆ0,1+Aˆ1,0 ,ρˆ
e0.6
urr + VL2 [Uˆ (g (t)σˆ−+g (t)σˆ−]ρˆ, (14)
nc0.4 ∆2κD fb 1 1 2 2
o
C
where g (t) are the effective cavity coupling constant for
0.2 i
each atom. They vary as g (t) = g cos(2πxi(t)), with
i max λ
0 x (t) being a Gaussian random position centred at zero
0 0.2 0.4 0.6 0.8 1 i
Time (seconds) x 10−4 withstandarddeviationofthetrap. Thiscorrespondsto
thecentreofthedistributionbeingatanantinodeofthe
standing wave. Figure 6 shows the simulation of Eq.(14)
FIG. 5: (Color online) Entanglement unconditioned on ei-
with the random position re-rolled at each time step.
ther spontaneous emission or cavity decay. With no feed-
back the concurrence is zero, but with feedback concurrence
is close to unity. These simulations use parameters taken 1
fromRusso[30](top)andKhudaverdyan[31](bottom),witha
detuning big enough to be in the adiabatic regime.
0.8
e
c
jumps, and hence control, occur is fixed by the atom- n
e0.6
cavity coupling and the cavity decay rate. If this jump r 0.84
r
rateistoolow,entanglementtakeslongertobuild,while u
c 0.82
if it is too fast, one could reach limitations in the band- n0.4
o
widthofeitherthephoto-detectorortheelectronicscon- C 0.8
6000 8000 10000
trollingthefeedbackpulses. Ontheotherhand,thefeed-
0.2
back rate in the Raman scheme can be changed by the
detuning or laser power, allowing full control over the
speed of the process. 0
0 2000 4000 6000 8000 10000
Time (in units of 1/2πg)
IV. EXPERIMENTAL LIMITATIONS
FIG. 6: (Color online) The concurrence of the two atom
system with the atoms trapped in a Gaussian distribution
A. Delocalised Particles in a Cavity Standing Wave
with standard deviation of 0.08 λ, which approximately cor-
responds to the trapping of an ion in Reference [32]: a 70nm
Anotherissuewiththeseschemesinvolvesthecoupling trapwithinastandingwavewithwavelengthλ=866nm. The
of the atoms to the optical mode of the cavity. These solid line shows the average concurrence over 100 runs, with
modelsrequirethecouplingstrengthsofthetwoatomsto the inset showing the maximum and minimum concurrence
beequal, asthisresultsintheantisymmetricstatebeing over these 100 runs with dashed lines.
a dark state of the system. The atoms are coupled to
standing waves in the cavity, so the coupling strength is ThereductioninconcurrenceobservedinFig. 6canbe
proportional to the amplitude of the standing wave (per better understood in terms of the conditional dynamics
photon)andassuch,variesfrommaximumtozeroinone based on the detection of photons leaving the cavity. As
quarter of a wavelength, typically a couple of hundred discussed in [20], the state √1 (eg ge ) is the steady
2 | (cid:105)−| (cid:105)
nanometres. The field strength varies more gradually state of the system when there is no spontaneous emis-
in the transverse direction, with the waist of the beam sion and the two coupling strengths are constant. When
6
thecouplingstrengthsofthetwoatomsdiffer, asymmet- This is simply saying that when the photon is detected
ric elements are introduced into the cavity emission dy- bythephotodetector(thetermwiththeη prefactor)the
namics. This stops the antisymmetric state being a dark feedback is applied, and when the photon is not detec-
state,andthestatecannowloseexcitationsthroughcou- tected (the term with the (1 η) prefactor) no feedback
−
pling with the environment (decay of the cavity mode), is applied. A cavity emission event is essentially a lost
andinthiswaytheconcurrencedrops. Thiscavityemis- opportunity to apply feedback, so the photodetector ef-
sion projects the ions into the light, unentangled state. ficiency is a scaling factor on the rate at which feedback
ThiscanbeseeninFig.7,wherethereisnospontaneous is applied.
emission, buttherearecavityemissioneventsevenwhen
thesystemisinthemaximallyentangledanti-symmetric
Asummaryoftheeffectsofdetectioninefficienciesand
state.
cavity coupling fluctuations can be seen in Fig. 8. The
steady-state concurrence for a range of trap sizes and
spontaneousemissionratesrunningthefullmasterequa-
1
tion simulations is shown. For reasonable parameters,
e γ = 0.1g and with ions trapped within .08 of a wave-
c0.8
n length, the concurrence is above 0.88. As the trap size
e
rr0.6 increases past 0.1 wavelengths the concurrence rapidly
u
c drops.
n
o0.4
C
0.2
0
0 1 2 3 4 5 6 7 8 9 10
Time (in units of 1/2πg) x 104
FIG.7: (Coloronline)Theconcurrenceandcavityemissions
fora systemwithnospontaneousemission and atrapsizeof
0.2 wavelengths. The solid line shows the concurrence, while
the circles show the times when photons leaked out from the V. CONCLUSIONS
cavity.
The mean time before a cavity emission from the dark
In this paper we have shown that a feedback control
state a is the inverse of the rate of the decay:
| 01(cid:105) scheme to generate entanglement using Raman three-
level atoms is mostly equivalent to a system using two-
T−1 = VL2 a (cid:0)g σ++g σ+(cid:1) level optical transitions. In the regimes where cavity
dark κ∆(cid:104) 01| 1 1 2 2 × mode and upper level can be adiabatically eliminated,
(cid:0)g σ−+g σ−(cid:1) a the final entanglement depends on the cooperativity pa-
1 1 2 2 | 01(cid:105)
2κ rameter that is the same for Raman or optical schemes.
Tdark = g2 g2. (15) While this equivalence is true while feedback is on, the
1 − 2 Raman system has two main advantages. First, the nec-
This is the mean time if the cavity coupling rates are
essary feedback bandwidth can be experimentally con-
kept constant. As the coupling constants vary, the rate
trolled, rather than having to operate at the lifetimes
of decay also varies. For the ions to remain in a highly
of optical transitions. Second, the Raman scheme has a
entangledstate,thistimescalemustbemuchlongerthan
higher entanglement lifetime if feedback is turned off.
the timescale over which the feedback drives the ions to
the dark state.
Feedback prepares unconditioned entangled states
withconcurrencegreaterthan0.88evenwhentakinginto
B. Non-unity cavity emission account a realistic delocalisation of the atoms within the
cavitymode. Entanglementcanbeevenhigher,andher-
alded, if the system is conditioned on the detection of
Theeffectofhavingphotodetectorswithoutperfectef-
photons leaving the cavity. Reliably produced pairs of
ficiencyisthesameintheRamansystemasitisinatwo
entangled states could be used to make entangled states
level system. When the photodetector has an efficiency
of more particles [33], with an ultimate aim of produc-
η, the master equation (9) is transformed into [20]
ing large cluster states for measurement based quantum
computing. This opens the perspective to produce mul-
i(cid:104) (cid:105)
ρˆ˙ = − Hˆ,ρˆ tipartiteentangledstatesdirectlythroughasimilarfeed-
(cid:126)
back setup, by choosing appropriate measurement and
+η VL2 [Uˆ (g (t)σˆ−+g (t)σˆ−]ρˆ controls.
κ∆2D fb 1 1 2 2
V2
+(1 η) L [(g (t)σˆ−+g (t)σˆ−]ρˆ
− κ∆2D 1 1 2 2
(cid:18) (cid:19)
(cid:88) (cid:88) (cid:2) (cid:3)
+ R ρˆ . (16)
i,j
D
i=s,aj=0,1
7
e 1
c
n
e
r
r0.5
u
c
n
o
C 0
0
2 0
γκ 0.05
g2 4 0.15 0.1
6 0.2 Trap size/wavelength
e 1
c
n
e
r
r0.5
u
c
n
o
C 0
0
2 0
0.05
γκ 4 0.1
g2 0.15
0.2
6 Trap size/wavelength
FIG. 8: (Color online) The top shows the steady state con-
currence of the two atom system as a function of the size of
the trap and the spontaneous emission rate. The x axis is a
measure of the standard deviation of the particles as a frac-
tionofthecavitymodewavelengthλ. Theyaxisisthetotal
spontaneous emission γ rate as a fraction of g. The z axis is
the steady state concurrence. The bottom figure shows the
same with a detector efficiency of 50%.
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