Exciting with Quantum Light. I. Exciting an harmonic oscillator

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by  J. C. López Carreño

Exciting with Quantum Light. I. Exciting an harmonic oscillator.
J. C. L´opez Carren˜o1 and F. P. Laussy2,1
1Departamento de F´ısica Teo´rica de la Materia Condensada,
Universidad Auto´noma de Madrid, 28049 Madrid, Spain
2Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia
(Dated: September 6, 2016)
We start a series of studies of the excitation of an optical target by quantum light. In this
first part, we introduce the problematic and address the first case of interest, that of exciting
the quantum harmonic oscillator, corresponding to, e.g., a single-mode passive cavity or a non-
interactingbosonicfield. WeintroduceamappingoftheHilbertspacethatallowstochartusefully
the accessible regions. We then consider the quantum excitation from single photon sources in the
form of a two-level system under various regimes of (classical) pumping: incoherent, coherent and
intheMollowtripletregime. Weclosethisfirstopuswithanoverviewofthematerialtobecovered
6
in the subsequent papers.
1
0
2
I. INTRODUCTION photo-luminescence level. At the quantum-optical level,

p however, they are the quintessence of quantum emis-
e sion. This picture has been confirmed experimentally
S Photonics1 has been highly successful in the engineer- by A. Mu¨ller’s group.33 A closer look at, and proper
ing of quantum sources since the proof-of-principle pro-
3 selection of, the photons emitted by quantum sources
duction of quantum light in the mid-seventies.2,3 Befit- thus appears fundamental for state-of-the-art quantum
] ting its status of the elementary brick of the light field,
h applications, their correlations being otherwise averaged
the single photon was the first type of genuinely non-
p overoftencompetingtypes. Thisviewpointoffrequency-
classical type of light and gave rise to the concept of a
- resolved photon correlations therefore poses in a new
t single photon source (SPS).4–6 Nowadays, SPS abound
n light the problem of the excitation of optical targets
with ever increasing figures of merit7–11 and are even
a withsuchinside-knowledgeofthefeaturesofthesources.
u commercially available. The motivations for SPS are
This revives a question put to close scrutiny by Gar-
q many,12 from metrology13,14 to input for quantum in- diner34 and Carmichael35 in 1993 following the emer-
[ formationprocessing,15–17passingbybio-technology,18,19 gence of sources of squeezed light,36 which appeared suf-
2 sensing and detecting,20,21 etc. In most cases, the tech- ficiently more elaborate as compared to SPS to warrant
v nology is still under development and quantum light is a direct investigation of how they would affect optical
7 not yet deployed on the market. For this reason, the fo-
targets as compared to conventional types of excitation
8 cus is still largely on the source itself rather than on its
(classical fields, possibly stochastic). Resonance fluores-
1 direct use as part of a technological component. There
cence in the squeezed vacuum has in fact been just re-
6
0 is however an increasing interest in using quantum light cently reported.37 Gardiner and Carmichael’s (indepen-
foractualapplications. Forinstance, therehavebeenre-
. dent) treatment of the problem of quantum excitation
1 cently converging propositions from independent groups in two consecutives Letters in the Physical Review34,35
0 to use quantum light for spectroscopy.22–26
6 achievedthesettingupofaformalism—namedthe“cas-
1 Thanks to the theory of frequency-resolved photon caded formalism” by Carmichael—that allows to excite
: correlations,27 itwasshownhowquantumsourcescanbe a system (which we will call the “target”) by an other
v
i greatlytunedintheircharacteristicsbyselectinglightin (the“source”)withoutback-actionfromthetargettothe
X astute frequency windows, revealed by the theory in the source. This permits to think separately of the quantum
r form of so-called “two-photon correlations spectra”.28,29 source, which properties can be first studied (through
a
This can be used to identify and exploit unsuspected thetwo-photonspectrum,forinstance)andthendirected
types of quantum correlations30 and/or enhance them onto a target. For historical accuracy, let us mention
by learned combination of timescales and frequencies,31 thattheproblemwasfirstcontemplatedbyKolobovand
merely by spectral filtering of a quantum emitter.32 Be- Sokolov38 whotackleditbyprovidingallthecorrelations
yondoptimizingparameterstofindthebestcompromises of the exciting quantum field. This was recognized as an
for sought applications, the theory also reveals that in- overkill by Gardiner and Carmichael34,35 (Gardiner had
terestingquantumsignalistypicallynotfoundattheex- made prior attempts along these lines). They proposed
pected spectral locations such as peaks, that correspond instead to model the quantum source dynamics as well
to classical-like de-excitation of the emitter between real as the response of the target, even if only the latter is
states at the one-photon level. Instead, strong quantum of interest. From our point of view, such a treatment
correlationsarisefrom,e.g.,two-photonde-excitationin- is essential since the frequency correlations of the source
volving intermediate virtual states. Such channels of de- are too complicated to be treated otherwise than fully
excitation were termed “leapfrog processes”.28,29 They and explicitly by solving the complete problem. Also,
are emitted in unremarkable frequency windows at the suchcorrelationsaredynamicalincharacter, andcannot
2
be well approximated by quantum states as initial con- Exciting with classical light
ditions. Instead, they must be dealt with through the
(a) (b)
full apparatus of dissipative quantum optics, feeding the
target with the complete treatment of virtual states and
other types of strongly correlated quantum input. The
cascaded formalism is therefore particularly apposite for
Incoherent excitation Coherent excitation
exciting optical systems with the knowledge of the two-
photon correlations spectrum of an emitter.
Despitetheconceptualimportanceofquantumexcita-
tion, there have been a moderate follow-up of this cas- Exciting with quantum light
caded formalism, which we believe is a deep and far-
(c)
reaching contribution to the problem of light-matter in- Incoherent
teraction. Eventhoughitbecametextbookmaterial(see SPS
the last chapter from one of the pioneering authors39)
and generated a sizable amount of citations, few texts
(d)
do actually fully exploit the idea. Gardiner and Parkins Coherent
(the formalism is sometimes also named after these two SPS
authors) undertook a more thorough analysis of various
typesofnon-standardstatisticsofthesource40 andCirac
(e)
Mollow
etal.usedittodescribeperfecttransmissionintheirpro-
PS
posal for a quantum network,41 but overall, the core of
the literature using the formalism focuses on specialized { {
particularcases, suchasdrivingwithsqueezedlight.42,43
Quantum Source Classical Target
Typically,thediscussionisthenheldatthelevelofcorre-
lations from a quantum state (namely a squeezed state),
FIG. 1. (Color online). Upper part: The two main types
as opposed to dynamical correlations from a quantum
of classical excitation. The harmonic oscillator can be im-
source. The other studies, already evoked, turned to ap-
plemented by a single-mode cavity, sketched here as two dis-
proximate or indirect approaches, quite similar to the
tributed Bragg reflectors facing each other. The SPS can be
earlier attempts before the cascaded formalism was set implementedbyaquantumdot,sketchedhereasalittlepyra-
up. The reasons for this is certainly a mix between con- mid, as it would be grown by self-assembly44. (a) Incoherent
venience of using well-known and established formalisms excitation, typically corresponding to thermal light. (b) Co-
and the as-yet unclear advantages of the alternative one. herent excitation, typically corresponding to driving the sys-
temwithalaser. Lowerpart: Upgradingtheclassicalsources
Inthisseriesoftexts,wemakeanextensivestudyofex-
of the upper panel with SPS. (c) is the counterpart of (a),
citingwithquantumlight(withnofeedbackofthetarget
to be referred to as the “incoherent SPS”. The coherent case
tothesource),andshowthatsomenewfeaturesoflight-
yields two very different types of quantum light at low and
matterinteractionemerge,makingtheoverallproblemin highpumpings: (d)thecoherentSPSand(e)theMollowPS
need of scrupulous attention. The formalism itself needs whichcanemitmorethanonephotonatatime. Othertypes
littlefurtherdevelopmentandwewillmainlyadaptitto of quantum sources and other types of targets constitute the
newcasesbutwiththeadditionalknowledgeprovidedby topic of the following texts in the series.
frequency-resolved correlations (we will extend the for-
malism in the following papers to consider sequences of
cascades and multiple sources). We briefly introduce its attention to this case. Section VII shows that the ad-
core machinery in a self-contained way in Section II as a vantages of the cascaded coupling over the conventional
convenienceforthereader,andrefertotheoriginalworks Hamiltonian coupling(cavity QED)remainpresenteven
fordetailsofthederivation34,35 ortotheSupplementary whenconsideringalternativedescriptionsofcouplingbe-
Material of Ref. 26, where it is cast in the problematic tween the source and the target. Section VIII draws the
of the present text. Section III gives an overview of the conclusions for the cases studied here while Section IX
many possibilities one can study as well as details of the does so for the wider picture of cascaded coupling and
configurations we focus on in the following of the series. introducestheothercasestobeinvestigatedinfollow-up
Afterwhatcanbeseenasalongintroduction,SectionIV papers.
introducesthefirstimportantresultsbyprovidingaway
to characterize quantum states of the Harmonic oscilla-
tor in a space of quantum-optical diagonal correlators,
II. THEORETICAL BACKGROUND
which will be helpful to later characterize the sources,
and with effect to disprove a popular criterion for single-
photon states. Section V and VI characterize the in- The coupling between two quantum systems is typi-
coherent and coherent SPS, respectively. As the latter cally given by an interaction Hamiltonian. In the second
will prove to be more interesting, we devote most of our quantization formalism, such a Hamiltonian reads in its
3
most simple form (we take (cid:126)=1 along the paper): The source must also be excited, which can be done ei-
ther by an incoherent or by a coherent (classical) type of
HI =g(c†1c2+c†2c1), (1) excitation. Theincoherentexcitationisdescribedsimply
byaddingtheLindbladterm(P /2)L ρtoEq.(3). The
where c1, c2 are annihilation operators describing the coherent excitation, however, rec1quiresc†1a subtler descrip-
particlesofthecoupledsystem,andg istheirinteraction
tion,forwhichoneusestheinput–outputformalism. The
strength. If the particles described by c have a decay
i coupling between a coherent field and the system (that
rate γ , and are freely evolving with a Hamiltonian H ,
i i latter will be used as the source of quantum excitation)
i=1,2, the master equation describing the dynamics of
happensthroughaninputchannelforthesaidsystem. If
this system is:
such an input channel is the only one available to excite
∂ ρ=i[ρ,H +H +H ]+ γ1L ρ+ γ2L ρ, (2) the source, then it follows that the only output channel
t 1 2 I 2 c1 2 c2 fromthesourcealsocontainsthecoherenceofthedriving
field. In this case, the target of the quantum exitation
whereL ρ=(2cρc†−c†cρ−ρc†c). Dependingonwhether
c is also driven by the coherent field, and its dynamics is
the coupling HI or the dissipation γi dominates, one given by Eq. (3) setting H = ω c†c −i√γ E(c† −c ),
speaks of strong or weak coupling, respectively. If one √ 1 1 1 1 1 1 1
and H =ω c†c −i γ E(c† −c ), i.e., the dynamics is
of the two systems, say, 1, is itself excited externally, for 2 2 2 2 2 2 2
ruled by the master equation:
instance being driven by a laser, or merely being given a
non-vacuum initial condition, then one has a crude pic-
(cid:104) √
ture of system 1 exciting system 2. This is a fairly ac- ∂ ρ=i ρ,ω c†c +ω c†c −i γ E(c† −c ) −
t 1 1 1 2 2 2 1 1 1
curate description in the weak-coupling limit where the
√ (cid:105) γ γ
dynamics seems irreversible, simply because excitations − i γ E(c† −c ) + 1L ρ+ 2L ρ−
2 2 2 2 c1 2 c2
are dissipated before they can cycle back (we discuss in √ (cid:16) (cid:17)
Section VII when this becomes exact). − γ1γ2 [c†2,c1ρ]+[ρc†1,c2] , (4)
Thecouplingbetweenquantumsystemsdoesnothave
tobereversible: itcaninsteadcorrespondtothescenario where E is the amplitude of the coherent field driving
of a source and its target. In this case, there is a deep the source. Note that the effective driving intensity, i.e.,
√
asymmetry between the coupled systems. For instance, Ω = γ E, depends on the decay rate of the system
1 1
one can remove the target from the beam of the source, that is being excited, in agreement with the fact that a
whichleavesthelatterunaffectedwhiletheformerpasses system that cannot emit cannot be excited either. To
from being irradiated to the vacuum. Note that such an preventthetargettobealsodrivenbythecoherentfield,
asymmetry does not have to hold on logical grounds. In onecanuseotherinput(andtheircorrespondingoutput)
fact, in electronics, while an ideal source should not be channels to excite the source (and also the target). Each
affectedbythecircuititpowers,inreality,thereisaload of these channels couples with an amplitude (cid:15) ≤1, with
i
(cid:80)
and every component affects all the others to some ex- the condition that (cid:15) =1, the sum being over all the
k k
tent. In photonics, the picture of a flying qubit, left to input channels. In this case, and considering only two
propagatelongenoughbeforeitmeetsitstarget,makesit input channels (with amplitudes (cid:15) and (cid:15) = 1−(cid:15) ) as
1 2 1
intuitivelyclearthatitshouldbepossibletoforbidback- wellasonlyoneinputchannelforthetarget(withampli-
action. Thiscouldalsoberealizedbytakingadvantageof tude 1), the dynamics of the system is given by Eq. (3)
√
the fast-growing field of chiral optics.45–52 Theoretically, with H =ω c†c −i (cid:15) γ E(c† −c ), H =ω c†c , and
this asymmetry is achieved by the cascaded coupling.39, 1 1 1 1 1 1 1√ 1 2(cid:112) 2 2 2
replacing the coupling strength γ γ by (1−(cid:15) )γ γ
1 2 1 1 2
wheretheequationsofmotionareexpressedinthequan-
in the second line of Eq. (3), i.e., the dynamics is now
tum Langevin form, thus allowing to set the output field
ruled by the master equation:
of one of the systems (the source) as the input field for
the other (the target). This can be brought to a mas- (cid:104) √ (cid:105)
∂ ρ=i ρ,ω c†c +ω c†c −i (cid:15) γ E(c† −c ) +
ter equation type of description, with both coherent and t 1 1 1 2 2 2 1 1 1 1
Lindbladtermsthatcontrivetodirecttheflowofexcita- γ γ √ (cid:16) (cid:17)
+ 1L ρ+ 2L ρ− (cid:15) γ γ [c†,c ρ]+[ρc†,c ] ,
tion from the source to target only. This makes all the 2 c1 2 c2 2 1 2 2 1 1 2
operators of the source independent from those of the (5)
target, while in turn those depend on operators of the
source. The generic case where the source (resp. target) where (cid:15)2 = 1−(cid:15)1. The additional input channel to the
isdescribedby theHamiltonian H (resp. H )andhas a sourcemakesthecouplingbetweenthecoherentfieldnot
1 2
decay rate γ (resp. γ ) is ruled by the following master as efficient as when there is only one input channel, thus
1 2
equation:39 reducing the effective driving intensity. For the target,
although the coupling strength is also reduced, now the
∂ ρ=i[ρ,H +H ]+ γ1L ρ+ γ2L ρ− drivingisuniquelyduetotheemissionfromthequantum
t 1 2 2 c1 2 c2 source.
−√γ γ (cid:16)[c†,c ρ]+[ρc†,c ](cid:17) . (3) Putting Eq. (3) in the Lindblad form contributes a
1 2 2 1 1 2 Hamiltonian part. The formalism thus corresponds to a
4
quantum coherent coupling, allowing the description of The SPS is a good starting point because it is the
continuous wave (cw) and resonant excitation of quan- paradigmofquantumlightandisthemostcommonlight
tum states. Importantly, in contrast to the Hamiltonian of this type in the laboratory. Exciting the harmonic os-
coupling in Eq. (1), the coupling strength is now fixed cillatormakestheproblembothsimpleandfundamental,
by the decay rate of the source and of the target. An so this is also a good starting point. While we have con-
infinitely-lived target cannot be excited. The stronger siderably restrained the possibilities, there still remains
onewishestomakethecouplingbetweenasourceandits much to be explored and the following will only address
target,thestrongerhastobetheir(geometric)meandis- themainresults. Thetwotypesofexcitationsaforemen-
sipation. This imposes some fundamental constrains on tionned (a–b) yield quite different types of SPS, that are
external driving (or driving without feedback). In con- introduced in the beginning of their respective Sections,
trast, Hamiltonian coupling sets the coupling strength and are summarized in table I. Note that in the table,
and decay rates independently. While it would there- the last row also includes the second one (which is the
foreappearthattheHamiltoniancouplinghastheupper limit of small pumping). They are however so distinct
hand,andthatoneshouldstriveforthestandardstrong- qualitatively that it is helpful to think of them as sep-
coupling regime, we will show in the following that the arate sources. As we will show, the second row (small
cascaded architecture can be superior to the other types coherent pumping) leads to the best antibunching in the
of coupling in some cases. target.
III. QUANTUM SOURCES AND OPTICAL IV. CHARTING THE HILBERT SPACE
TARGETS
Prior to considering which quantum states of the har-
The general problem of excitation with quantum light monicoscillatoronecanaccessbyexcitingitwithvarious
has obviously numerous ramifications. To the already types of SPS, one needs a roadmap to characterize all of
large variety of optical targets, one now has to combine the states at a glance. The Hilbert space is a big place.
a much enlarged set of quantum sources. This literally Human’s capacity of abstraction gives a deceivingly sim-
opens a new dimension to optics. Indeed, classical exci- ple picture of it. The canonical basis of Fock states |n(cid:105)
tation could arguably be limited to a rather small set of withn∈Nprovidesacomprehensiveandconcisemapof
categories: thestatesintheharmonicoscillator’sHilbertspaceH∞:
∞
(a) Coherent excitation (driving with a laser) (cid:88)
|ψ(cid:105)= α |n(cid:105) , (6)
n
(b) Incoherent excitation (incoherent pumping, Boltz- n=0
mann dynamics, thermal baths, equilibrium, etc.) withα ∈Csuchthat(cid:80)∞ |α |2 =1. Thisisasprecise
n n=0 n
asitismisleading,sincethisfailstoprovideaqualitative
(wepostponetopartVthediscussionoftime-dependent
map of the possible states. Instead, one uses families of
and pulsed excitation and consider until then the case
particular cases. Beyond Fock states per se, one makes
of continuous wave excitation, cf. Section IX). Quan-
tum light, on the other hand, encompasses not only the a great use of the coherent s√tates of classical physics, for
which α = αnexp(−|α|2)/ n! for some complex num-
above—if only because it is a more general case that in- n
ber α that defines the amplitude and phase of a classical
cludes classical excitation as a particular case—but also
field. The next family of important states requires the
comes with many more and higher degrees of freedom.
concept of mixity that involves statistical averages and
This did not lead so far, to the best of our knowledge, to
upgrades the wavefunction to a density matrix:
a classification. Tentatively, this could be provided in a
first approximation by g(2) (antibunching, uncorrelated, ∞
(cid:88)
bunching, superbunching). Since squeezed states have ρ= β |n(cid:105)(cid:104)m| . (7)
nm
thecorrelationsofcoherentstates,however,andtheyare
n,m=0
precisely the type of input that motivated a new formal-
ism, it is clear that this is still far from appropriate. We This allows the introduction of thermal states, for which
decidedtoapproachthisgeneralproblembyconsidering: βnm = (1−θ)θnδnm for some reduced temperature θ ∈
[0,1]. Thesetwofamiliescanbeunitedintoalargerclass
• the same optical target (here an harmonic oscilla- of“cothermal”53 statesthatinterpolatebetweenthetwo
tor, in the next paper a two-level system), and describe, e.g., single-mode Bose condensates not too
far from threshold. From there on, one essentially deals
• the same type of quantum source (here and in the
withquantumstatesoflightwithpopularexamplessuch
nextpaper,aSPS,inthethirdpaper,anN-photon
assqueezedstates,54catstates,55etc.,andotherlesspop-
emitter),
ularsuchasbinomialstates56 ordisplacedFockstates.57
• both coherent and incoherent regimes,
Such a zoology of states is familiar to every quantum
• both low and high pumping regimes. physicist, but it fails to provide the sought mapping of
5
Source Pumping Population n g(2)(τ) Lineshape Linewidth
σ σ
(cid:0) (cid:1)
Incoherent SPS P P /(γ +P ) 1−exp −(γ +P )τ Lorentzian γ +P
σ σ σ σ σ σ σ σ
Coherent SPS small Ω 4Ω2/γ2 (1−exp(−γ τ/2))2 Lorentzian γ
σ σ σ σ σ
Coherent SPS large Ωσ 4Ω2σ/(γσ2+8Ω2σ) 1−e−3γ4στ(cid:104)cosh(Γ4στ)+ 3Γγσσ sinh(Γ4στ)(cid:105) Triplet (cid:26)3γσγ/σ2((cseantterlalilt)es)
TABLE I. Characteristic of a SPS driven incoherently or coherently. In the latter case, for clarity, we also consider separately
the low-driving intensity (middle row) from the general case (bottom row) that covers all pumpings, to highlight the strong
qualitative features of the Mollow regime.
theHilbertspace. Inparticular, itisrestrictedtoknown most formulations. Here we make this notion precise
(and popular) cases and leaves most of Eqs. (6–7) un- by providing the complete picture, in Fig. 2, along with
charted. the closed-form expression for the boundary that sepa-
We introduce a map of the Hilbert space based on im- rates accessible combinations from impossible ones (this
portant observables for the quantum state, namely, the is proved in Appendix A):
n-th order correlation functions:
(cid:98)n (cid:99)(2n −(cid:98)n (cid:99)−1)
(cid:104)a†nan(cid:105) g(2) = a a a . (11)
g(n) = . (8) n2a
(cid:104)a†a(cid:105)n
This boundary is provided by superpositions of contigu-
Since we consider only the quantum states and not their
ous Fock states, i.e., of the type:
dynamics, normalization makes g(1) trivially unity. We
thereforeuseinsteadthenormalizationitself,whichisan (cid:112)p(n)|n(cid:105)+(cid:112)1−p(n)eiθ|n+1(cid:105) , (12)
important observable, the population of the state (aver-
age number of excitations), for which we introduce the for n ∈ N, p(n) ∈ [0,1] and θ ∈ [0,2π[ an (irrelevant)
notation: phase. (As already commented, this includes also mixed
states of the type p(n)|n(cid:105)(cid:104)n|+(1−p(n))|n+1(cid:105)(cid:104)n+1|
n ≡(cid:104)a†a(cid:105). (9)
a and all others with the same diagonal but different off-
Note that only the diagonal elements p(m) = |α |2 are diagonalelements. Wewillnotmakethisdistinctionany-
m
more in the following.) We call these states, Eq. (12),
neededsothereisnodistinctionbetweenpureandmixed
“Fock duos”. They set the continuous lower limit of the
statesinthisdiscussion. Then-thordercorrelatorreads:
space of available quantum states in our charted Hilbert
(cid:80)∞ m!p(m)/(m−n)! space. Theirantibunching(11)isageneralizationtonon-
g(n) = m=0 . (10)
(cid:0)(cid:80)∞ mp(m)(cid:1)n integer na of the formula g(2) = 1−1/na for the Fock
m=0 state |n (cid:105) to which it reduces for n ∈ N. It shows that
a a
A convenient mapping of the Hilbert space is to chart the popular “single-particle criterion” that asserts that
it with the n and g(2) “rulers”, that is, tag the possible g(2) < 0.5 ensures a one particle state4,58–61 is wrong,
a
quantum states through their joint statistical properties as demonstrated, e.g., for the case n = 1.5 for which
a
and population. The value of g(2) allows to tell apart g(2) =4/9≈0.44. The generalization to higher orders is
classicalfromquantumstatesdependingonwhetherg(2) straigthforward. Namely, the nth-order correlation fron-
islarger(bunching)orsmaller(antibunching)thanunity. tier for real n is given by:
a
This is meant in the sense of whether a classical, possi-
(cid:20) (cid:21)
bly stochastic, description is possible, or whether some g(n) = (cid:98)na(cid:99)! 1+ n(na−(cid:98)na(cid:99)) , (13)
“quantum”featuressuchasviolationofCauchy–Schwarz ((cid:98)n (cid:99)−n)!nn (cid:98)n (cid:99)+1−n
a a a
inequalities or negative probabilities in the phase space
are manifest. The population gives another meaning that generalizes the Fock state formula g(n) =
to quantumness, in the sense of few-particle effects vs (1/nn)m!/(m−n )! valid for integer n where it agrees
a a a
macroscopic occupation. It is therefore particularly in- with Eq. (13). Interestingly, the lower limit is also set,
teresting to contrast these two attributes and consider, for all n, by Fock duos (not “n-plets”).
e.g., highly occupied genuinely quantum states. There is OnecanalsoconsiderotherchartsoftheHilbertspace,
aphysicallimittosuchstatesandforpopulationsn >1, such as (g(2),g(3)), this time contrasting two- and three-
a
some values of antibunching are out of reach. This is ex- particle correlations together. This time there are no
pected on physical grounds from familiar features of the boundaries in this space (one can find state with any
Fock states popularly known as being the “most quan- jointvaluesofg(2) andg(3)). Theproofofthisstatement
tum” states, with their g(2) tending to one as the num- is given in Ref. 62. Since the excitation of an harmonic
ber of excitations increases. Also, macroscopic quantum oscillatorbytheSPSleadstostrongcorrelationsbetween
states, such as a BEC, are essentially coherent states in various g(n), we will focus on the (n ,g(2)) space.
a
6
FIG. 3. (Color online). Trajectories in the Hilbert space
FIG. 2. (Color online). Charting the Hilbert space. The chartedbyna andg(2) forthestatesexcitedbyanincoherent
shadedareashowstheregionwhereaphysicalquantumstate SPS, for various values of γa/γσ. The implicit parameter is
existswiththecorrespondingjointpopulationandantibunch- pumping. Highlighted are the cases γa/γσ = 10−1 (blue),
ing. ItisdelimitedbytheboundaryEq.(11)of“Fockduos”, 1 (green), 10 (yellow) and 102 (red). The dark green thick
defined by Eq. (12). Higher order correlators (thin dashed) envelope shows the closest one can get in this configuration
set other boundaries, also delimited by Fock duos. to the ideal antibunching, which is zero. All the states, and
only these states, above this line and below g(2) = 2, are
accessible with an incoherent SPS.
V. EXCITING WITH AN INCOHERENT SPS
These trajectories in the Hilbert space are plotted in
Now that we can conveniently characterize the states Fig. 3. The curves start from the point:
of the harmonic oscillator that one can excite, we come (cid:18) (cid:19)
2
back to the dynamical problem of the quantum driving n =0,g(2) = , (16)
a 1+3γ /γ
of an harmonic oscillator and how close can one get to a σ
the limit set by Eq. (11). We first consider the case of at vanishing pumping, reach a turning point with coor-
excitation with an incoherent SPS, i.e., where the two- dinates:
levelsystem(2LS)thatactsasthesource,isitselfdriven
(cid:32) (cid:112)
incoherently, as sketched in Fig. 1(c). Namely, there is a 4(2+(γ /γ )−2 1+(γ /γ ))
a σ a σ
n = ,
constant rate Pσ at which the 2LS is put in its excited a (γa/γσ)2
state, and is otherwise left to decay. The system is thus
(cid:33)
describedbythemasterequations(2–3),withc =σ the 2
1 g(2) = , (17)
2LS and c = a the harmonic oscillator operators, re- (cid:112)
2 3 1+γ /γ −2
a σ
spectively. The 2LS pumping is described by a Lindblad
term (Pσ/2)Lσ†ρ. Thanks to the absence of feedback, when Pσ = (cid:112)γσ(γa+γσ) and converge at (na =
the dynamics is ruled by closed equations which allows 0,g(2) = 2) at large pumpings, where the source gets
us to obtain exact solutions for the observables of inter-
quenched. For each value of γ /γ , there are two values
a σ
est. Namely, we find that the cavity population n and
a of pumping that result in the same population but two
statistics g(2) of the target are given by (cf. Table I for values of g(2). The curve of Eq. (15) is fairly constant
the source):
till the turning point and, from Eq. (16), leads to a gen-
uinequantumstate(i.e., featuringantibunching)aslong
4P γ
na = (γ +P )(γσ +σP +γ ), (14a) as γa/γσ >1/3, which means that, with a SPS, one can
σ σ σ σ a imprintantibunchinginasystemthathasasubstantially
2(γ +P ) longer lifetime. The optimum antibunching/population
g(2) = σ σ . (14b)
γσ+Pσ+3γa for a given γa/γσ is achieved slightly earlier than the
turning point, namely, when P = γ . The envelope of
σ σ
Onlytwoparametersarerequiredtodescribefullythe all the curves in Fig. 3 is the closest one can get to the
system: i) the ratio between the decay rate of the cavity Fock-duos limit, and is given by g(2) = 2n /(3−2n ).
a a
andtheemissionrateofthe2LS,γ /γ ,andii)theinten- This is the thick dark green line in Fig. 3.
a σ
sity(orpumpingrate)ofthe2LS,P ,alsonormalizedto The above solution gives an already substantial de-
σ
γ tokeeptheparametersdimensionless. EliminatingP scription of the response of an harmonic oscillator to
σ σ
from Eqs. (14), this gives the equation for the trajectory an incoherent SPS. We can complement it with alter-
inthe(n ,g(2))spaceasfunctionoftheparameterγ /γ : native descriptions that approach the solution from dif-
a a σ
ferentviewpointsandmanifesttheadvantageofcascaded
2(2−g(2))(3g(2)(γ /γ )−(2−g(2))) coupling over other types of excitation. A natural com-
n = a σ . (15)
a 3 g(2)(1+g(2))(γ /γ ) parison is with the standard Hamiltonian coupling, that
a σ
7
Cascaded Coupling Hamiltonian Coupling with Decay Hamiltonian Coupling Minimum
Decay
with
miltonian Hamiltonian
Ha
Cascaded
Fock duos
Hamiltonian with Decay
Cascaded Hamiltonian
Fock duos
FIG.4. (Coloronline). Upperrow: Populations(isolines)andg(2) (color)ofthetargetexcitedbyanincoherentSPS,through
(a)thecascadedcouplingor(b,c)theHamiltoniancouplingwith(b)andwithout(c)decayofthe2LS.(d)Minimumg(2) that
can be reached at a given population for all these cases. The color circles mark the parameters P /γ and γ /γ at which
σ σ a σ
the population isoline meets the minimum g(2). Lower row: Same but for a coherent SPS, with (e) the cascaded coupling and
(f,g) the Hamiltonian coupling with (f) and without (g) decay of the 2LS. (h) Minimum g(2) that can be reached at a given
population for all these cases. The color circles mark the parameters Ω /γ and γ /γ at which the population isoline meets
σ σ a σ
the minimum g(2).
istheusualwaytocoupledifferentsystems. Forinstance, toniancouplingwithnodecayofthe2LS,case(c),differs
it is convenient to lay out the observables as function of qualitatively from the former cases and provides a sub-
the parameters in 2D density plots, as shown in the up- stantiallybetterantibunchingthanHamiltoniancoupling
perrowofFig.4. Panels(a–c)showthephotonstatistics with decay of the 2LS. It even becomes better than the
(colorcode)andthepopulationofthecavity(isolines)us- cascaded coupling when n (cid:38)2/3. This situation can be
a
ingthe(a)cascadedcouplingand(b,c)theHamiltonian much improved with the coherent SPS.
coupling. For the later case, we consider two configura-
tions: (b)whenthe2LShasthesamedecaytermaswhen
acting as a source in the cascaded scheme, and (c) when
VI. EXCITING WITH A COHERENT SPS
the2LShasnodecayterm,sothatallitsexcitationisdi-
rectedtowardsthecavity,correspondingtotheidealexci-
We now consider excitation by a coherent SPS, i.e.,
tation of the cascaded scheme. Interestingly, (a) and (b)
whenthe2LSthatactsasthesourceisdrivencoherently
types of excitation present a qualitatively similar pho-
byanexternallaser,assketchedinFig.1d). Thisdriving
ton statistics layout, but the topography of the resulting
is described by the Hamiltonian:
population yields largely differing states, as evident once
reported in the (n ,g(2)) space, panel (d). The color cir- √
a H =ω σ†σ+Ω (cid:15) (σ†e−iωLt+σeiωLt), (18)
clesalongtheisolinesmarkthepointswithstrongestan- σ σ σ 1
tibunching (i.e., smallest g(2)) for a given population. It
where ω is the energy of the 2LS, and ω and Ω =
is seen that for a given signal, the antibunching is signif- √ σ L σ
γ E arethefrequencyandintensityofthedrivinglaser
icantly larger with the cascaded coupling. In both cases σ
(which drives a 2LS of decay rate γ with a coherent
the antibunching is marred before the cavity population σ √
field of intensity E), respectively. The coefficient (cid:15) is
reaches one photon on average. At larger populations, 1
put here so that the cavity is driven only by the emis-
the photon statistics of the cavity become bunched, i.e.,
sion of the 2LS and not by the external laser. The
g(2) = 2. On can contemplate other configurations but
system is otherwise still described by the master equa-
theyresultinworseresults: detuningthecavityfromthe
tions (2) and (5), with c = σ the 2LS (source) and
2LS leaves the photon statistics unchanged, but reduces 1
c = a the harmonic oscillator (target), and reducing
the population of the cavity, so the antibunching in lost 2 √
the coupling strength by a factor 1−(cid:15) , i.e., the cou-
at even smaller cavity populations. Dephasing the 2LS (cid:112) 1
pling is given by (1−(cid:15) )γ γ . Here again, one can
increases the broadening of the emission peak, but also 1 a σ
obtain the cavity observables in closed-form, although it
drives the cavity towards a coherent state faster. Hamil-
takes slightly more cumbersome expressions. At reso-
8
(a) Exciting at resonance (b) Exciting with leapfrog
(c) Exciting with a satellite (d) Scanning in frequency
FIG. 5. (Color online). Same as Fig. 3 but for the excitation from a coherent SPS. In (a,c) the implicit parameter is pumping
and the green line shows the closest one can get from the Fock-duos limit from the case at resonance. In (d) the implicit
parameter is the detuning between the source and its target. (a) Exciting at resonance. The optimum antibunching is the
envelopeinthisconfigurationofthefamilyofcurvesvaryingwithγ /γ . Highlightedarethecasesγ /γ =10−2 (blue),10−1
a σ a σ
(green),1(chartreuse),10(yellow)and102(red). (b-d)Excitinginotherconfigurations,namely(b)withtheleapfrogprocesses,
(c) with the satellite and (d) as a function of frequency. Highlighted are the cases γ /γ = 10−1 (blue), 0.5 (chartreuse), 1
a σ
(green), 5 (orange) and 10 (red). The dashed-dotted line in panel (b) and the dashed line in panel (c) show the limit of large
intensitiesandaremerelyavisualaidratherthanaphysicalboundary. Theyarealsoshowninpanel(d),alongwiththeblack
solid line for the envelope of the curves at ω =ω .
a σ
nance (out-of resonance cases can also be obtained but get even more bulky):
(cid:16) (cid:17)
16(1−(cid:15) )γ Ω2 γ2 γ +8γ Ω2
n = 1 (cid:102)01 0 (cid:102)11 (cid:102)12 (cid:102)10 0 , (19a)
a γ γ (γ2 +8Ω2)(γ γ +16Ω2)
(cid:102)10 (cid:102)11 (cid:102)01 0 (cid:102)11 (cid:102)12 0
g(2) =
γ (γ2 +8Ω2)(γ γ +16Ω2)(γ γ2 γ2 γ γ +8γ γ (17γ3 +29γ2 γ +18γ γ2 +4γ3 )Ω2+192γ2 γ Ω4)
(cid:102)11 (cid:102)01 0 (cid:102)11 (cid:102)12 0 (cid:102)11 (cid:102)21 (cid:102)31 (cid:102)12 (cid:102)32 (cid:102)10 (cid:102)31 (cid:102)10 (cid:102)10 (cid:102)01 (cid:102)10 (cid:102)01 (cid:102)01 0 (cid:102)10 (cid:102)21 0 ,
γ γ (γ γ +8Ω2)(γ γ +16Ω2)(γ2 γ +8γ Ω2)2
(cid:102)21 (cid:102)31 (cid:102)11 (cid:102)21 0 (cid:102)31 (cid:102)32 0 (cid:102)11 (cid:102)12 (cid:102)10 0
(19b)
where we have introduced the notation γ = iγ +jγ P[n] is an n-th order polynomial of the variable n . The
i(cid:101)j√ a σ a
(e.g.,γ =3γ +γ ),andwehavesetΩ = (cid:15) Ω. Elim- ±termsprovidethetwobranchesofthecurveasseenin
inating(cid:102)3Ω1 2 froma Eqσs. (19) yields, for a g0iven γ 1/γ , solu- Fig.5(a). Thelowerbranchcorrespondstolowpumping,
a σ
tions of the type g(2) = (P[3]±(cid:112)P[6])/(P[2]n2) where where the PL of the source is still a single line as shown
a
9
in the bottom inset, while the upper branch corresponds in a small region (n ≈ 0.6,g(2) (cid:46) 2). Panel (d) shows
a
to high pumping where the PL has split into a Mollow the trajectories when varying the frequency of emission,
triplet, shownintheupperinset. Eachcurvestartsfrom where the cases just discussed—resonance, leapfrog and
the point: satellites—appearasboundariesofthecompletepicture,
(cid:18) (cid:19) thus showing that these configurations already give ac-
1
n =0,g(2) = , (20) cess to all the accessible states.
a (1+γ /γ )2
a σ We can also compare in the parameter space the co-
showing that coherent SPS provide a much stronger an- herent SPS with its Hamiltonian counterpart, as we did
tibunching than their incoherent counterpart, Eq. (16), for the incoherent SPS. This is shown in the bottom row
alreadyatvanishingpumping. Athighpumping, incon- of Fig. 4, that is also usefully compared with the upper
trast to the incoherent case that vanishes the population row(incoherentSPSexcitation). ForthecoherentSPSas
of its target, the coherent SPS quenches it to a nonzero well, both cascaded and Hamiltonian coupling are qual-
value and featuring bunching: itatively similar in their statistical layout when the 2LS
(cid:18) (cid:19) has a decay term, but also differ notably from the to-
1−(cid:15) 1+γ /γ
n = 1 ,g(2) =3 a σ . (21) pography of the associated intensities. This is, again,
a 1+γ /γ 1+3γ /γ
a σ a σ clear on the (n ,g(2)) space, with the maximum anti-
a
Eliminating γ /γ here leads to the curve g(2) = 3(1− bunchingthatcanbeobtainedforagivencavitypopula-
a σ
tionwithcascadedcoupling(greenline)andHamiltonian
(cid:15) )/[3(1 − (cid:15) ) − 2n ], which for (cid:15) = 1/2 simplifies to
1 1 a 1
g(2) = 3/(3−4n ) as shown in thick black in Fig. 5(a). coupling (red line). The Hamiltonian coupling with no
a
decay of the 2LS also results in qualitative differences
The expression for the lower envelope of maximum anti-
and an enhanced antibunching. However, in this case,
bunching, in blue, is too complicated to be derived fully
theHamiltoniancouplingnevercomestosurpassthecas-
inclosed-formbutwecanfindsimpleasymptoticexpres-
sionsbyseriesexpansionofEqs.(19),namely,g(2) ≈5n2 cadedscheme. Instead,inalltherangeofpopulation,the
a
when n (cid:28)1 and g(2) ≈1−1/(3(n +5)) when n (cid:29)1. antibunching obtained through the cascaded coupling is
a a a
larger than the one obtained with any type of normal
The states thus get close to the ideal Fock limit for
(Hamiltonian) coupling, for a given population. When
large populations, approaching unity as 1/(3n ) rather
a
the cavity is detuned from the 2LS, the antibunching is
than 1/n . There is no limit in the population that can
a
lost before the cavity’s population reaches one photon.
be excited in the target by the coherent SPS, in stark
Also, when the cavity is in resonance with the leapfrog
contrasttotheincoherentSPSthatisboundedbyunity.
Although the accessible states in the (n ,g(2)) space re- processemission,thephotonstatisticsofthecavityissu-
a
perbunched but its population remains well below unity.
main quite some distance away from the ideal limit, the
Includingadephasingratetothedynamicsofthe2LSin-
coherent SPS provides a much better antibunching than
creasesthebroadeningofeachemissionlineinthetriplet,
its incoherent counterpart, as can be seen by comparing
and even a small dephasing rate spoils the antibunch-
Figs. 3 and 5. Namely, it is still antibunched when pop-
ing. Thus, the best antibunching with a coherent SPS
ulation is unity (with a maximum antibunching slightly
is obtained exciting the cavity at low pumping and in
over1/2)andwehavealreaycommentedhowitallowsfor
resonance with the 2LS, using the cascaded coupling.
arbitrary large populations, that still feature antibunch-
ing. This is achieved by exciting at resonance targets of One of the departing features of the coherent SPS ex-
very long lifetimes as compared to the source. citation, is that it allows to populate its target with
This describes the resonant situation. Since the co- more than one excitation on average, therefore trigger-
herent SPS has a rich spectral structure, it opens new ing Bose stimulation effects. There is a phase locking
configurations of excitation in the Mollow triplet regime (of 0, as determined by the Hamiltonian) of the incident
beyondthecentralpeak, suchasexcitingwithasatellite photonsfromthequantumsourcethataccumulatewhile
peak(asshowninFig.5(c))orwiththeleapfrogwindow still exhibiting some features of Fock states, in partic-
(Fig. 5(b)). In these cases we show the results in log-log ular being antibunched. A complete picture of the re-
plots as only small populations are within reach (unlike sulting quantum states, beyond the diagonal elements
thecaseofresonance)and,inthecaseofleapfrogexcita- only, is given Fig. 6 for the states marked with black
tion,alsoahugebunchingcanbeimpartedtothetarget. circles in Fig. 4(f), both through the Wigner represen-
The envelope of optimum antibunching as obtained in tation, Fig. 6(a-c), and through their matrix represen-
resonance is reported in these panels (also in green) for tation, Fig. 6(d-f). The Wigner representation exhibits
comparison, showing that these alternative schemes do negative values at the origin, a mark of a genuine quan-
not enhance antibunching. The excitation with leapfrog tum state with no classical counterpart, for the states
processesallowstoaccessnewregionsoftheHilbertspace with n =1 and n =1.5 (the blue spot at the center in
a a
related to superbunching, so it is a configuration that Fig.6(b)). Itis,however,positiveeverywhereforn =3,
a
presents its own interest. This is achieved at the price although the state is still antibunched. Also, one can see
of small populations. Exciting with a satellite peak also that the phase uncertainty decreases as the population
conquersnewterritoriesinthe(n ,g(2))spacenotacces- increases, in agreement with the “classical limit”. More-
a
siblethrougheitherthecentralpeakorleapfrogemission over, the matrix elements in these cases also satisfy the
10
(a) (a)
Thermal Leapfrog
Coherent
C entral peak Fock duos
(b) (e)
(b)
k
a
pe
Thermal ntral
(c) hermal Ce
Cot
Coherent L e a pfr o g
Fock duos
FIG. 7. (Color online). States accessible in the (g(2),g(3))
spacebycoherentSPSexcitation,atresonancewiththecen-
tral peak (orange line) or with the leapfrog emission (red
FIG. 6. (Color online). Wigner function (a–c) and density
line). The dashed blue line corresponds to the “Fock duos”,
matrix (d–f) for the states marked by black circles along the
Eq.(A1). Thesameresultisshowninboth(a)log-logand(b)
greenlineinFig.4(h). Thesestateshavepopulationsn =1,
a
linear scales.
1.5 and 3 from top to bottom and all feature antibunching.
cesses,onthecontrary,allowonetoreachlargejointval-
relation ρ >max(ρ /2,ρ ), which means that fluctu- uesofg(2) andg(3). Interestingly,whilethepointsacces-
22 11 33
ationsaresmallerthancouldbeexpectedfromrateequa- sible in the (n ,g(2)) space extend over 2D areas, in the
a
tion arguments, since two excitations are twice as much (g(2),g(3))spacetheyareconfinedto1Dcurves, showing
likelytodecaythanoneexcitation,makingtheprobabil- how these correlators are strongly inter-dependent, and
ity to find one excitation only in principle at least twice thus making the (n ,g(2)) representation more promi-
a
as large than double occupation.26 nent.
WeconcludethisSectionondrivinganharmonicoscil-
lator with the coherent SPS by considering states in the
(g(2),g(3))space. ThisisshowninFig.7forthemostno- VII. VARIATIONS IN THE TYPE OF
table configurations of exciting with the central peak of COUPLING
theMollowtripletortheleapfrogregionbetweenthecen-
tral peak and a satellite. All points are physical on this The comparison between both rows of Fig. 4 shows
map, including cases with g(2) (cid:28) 1 and g(3) (cid:29) 1 (and that the coherent SPS is a much better quantum drive
vice-versa), although the accessible regions are strongly togeneratebrightantibunchingthantheincoherentone,
confined along curves that we can fit in monomial form, evenintheregionwheren <1. ThecoherentSPSexcit-
a
leading in good approximation to g(3) ≈0.2[g(2)]2 at low ing the cavity still remains some distance away from the
pumping (in the antibunching corner) and g(3) ≈4.5g(2) Fock duos boundary, but it gets significantly closer than
atlargepumpings(inthesuperbunchingcorner). Wein- with the Hamiltonian coupling. We complete our juxta-
dicate positions of popular quantum states such as Fock positionbetweenthetwotypesofcoupling,cascadedand
states, coherent states, thermal states and their combi- Hamiltonian, by comparing with other variations in the
nations. Driving with a SPS spans over great distances way the source couples to its target.
in this space, especially when exciting with leapfrog pro- First, we consider another source of decay for the
cesses, as seen in panel (a) of the figure. This shows, source in the cascaded coupling scheme, i.e., with an ex-
again, how the central peak remains confined essentially tra term (γ∗/2)L ρ in the master equation, leaving the
σ σ √
to the antibunched corner of the map (excursions to coupling strength constant and equal to γ γ /2. This
a σ
bunching cases are up to g(2) = 3). Panel (b) is a zoom describesthesituationwherenotallthelightthatislost
around small values in linear scale. The leapfrog pro- from the source is redirected to the target. Figure 8(a-