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Cohe ence o quan um non-Gaussian s a es ia
nonlinea abso p ion o quan a
To ci e his a icle: Kingshuk Adhika y
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2025
Quan um Sci. Technol.
10 035048
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PAPER
Cohe ence o quan um non-Gaussian s a es ia nonlinea
abso p ion o quan a
Kingshuk Adhika y, Da en W Moo e∗and Radim Filip
Depa men o Op ics, Palack´
y Uni e si y, 17. lis opadu 1192/12, 779 00 Olomouc, Czech Republic
∗Au ho o whom any co espondence should be add essed.
E-mail: [email p o ec ed]
Keywo ds: quan um cohe ence, quan um non-Gaussian, quan um enginee ing and echnology
Abs ac
The linea and phase insensi i e abso p ion o a single quan a ia cohe en in e ac ions wi h a
sa u able sys em, e en a single g ound s a e qubi , is su icien o de e minis ically gene a e
quan um non-Gaussian s a es in an oscilla o , e en s imula ed me ely by inc easing he mal
oscilla o ene gy. Howe e , he esul an s a es only app oach Fock s a es and he e o e do no
exhibi quan um cohe ence. He e we o e come his limi a ion using a minimal s ep: a nonlinea
phase-insensi i e abso p ion p ocess added o he linea one. The cohe en addi ion o such
indi idually passi e p ocesses allows cohe ence o eme ge and inc ease in phase space wi hou an
ex e nal d i e and wi h minimal in e ac ion equi emen s. The cohe ence o quan um
non-Gaussian s a es eme ges because he linea and nonlinea abso p ion p ocesses a e no
mu ually passi e. In he simples case o a ionally symme ic Wigne unc ions o he oscilla o
Fock s a es con e hei many nega i e egions o an ex emely complex asymme ic s uc u e in
sha p con as o he o a ional symme y o hose ob ained by he indi idual in e ac ions. We
ex end his case o include an unsa u able abso be (oscilla o ) and analyse swi ching be ween
linea and nonlinea abso p ions, sui able o b oad classes o expe imen s.
1. In oduc ion
A co e p ope y o quan um heo y is he capaci y o an indi idual sys em o o m cohe en
supe posi ions [1,2]. While e e y quan um s a e is a supe posi ion in some basis, some supe posi ions in
pa icula con ex s a e excep ionally undamen al o ele an o applica ions. Quan um cohe ence
unde goes in ensi e esea ch in a la ge a ie y o con ex s, such as open sys ems whe e only ce ain
supe posi ions su i e decohe ence [3,4] o whe e obse able cohe ences a e gene a ed ia ex e nal
d i ing [5,6], o as a esou ce in quan um echnologies [7,8] and quan um he modynamics [9]. Especially
undamen al and ele an a e supe posi ions o ene gy eigens a es and he ways in which hey can a ise. They
al eady ha e di e se and expe imen ally demons a ed applica ions in quan um phase sensing [7,10] and
bosonic quan um e o co ec ion [11,12] o quan um compu ing and communica ion.
Su p isingly, basic quan um in e ac ions de e minis ically gene a e quan um non-Gaussian oscilla o
s a es ia linea cohe en abso p ion o quan a om a he mal oscilla o [13,14]. Speci ically, phase
insensi i e and ene gy conse ing abso p ion o ene gy om a he mal oscilla o by a g ound s a e qubi
uncondi ionally gene a es s a es ha can app oach Fock s a es, wi hou any ex e nal p ocesses such as
measu emen , eed o wa d o d i en/dissipa i e enginee ing. Howe e , hey canno gene a e local oscilla o
cohe ence e en i hey do gene a e ligh -ma e en anglemen [15]. Indeed his limi a ion is a gene al
p ope y o mul ipho on Jaynes–Cummings (JC) models [16] (see igu e 1).
In his pape we o e come his limi a ion using a nonlinea phase-insensi i e abso p ion added o he
linea one [13,14] in a ully quan um mechanical way. Thus, a combina ion o wo di e en
phase-insensi i e abso p ion p ocesses, o simplici y es ed on Fock s a es, indi idually ene gy conse ing
© 2025 The Au ho (s). Published by IOP Publishing L d
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
Figu e 1. Cohe ence o quan um non-Gaussian s a es ia a combina ion o linea and nonlinea abso p ion by a single qubi : A
g ound s a e qubi abso bing ixed quan a o ene gy (one o wo in his illus a ion) om an oscilla o p epa ed in a Fock s a e
p oduces only mix u es o Fock s a es. These s a es a e o a ionally symme ic in phase space and he e o e he cohe ence s ic ly
anishes. In con as , i bo h in e ac ions a e simul aneous hen a supe posi ion o abso p ions esul s which begins o b eak he
o a ional symme y, indica ing he eme gence o quan um cohe ence. Wigne unc ion ans o ma ions: The ans o ma ion o
he Wigne unc ion o Fock s a e |2⟩ emains an incohe en mix u e o Fock s a es o he indi idual in e ac ions. The
combina ion howe e esul s in he eme gence o cohe ence, C=0.08 and he loss o o a ional symme y. The Wigne unc ion
in he igu e is ound o in e ac ion s eng hs g(2)
g(1)=0.1 and sho in e ac ion ime τ=0.157.
and incapable o p oducing cohe ence, join ly esul s in cohe en quan um non-Gaussian s a es. The
combina ion is essen ial o c ea e us a ion be ween he condi ions equi ed o he in e ac ion o be
passi e, i.e. ene gy conse ing, hus allowing oscilla o supe posi ions o de e minis ically de elop e en a e
igno ing he inal s a e o he qubi . The esul an supe posi ions in he Fock basis show subs an ial quan um
cohe ence and quan um non-Gaussian ea u es, e aining he Wigne nega i i y o he o iginal Fock
s a es [13,14] (see igu e 1). In wha ollows we demons a e he s iking ex en o which his appa en ly
simple compund in e ac ion gene a es ex emely quan um non-Gaussian s a es wi h subs an ial cohe ence,
compa e i o he classical incohe en oscilla o s a es, ex end his idea o an oscilla o abso be , and sugges a
easible expe imen o e i y he eme gence o cohe ence om nonlinea abso p ion. We close wi h some
discussion o he na u e o he nonlinea abso p ion in e ac ion wi h espec o cohe ence gene a ion.
2. Resul s
To ad ance he esul s in [13,14], we add ess he eme gence o local oscilla o cohe ence ia abso p ion o
ene gy om he oscilla o , s a ing om he pu e incohe en Fock s a es app oached by hose me hods. The
ele an basis in which o examine non i ial supe posi ions is he e o e he ene gy eigenbasis o he
oscilla o , gi en by he Fock s a es |n⟩, he eigens a es o he ha monic oscilla o Hamil onian Hω=ωb†b,
wi h cons an equency ω. Na u ally, he ee e olu ion o he oscilla o does no c ea e supe posi ions om
such a se up so he oscilla o mus in e ac wi h a new subsys em wi h ee e olu ion HΩ, i sel p epa ed in
an incohe en s a e, diagonal in he ene gy eigenbasis. We will ypically ake his o be he g ound s a e o HΩ,
which can be app oached by cooling. Fo a qubi subsys em we ha e HΩ=Ω
2σz, wi h σza Pauli ma ix.
Le us now be mo e p ecise. To quan i y he o e all cohe ence o an oscilla o we use he ela i e en opy
o cohe ence [17] de ined as
C(ρ) = S(ρdiag)−S(ρ),(1)
whe e Sis he on Neumann en opy and ρdiag is he diagonal ma ix con aining he p incipal diagonal o he
co esponding densi y ma ix ρ. Fo JC-like in e ac ions he oscilla o and abso be Hamil onians which se
he ene gy eigenbases a e Hω(abo e). The k h o de JC in e ac ion akes he o m
V(k)=g(k)(σ+bk+σ−(b†)k),(2)
whe e σ±a e he qubi aising and lowe ing ope a o s. As said, such in e ac ions a e known o p ese e he
local incohe ence o incohe en ini ial s a es [16]. Indeed he ee e olu ion H0=Hω+HΩcommu es wi h
V(k) o all k, p o ided ce ain equency condi ions a e me . Mo e p ecisely,
[H0,V(k)]=g(k)(kω−Ω)(σ−(b†)k−σ+bk),(3)
which is only ze o o Ω = kω. Tha is, he ene gy N=kσ+σ−+a†ais a conse ed quan i y. When he
in e ac ion Hamil onian commu es wi h he ee Hamil onian no local oscilla o cohe ence eme ges. We
2
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
no e in passing ha a de uned model, wi h Hamil onian H=−∆a†a+Ω
2σz+V(k)and equency ∆, s ill
does no gene a e local oscilla o cohe ence. A simple me hod o o e come his limi a ion and p oduce
oscilla o cohe ences is o combine, in a ully quan um way, wo o hese ene gy conse ing in e ac ions
(equa ion (2)) wi h di e en k.
2.1. Sho - ime eme gence o cohe ence and sequen ial app oach
Le ’s examine he simples case o k=1 and k=2, so ha he in e ac ion has he o m V=V(1)+V(2). This
combines linea (k=1) and nonlinea (k=2) abso p ion by he qubi . Apa om simplici y his is also he
mos ele an expe imen ally, as i in ol es he al eady ex emely well cha ac e ised lowes o de JC
in e ac ions [18]. Fo such in e ac ions he numbe o exci a ions is no longe conse ed, ha is, he e is
equency us a ion be ween he compe ing abso p ion p ocesses. Despi e he analy ically in ac able
na u e o he sys em, some insigh in o he eme gence o cohe en quan um non-Gaussian s a es can be
gained by conside ing he sho ime e olu ion o he sys em. An illumina ing app oach is o conside he
i s o de expansion o he uni a y e olu ion ia he Bake -Campbell-Hausdo heo em. Tha is, o i s
o de in ≪1 we ha e U=e−i(V(1)+V(2)) =e−iV(1) e−iV(2) +O( 2). Te ms beyond his app oxima ion
u he inc ease he cohe ence. Fo his app oxima ion, cohe en quan um non-Gaussian s a es al eady
eme ge. Ad an ageously, his app oxima ion also mo i a es Hamil onian swi ching be ween in e ac ions V(1)
and V(2)o ice e sa. This sequen ial me hod p o ides an al e na i e and immedia ely accessible p ocedu e
o p oduce cohe ence by combining passi e and phase insensi i e in e ac ions cohe en ly abso bing
indi idual quan a. Fo example, in he con ex o apped ions such in e ac ions a e gene a ed by
illumina ing he ion a he k− h sideband. The e o e implemen a ion o he swi ching p o ocol equi es only
ha wo such sidebands a e independen ly a ailable and con ollable in he same se up [7].
In igu e 2we show he eme gence o cohe ence a he le el o he Wigne unc ions. S a ing wi h he
V(1)in e ac ion and he s a e |g⟩|n⟩≡|g,n⟩,n>0, he s a es |g,n⟩and |e,n−1⟩become coupled, and he
ypical s a e is a supe posi ion o hese wo. T acing ou he qubi does no p oduce any oscilla o cohe ence.
When he Hamil onian is swi ched o V(2), hese wo s a es decouple and couple o new s a es: |g,n⟩couples
o |e,n−2⟩, and |e,n−1⟩couples o |g,n+1⟩. A ypical s a e is now a supe posi ion o hese ou , and he
a e age o al numbe o exci a ions has changed. In ac , we can explici ly w i e he s a e as
|Ψ1⟩=|g⟩(α( )|n⟩+β( )|n+1⟩) + |e⟩(γ( )|n−1⟩+δ( )|n−2⟩)(4)
whe e
α( ) = cos(g(2)√n(n−1) )cos(g(1)√n )(5)
β( ) = −sin(g(2)√n(n+1) )sin(g(1)√n )(6)
γ( ) = −icos(g(2)√n(n+1) )sin(g(1)√n )(7)
δ( ) = −isin(g(2)√n(n−1) )cos(g(1)√n ).(8)
When he qubi is now aced ou he emaining oscilla o is ypically in a supe posi ion o Fock s a es.
Indeed each qubi eigens a e is coupled o a nono e lapping supe posi ion o Fock s a es so ha he ene gy o
he qubi no longe speci ies he ene gy o he oscilla o . Simila analyses hold o he in e ed o de o
sequen ial ope a ions, albei wi h a di e en cons ain on ini ial n.
The Wigne unc ion co esponding o he maximum cohe ence o he sequen ial cohe ence eme gence
is shown in igu e 2. The esul is a maximum qubi cohe ence o C ≈ln2 while he Wigne unc ion loses
o a ional symme y. Equa ion (4) shows ha acing ou he qubi esul s in a mix u e o supe posi ions,
each om a wo dimensional subspace. Due o his s uc u e he cohe ence is no inc eased by inc easing n,
in con as wi h wha ollows in he long ime eme gence o cohe ence. Thus, he cohe ence o he oscilla o
in equa ion (4) is indeed bounded by he wo dimensional subspace. Howe e , cohe ence only mildly
inc eases when epea ing he swi ching p ocedu e. Fu he mo e a ying he indi idual in e ac ion imes o
each s ep does no inc ease he cohe ence, and changing he o de o s a wi h V(2)dec eases he anges o
imes o which he maximum cohe ence eme ges. We now mo e o he mo e au onomous dynamics
wi hou swi ching, whe e hese limi a ions a e su passed.
3
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
Figu e 2. Sequen ial eme gence o cohe en quan um non-Gaussian s a es, C=0.7 by wo sequen ial linea and nonlinea
abso p i e ope a ions. The i s s ep, linea abso p ion, p epa es en anglemen be ween qubi and he oscilla o bu ails o
p oduce cohe ence in he oscilla o (see Wigne unc ion). A e he second s ep wi h a nonlinea abso p ion cohe ence al eady
eme ges and he o a ional symme y is s ongly b oken o highe Fock occupa ions. The ini ial s a e is he Fock s a e |7⟩, he
in e ac ion ime is =1.57, he same o bo h s eps, and he emaining de ailed pa ame e s a e gi en in igu e 3.
Figu e 3. The eme gence o cohe en quan um non-Gaussian s a es om equency us a ed nonlinea abso p ion, s imula ed
by ini ially incohe en Fock s a es in ba a ixed coupling a io g(2)
g(1)=0.1. The ini ial (τ=0) highly non-Gaussian s a e is he
Fock s a e |7⟩. The igh mos s a e co esponds o he maximum cohe ence, C≈4, achie ed o e he in e al 0 ⩽τ⩽2πa ime
τ=3.32 and mo e han 4 imes la ge han he sho ime app oxima ion discussed in he main ex . The cen al s a es
co espond o an example o a s a e wi h hal he maximum cohe ence, in his case achie ed a τ=0.95. The s a es emain
adically non-Gaussian, con aining many nega i e egions and o a ional symme y is comple ely los . The co esponding densi y
ma ices, wi h en ies ρnm, below he Wigne unc ions show ha V ends o gene a e s a es wi h supe posi ions be ween la ge
and small Fock s a es wi h en ies e y a om he o iginal Fock s a e, and only small con ibu ions om he g ound s a e. Mo e
de ails on he pa ame e choices a e gi en in he appendix.
2.2. Long- ime eme gence o cohe ence
Figu e 3shows ypical examples o he oscilla o a se e al s ages a e he compound in e ac ion
simul aneously in ol ing V(1)and V(2). The ini ial s a e is always aken o be he incohe en s a e |g⟩|n⟩,
whe e |g⟩is he g ound s a e o he abso be and |n⟩is a Fock s a e o he oscilla o . The a io o coupling
s eng hs is se o g(2)
g(1)=0.1, whe e he linea abso p ion s ill domina es, o e he ange 0 ⩽τ⩽2πwhe e
τ=g(2) is a scaled ime. De ails on hese pa ame e choices o he cohe ence dynamics a e gi en in an
appendix. Once us a ion o he ene gy conse a ion condi ions is in oduced ia he combined linea and
nonlinea abso p ion p ocesses subs an ial cohe ence is g adually gene a ed alongside s ikingly complex
Wigne dis ibu ions wi h s ongly b oken o a ional symme y and la ge densi y ma ix cohe ences
(o -diagonal elemen s o he densi y ma ices). As he Wigne unc ions de elop, hei p ominen nega i e
egions pe sis despi e he mixedness in oduced by he acing ou o he esonan abso be . Tha is, he
s a es p oduced do no belong o he class o s a es de ined by he con ex mix u e o Gaussian s a es. S a es
om his class may be non-Gaussian [19], bu hey do no possess any Wigne nega i i y. Impo an ly, he
o a ional symme y is g adually b oken in ime, as isible in igu e 3, and he Wigne unc ion app oaches a
comple ely new opology in phase space, going e en beyond he complexi y o hose cu en ly measu ed in
nonlinea po en ials [20]. The b eaking o he o a ional symme y combines classical cohe en displacemen
in phase space wi h quan um non-Gaussian symme y b eaking o he nega i e pa s o in e e ence e ec s
4
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
Figu e 4. The sp ead in o he Fock basis and us a ion o ene gy conse a ion is cap u ed by he ise in mean ene gy o he
sys em ⟨N⟩, accompanied by a la ge inc ease in he s anda d de ia ion ∆N. The maximum cohe ence occu s a he dashed
e ical line. The ba cha shows he maximum cohe ence gene a ed wi h g(2)
g(1)=0.1 o e he ange 0 ⩽τ⩽2πas a unc ion o
he ini ial Fock s a e |n⟩. The maximum cohe ence gene ally inc eases wi h nup o sa u a ion a n=7. The blue ba s indica e he
emo al o he Gaussian shell ia displacemen and squeezing ope a ions (de ailed in ex ). The cohe ence pe sis s and is hus well
beyond he co a iance ma ix app oxima ion.
in he Wigne unc ion. Such complex symme y b oken s uc u es appea close o he maximum o he
mean numbe o quan a along wi h a la ge inc ease in he noise.
The alue o he cohe ence gi es only an o e all iew o he Fock s a e supe posi ions con ibu ing o he
cohe ence. Examining he densi y ma ix cohe ences in igu e 3, hey sp ead deeply in o he Fock basis,
coupling low and high Fock s a es. This ea u e is no cap u ed by he sho - ime app oxima ion, no when
ex ended o a sequen ial scheme whe e he sho - ime app oxima ion ope a o s a e epea edly applied.
Addi ionally, he e is only a ma ginal con ibu ion om he g ound s a e. The mean numbe o quan a
p oduced in he ull dynamics (see igu e 4) is subs an ially highe han ha o sequen ial me hod and he
sp ead in o he Fock basis a beyond he ini ial occupa ion numbe is e lec ed in he g ow h o he mean
ene gy o he sys em ⟨N⟩≫7. This e ec is al eady known o linea abso p ion [21] bu he e is
accompanied by he eme gence o cohe ence. Tha is, linea abso p ion can esul in an inc ease in mean
ene gy, i he linea abso p ion is associa ed wi h blue-de uned in e ac ions. Howe e , i also esul s in a
educ ion o he noise in ene gy, as he ou pu s a es closely app oxima e Fock s a es. The addi ion o
nonlinea abso p ion esul s in a simul aneous inc ease in bo h mean ene gy ⟨N⟩and noise ∆N, which
allows o he eme gence o cohe ence.
Figu e 4also shows he inc ease in maximum cohe ence achie ed o e he ange 0 ⩽τ⩽2πas a
unc ion o ini ial Fock s a e. The e is a no able inc ease in he maximum achie able cohe ence wi h
inc easing n, up o sa u a ion a n=7. The blue ba s show he cohe ence a e he emo al o he Gaussian
app oxima ion, i.e. displacemen and squeezing a e applied un il he mean alues o X=1
√2(b+b†)and
P=i
√2(b†−b)a e ze o and he co a iance ma ix is diagonal wi h equal en ies. Quan um cohe ence due o
Gaussian displacemen /squeezing is hus emo ed indica ing ha he cohe ence beyond he Gaussian
app oxima ion is subs an ial. A nega i e Wigne unc ion emains nega i e unde Gaussian ope a ions hus
he cohe ence is s ongly connec ed o he quan um non-Gaussiani y o he s a e.
3. Discussion
3.1. Weak coupling egimes, dephasing and classical ini ial s a es
To ex end his esul o many possible expe imen al scena ios, we analyse wo signi ican po en ial obs acles
in e en well-isola ed oscilla o s: he p esence o ee e olu ion alongside he compound in e ac ion and
ex e nal dephasing p ocesses. Thus a , o simplici y, hese discussions ha e aken place in he ul a-s ong
coupling egime, in which he ee mo ion can be neglec ed. Rein oducing he ee mo ion adds signi ican
complexi y o an al eady in ac able p oblem. Howe e we a e in e es ed in he eme gence o cohe ence,
a he han i s op imisa ion. The e o e we compa e he maximum cohe ence when he ee mo ion is
ele an wi h he maximum cohe ence ob ained om ou example in igu e 3, keeping he emaining
pa ame e s unchanged. Below he sa u a ion obse ed a n=7 i is possible o ind egions ou side he
5
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
ul as ong coupling egime whe e he cohe ence can be enhanced. This e lec s a simila inding o qubi
cohe ence in he Rabi model eme ging om a simila uni a y se ing [22]. Tha is, his ema kable
eme gence o cohe ence is no es ic ed o he expe imen ally challenging ul as ong coupling egime, bu
is much mo e common and may e en be g ea e ou side i . Abo e n=7 i is ypical o he maximum
cohe ence o be in he ul as ong coupling egime. Howe e e en when he cohe ence is lowe ed he ee
mo ion does no signi ican ly impac he quan um non-Gaussiani y o complexi y and nega i i y o he
esul ing Wigne unc ions (examples in appendix). Simila ly, he densi y ma ices s ill show a subs an ial
sp ead in o he Fock basis cohe ences.
As expec ed, coupling o a dephasing en i onmen s ongly educes he cohe ence. We gi e an example in
he appendix whe e quan um non-Gaussian ea u es con i med by nega i i y o he Wigne unc ion appea
a sho e imes bu a e e en ually supp essed by he decohe ence p ocess, despi e lea ing a non-Gaussian
s a e wi h non i ial cohe ence. One may wonde i he eme gence o cohe ence om nonlinea abso p ion is
due o he nonclassical ea u es o he Fock s a es o o he sa u abili y o he qubi abso be , bo h o which
we ha e elied on h oughou . In ac he quali a i e ea u es o ou esul s hold o ini ial s a es which a e
classical mixed s a es showing only he mal noise in he exci a ion numbe , as well as when he qubi is
eplaced by an unsa u able oscilla o (see appendix). S ikingly, he nega i e ea u es o he Wigne unc ion
a e mo e obus o ini ial he mal noise han o decohe ence.
3.2. Ex ension o us a ion o ene gy conse a ion o o he cases
To c ea e cohe ence in a single oscilla o om an incohe en s a e he oscilla o ene gy mus change and he
in e ac ion Hamil onian mus no commu e wi h he oscilla o ee mo ion, [HΩ,V]=0. Wi h a o al
Hamil onian H=Hω+HΩ+V, he e a e wo dis inc possibili ies. Ei he (i) [Hω+HΩ,V] = 0 o (ii)
[Hω+HΩ,V]=0. Fo he i s case, i ollows ha [HΩ,V] = −[Hω,V]=0. In his case sum o he ene gies
o he subsys ems is conse ed, so ha o e all Vdesc ibes a globally passi e p ocess. In his case, e en hough
he local exci a ion numbe o he oscilla o can change, no cohe ence eme ges. Since he o al exci a ion
numbe is conse ed, any change in he ene gy o subsys em Hωis di ec ly compensa ed o by gain o loss o
ene gy in subsys em HΩ. Tha is, ene gy exchange be ween he subsys ems can be media ed passi ely by he
in e ac ion V, wi hou any ne exchange o ene gy s o ed in he in e ac ion, so ha he e is no unce ain y in
he oscilla o ene gy. This explains why phase insensi i e in e ac ions such as JC, beamspli e s, o e en
ilinea in e ac ions do no p oduce oscilla o cohe ence. Passi e in e ac ions do no p oduce cohe ence in
he oscilla o and his holds e en when he in e ac ion is locally ac i e.
Fo he second case he e a e wo subcases: (a) [HΩ,V] = 0 and (b) 0 = [HΩ,V]=−[Hω,V]. Fo case (a)
i is s ill possible o gene a e cohe ence. Fo example, he op omechanical in e ac ion a†a(b+b†)[23] will
gene a e cohe ence in he mechanical mode be en hough he op ical mode’s ee Hamil onian commu es
wi h he in e ac ion and simila ly o he dispe si e Rabi in e ac ion σz(b+b†)in supe conduc ing
ci cui s [24–27] and spin-mechanics [28]. This occu s e en i he op ical o qubi sys ems a e no p epa ed in
he g ound s a e. Since hei ene gy emains cons an , ye he oscilla o mode gains o loses ene gy, he e
mus be an ac i e con ibu ion om he in e ac ion. Fo he wo cases abo e, i comes because o he
coun e o a ing e ms in he in e ac ion; howe e , o ou case he e, we combine in e ac ions which a e each
sepa a ely in he o a ing wa e app oxima ion. This becomes e en mo e appealing in case (b), which
con ains he nonlinea abso p ion in e ac ion s udied in his manusc ip ; mo eo e as wi h he
op omechanical and dispe si e Rabi in e ac ions he nonlinea abso p ion me hod does no need o depend
on he sa u abili y o he abso be (see appendix), as in ou simple example o equa ion (2). Addi ionally ou
in e ac ion is no limi ed o he simples case we selec ed in ol ing k=1 and k=2. Any pai o kwill
con inue o be con ained in case (b), and p oduce cohe en quan um non-Gaussian s a es. The us a ion o
he condi ions o commu a ion o hold p e en s bo h subsys ems om conse ing hei ene gy in a
non i ial way: allowing only mu ually ac i e ans o ma ions ha may esul in oscilla o cohe ence.
4. Conclusion
Clea ly, non-Gaussian quan um supe posi ions in oscilla o s equi e ce ain minimal condi ions o be me in
o de o a ise wi hou ini ial cohe ence, di ec ex e nal cohe en d i ing o he subsys ems and
coun e o a ing e ms in he in e ac ion Hamil onians. He e we ha e used expe imen ally easible
phase-insensi i e in e ac ions o demons a e some o hese equi ed condi ions. Tha is, o his minimal
case, wi h all subsys ems p epa ed in incohe en s a es, he esul ing e olu ion mus in ol e a mu ually ac i e
ans o ma ion in o de o quan um cohe ence o eme ge. This is no a su icien condi ion, bu a necessa y
one, so any pa icula Hamil onian used like his mus also be ho oughly explo ed o such e ec s.
6
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
The necessi y o a mu ually ac i e ans o ma ion implies, h ough conse a ion o ene gy, ha ano he
physical sys em is a leas e ec i ely p esen and dona ing o ecei ing ene gy om he oscilla o (see
appendix o discussion). In many ecen cases o quan um echnology, his ex a physical sys em is in ac
he ex e nal d i e p esen du ing s a e gene a ion ha we ha e a oided h oughou he discussion. C ucially,
he e a e many sys ems in which an e ec i e Hamil onian dynamics can be de i ed which allows his
sou ce/sink o ene gy o be ully ex e nalised. The e ec i e Hamil onian obscu es he o igin o quan um
cohe ence which may equi e he comple ion o he ull Hamil onian as ou lined in he appendix. Ins ead o
sea ching o sys ems which can be ex e nalised in his way, one may look o na u ally occu ing o ces o
echnological a angemen s o ma e whose in e nal s uc u e con ains he equi ed ene gy sou ce/sink o
gene a e cohe ence wi hou ex e nal d i es o coun e o a ing e ms du ing he s a e gene a ion. Such o ces
o echnology may hen p o ide a minimal app oach o ese oi enginee ing, in which he in e nal s uc u e
eplaces he ex e nally d i en enginee ed en i onmen . This al e na i e ansien me hod is a new s a ing
poin in ese oi enginee ing me hods. Again, and in con as o hem, his me hod does no con ain ei he
coun e - o a ing e ms o ex e nal cohe en d i es du ing he s a e p epa a ion applied o incohe en
s a es [29]. Con inuing om he s a ing poin , such minimal mechanisms can be ex ended o he
gene a ion o quan um non-Gaussian s a es wi hou he abo e men ioned ools ypically used in
supe conduc ing ci cui s [25,30], apped ions [31], op omechanical sys ems [32,33], and wo and
mul i-mode non-Gaussian en anglemen clea ly dis inguishable om p e iously analysed cases [34].
Sea ching o such possibili ies beyond spin-mechanical in e ac ions wi h coun e o a ing e ms [28,35],
and using mode n echnology wi h quan um sys ems may open many exci ing doo s in a ious quan um
echnologies equi ing cohe en quan um non-Gaussian s a es [36–40].
The high-quali y Fock s a es used o ini ia e hese e ec s a e ou inely a ailable o apped ions and
supe conduc ing ci cui s [21,41]. Howe e , hey can also be ob ained wi h a pu ely linea cohe en
abso p ion p ocess wi hin he o a ing wa e app oxima ion [13,14], and a e hus a ailable o es ing hese
minimal condi ions. These p ocesses end o esul in he equi ed Fock s a e in an admix u e wi h he
g ound s a e. E en o e y high con amina ion his does no p e en he eme gence o cohe ence o he
quan um non-Gaussian ea u es we ha e discussed (see appendix). Mo eo e we a e no limi ed o such Fock
s a es, o impe ec e sions he eo , as a di ec obse a ion o hese e ec s can also eme ges om ini ially
incohe en he mal o Poissonian oscilla o s a is ics (see appendix) which a e also eadily p epa ed in
sys ems such as apped ions and supe conduc ing ci cui s.
Da a a ailabili y s a emen
The da a ha suppo he indings o his s udy a e openly a ailable a he ollowing URL/DOI: h ps://
zenodo.o g/ eco ds/15392092 [42].
Acknowledgmen
The au ho s acknowledge unding om P ojec No. GA22-27431S o he Czech Science Founda ion and he
p ojec CZ.02.01.01/00/22_008/0004649 (QUEENTEC) o EU and MEYS Czech Republic. R.F. was also
suppo ed by he Eu opean Union’s HORIZON Resea ch and Inno a ion Ac ions unde G an Ag eemen
no. 101080173 (CLUSTEC) and he Quan e a p ojec CLUSSTAR (8C24003) o MEYS Czech Republic.
P ojec CLUSSTAR has ecei ed unding om he Eu opean Union’s Ho izon 2020 Resea ch and Inno a ion
P og amme unde G an Ag eemen No. 731473 and 101017733 (Quan ERA).
Appendix A. Sequen ial me hod
We expand on he sequen ial scheme p oposed in sec ion 2.1. The sho ime app oxima ion using he i s
e ms o he BCH heo em is a i s s ep which al eady p oduces cohe ence and b eaks o al exci a ion
numbe conse a ion. Fi s , we no e he quan um non-Gaussian s a es ha eme ge om his dynamics and
compa e wi h he main esul in igu e 3. In igu e 5we show he e ec o his Hamil onian swi ching wi h
V(2)ini ia ing, n=7, and g(2)
g(1)=0.1. Repea ed swi ching does no subs an ially inc ease he a ailable
cohe ence, no does inc easing he ini ial Fock s a e occupa ion.
7
Quan um Sci. Technol. 10 (2025) 035048 K Adhika y e al
Figu e 5. An example o he Hamil onian swi ching p ocess using V(2) o ini ia e he qubi -oscilla o en anglemen . Each
in e ac ion has he same ime in e al , he ini ial oscilla o occupa ion is n=7 and he a io o coupling s eng hs is again
g(2)
g(1)=0.1.
Figu e 6. Cohe ence as a unc ion o ime o se e al coupling s eng h a ios and ini ial Fock s a es. F om le o igh ,
g(2)
g(1)=10,1,0.1. Fo ela i ely la ge g(2) he cohe ence dynamics has an oscilla o y cha ac e , which is los o lowe g(2)bu wi h
a la ge gain in achie able cohe ence. The sa u a ion wi h inc easing nis al eady isible he e.
Appendix B. Cohe ence dynamics
The cohe ence o he nonlinea abso p ion has a complex ime e olu ion and dependence on he ini ial s a e
and coupling s eng hs, displayed in igu e 6. In he main ex we ha e selec ed g(2)
g(1)=0.1 as i p oduces e y
la ge cohe ence. The oscilla o y beha iou is los , bu he magni ude o he cohe ence is g ea ly enhanced by
his choice. The maximum cohe ence gene ally appea s a τ=π. When his in e ac ion ime is ixed, i
becomes clea ha he maximum cohe ence ends o occu a ound g(2)
g(1)=0.1 as ninc eases, as in igu e 7.
Al hough some highe alues o nde ia e om his, he inc ease in cohe ence compa ed o his choice o
coupling s eng hs is negligible, as can be seen by compa ing igu es 7and 6. A e ixing his choice, he
maximum alues o cohe ence used in he main ex a e hose op imised o e he ime in e al 0 ⩽τ⩽2π.
Figu e 7also shows he imes a which hese maximum cohe ences occu as a unc ion o he ini ial Fock
occupa ion n. Al hough he e a e luc ua ions due o mino changes in maximum cohe ence, his con i ms
he in ui ion ha he maximum occu s a ound τ=π.
Appendix C. Weak coupling and decohe ence
He e we gi e g ea e de ails on he poin s made in he i s subsec ion om he discussion. Fi s ly, ou side he
ul as ong coupling egime he e ec s o he ee mo ion e ms o he Hamil onian a e ele an o he
cohe ence dynamics. As discussed in he main ex he e ec s we ha e desc ibed a e no limi ed o egimes
wi h such la ge in e ac ion s eng hs. Figu e 8shows he Wigne unc ion and densi y ma ix o he
maximum cohe ence o n=7 and g(2)
g(1)=0.1 wi h ω=Ω=1, esul ing in C=3.5. This well app oxima es
he maximum cohe ence achie ed in he ul as ong coupling egime bu p oduces qui e di e en ou pu
s a es. Ne e heless he quali a i e ea u es emain: la ge cohe ence, supe posi ions be ween dis an Fock
s a es, and complex Wigne unc ions wi h mul iple nega i e egions.
8
