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, Palacký Uni e si y, 17. lis opadu 1192/12, 771 46 Olomouc, Czech Republic
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.
I. INTRODUCTION
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 ba-
sis, 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 commu-
nica ion.
Su p isingly, basic quan um in e ac ions de e minis i-
cally 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 os-
cilla 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 o-
cesses such as measu emen , eed o wa d o d i en/dissi-
pa 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) mod-
els [16] (see Fig. 1).
In his pape we o e come his limi a ion using a non-
linea phase-insensi i e abso p ion added o he linea
∗kingshuk.adhika [email protected]
†da en.moo [email protected]
‡[email p o ec ed]ol.cz
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 idu-
ally ene gy conse ing 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 su-
pe 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 Fig. 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 non-
linea abso p ion in e ac ion wi h espec o cohe ence
gene a ion.
II. RESULTS
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 en-
e 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 os-
cilla 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
2
Ini ial S a es and In e ac ions Wigne Func ion T ans o ma ions
No Cohe ence!
No Cohe ence!
Cohe ence & QNG!
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|2i
<la exi sha1_base64="2WFZNUK SLwH/uWIe2aiPe /w3c=">AAACAXicbVDLSgNBEJyN xh UY9eBoPgKeyqRI9BLx4jmAckS5id9CZDZh/M9Aphyck 8Kp 4E28+iV+gP/hbLIHk1jQUFR1093lxVJo O1 q7C2 G5Vdwu7ezu7R+UD49aOkoUhyaPZKQ6H MgRQhNFCihEy ggSeh7Y3 M /9BEqLKHzESQxuwIah8AVnaKR2bwyYXk775Ypd Wegq8TJSYXkaPTLP71BxJMAQuSSad117BjdlCkUXMK01Es0xIyP2RC6hoYsAO2ms3On9MwoA+pHylSIdKb+nUhZoPUk8ExnwHCkl71M/M/ Juj uKkI4wQh5PNF iIpRjT7nQ6EAo5yYgjjSphbKR8xxTiahBa2eIqZZLJcnOUUVkn ou Uq WHq0 9Nk+oSE7IKTknD kmdXJPGqRJOBmTF/JK3qxn6936sD7n QU nzkmC7C+ gEOKp u</la exi >
|3i
<la exi sha1_base64="ZCEupdPG79U XRlT7ONpQoEX yo=">AAACAXicbVDLSsNAFJ34 PVVdekmWARXJRGpLo uXFawD2hDmUx 2iGTSZi5EU oyi9wq1/gT z6JX6A/+GkzcK2H hwOOde7 3HTwTX6Dj 1 6xubWdmmn Lu3 3BYOTpu6zhVDFosF Hq+lSD4BJayFFAN1FAI19Axw/ c /zBE zWD7iJAE oiPJA84oGqnTDwEzZzqoVJ2aM4O9S yCVEmB5qDy0x/GLI1AIhNU657 JOhlVCFnAqbl qohoSykI+gZKmkE2s m507 c6MM7SBWpiTaM/X REYj SeRbzojimO97OXi 14 xeDGy7hMUgTJ5ouCVNgY2/n 9pA YCgmhlCmuLnVZmOqKEOT0MIWX1GTTJ6Lu5zCKml 1 x6 5wVW3cFgmVyCk5Ix EJdekQe5Jk7QIIyF5Ia/kzXq23q0P63PeumYVMydkAdbXLwlil+s=</la exi >
|0i
<la exi sha1_base64="dWZ1ydzaS ee9Y/1qBs3awVXEVI=">AAACAXicbVDLSsNAFJ34 PVVdekmWARXJRGpLo uXFawD2hDmUx 2iGTSZi5EU oyi9wq1/gT z6JX6A/+GkzcK2H hwOOde7 3HTwTX6Dj 1 6xubWdmmn Lu3 3BYOTpu6zhVDFosF Hq+lSD4BJayFFAN1FAI19Axw/ c /zBE zWD7iJAE oiPJA84oGqnTDwEzdzqoVJ2aM4O9S yCVEmB5qDy0x/GLI1AIhNU657 JOhlVCFnAqbl qohoSykI+gZKmkE2s m507 c6MM7SBWpiTaM/X REYj SeRbzojimO97OXi 14 xeDGy7hMUgTJ5ouCVNgY2/n 9pA YCgmhlCmuLnVZmOqKEOT0MIWX1GTTJ6Lu5zCKml 1 x6 5wVW3cFgmVyCk5Ix EJdekQe5Jk7QIIyF5Ia/kzXq23q0P63PeumYVMydkAdbXLw 6l+w=</la exi >
|1i
<la exi sha1_base64="3mgo8sH6QRu00m RdjyoYTMlC8k=">AAACAXicbVDLSgNBEOyN xh UY9eBoPgKewGiR6DXjxGMA9IljA7mU2GzD6Y6RXCkpN 4FW/wJ 49U 8AP/D2WQPJ Ggoajqp Li6XQaN VmFjc2 7p7hb2 s/ODwqH5+0dZQoxlsskpHqelRzKULeQoGSd2PFaeBJ3 Emd5n eeJKiyh8xGnM3YCOQuELR FIn 6EY1qbDcoVu2 PQdaJk5MK5GgOyj/9YcSSgI IJNW659gxuilVKJjks1I/0TymbEJH GdoSAOu3XR+7oxcGGVI/EiZCpHM1b8TKQ20ngae6Qwoj Wql4n/eb0E/Rs3FWGcIA/ZYpG SIIRyX4nQ6E4Qzk1hDIlzK2EjamiDE1CS1s8RU0yWS7Oag pF2 O Vq/eGq0 jNEy CGZzDJThwDQ24hya0gMEEXuAV3qxn6936sD4X QU nzmFJVh wySl+0=</la exi >
|2i
<la exi sha1_base64="2WFZNUK SLwH/uWIe2aiPe /w3c=">AAACAXicbVDLSgNBEJyN xh UY9eBoPgKeyqRI9BLx4jmAckS5id9CZDZh/M9Aphyck 8Kp 4E28+iV+gP/hbLIHk1jQUFR1093lxVJo O1 q7C2 G5Vdwu7ezu7R+UD49aOkoUhyaPZKQ6H MgRQhNFCihEy ggSeh7Y3 M /9BEqLKHzESQxuwIah8AVnaKR2bwyYXk775Ypd Wegq8TJSYXkaPTLP71BxJMAQuSSad117BjdlCkUXMK01Es0xIyP2RC6hoYsAO2ms3On9MwoA+pHylSIdKb+nUhZoPUk8ExnwHCkl71M/M/ Juj uKkI4wQh5PNF iIpRjT7nQ6EAo5yYgjjSphbKR8xxTiahBa2eIqZZLJcnOUUVkn ou Uq WHq0 9Nk+oSE7IKTknD kmdXJPGqRJOBmTF/JK3qxn6936sD7n QU nzkmC7C+ gEOKp u</la exi >
|3i
Linea Abso p ion
Nonlinea Abso p ion
Combined Abso p ion
<la exi sha1_base64="wdwwa0yWWcdalpzBqNx4Ie2T KQ=">AAAB/nicbVDLSsNAFJ3UV62 qLhyM1gEoRASkeqy6MZlB uAJoTJdNIOnTyYuRFLKPg blwo4 b cO OG2z0NYDFw7n3Mu99wSp4Aps+9so ayu W+UNy b2zu7e+b+QVslmaSsRRORyG5AFBM8Zi3gIFg3lYxEgWCdYHQz9TsPTCqexPcwTpkXkUHMQ04JaMk3j1zFBxHxa6TmAnuE GhRa+KbVduyZ8DLxClIFRVo+uaX209oF EYqCBK9Rw7BS8nEjgVbFJxM8VSQkdkwHqaxiRiys n50/wqVb6OEyk hjwTP09kZNIqXEU6M6IwFA elPxP6+XQXjl5TxOM2AxnS8KM4EhwdMscJ9LRkGMNSFUcn0 pkMiCQWdWEWH4Cy+ Eza55ZT +p3F9XGdRFHGR2jE3SGHHSJGugWNVELUZSjZ/SK3own48V4Nz7m SWjmDlE 2B8/gCio5VL</la exi >
+a+ h.c.
<la exi sha1_base64="+ 9iF30Bz5sBOqeTb9dJ6 0wDaQ=">AAACAHicbVDLSsNAFJ3UV62 qAsXboJFEAohKVJdF 24 GA 0MQwmU7aoZMHMzdiCdn4K25cKOLWz3Dn3zh s9DWAxcO59zL 4CWcSLO bK62s q1 lDc W9s7u3 6/kFHxqkg E1iHouejyXlLKJ YMBpLxEUhz6nXX98P W7D1RIFkd3MEmoG+JhxAJGMCjJ048cyYYh9m 4 l5zgD5CNjKJmX 61TK GYxlYhekigq0PP3LGcQkDWkEhGMp+7aVgJ hAYxwmlecVNIEkzEe0 6iEQ6pdLPZA7lxqpSBEcRCVQTGTP09keFQyknoq84Qw0guelPxP6+ QnDpZixKUqARmS8KUm5AbEzTMAZMUAJ8oggmgqlbDTLCAhNQmVVUCPbiy8ukUz h m4Pa82 4o4yugYnaAzZKML1EQ3qIXaiKAcPaNX9KY9aS/au/Yxby1pxcwh+gP 8w VGJX </la exi >
+a2+ h.c.
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, indi-
ca 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 e-
sul 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) = 0.1and sho in e ac-
ion ime τ= 0.157.
Wigne Func ion T ans o ma ions
Cohe ence & QNG!No Cohe ence!
<la exi sha1_base64="cgFQ+X8QFamg786GQ LEaxpNPbc=">AAAB8HicbVDLTgJBEOzFF+IL9ehlIzHxRHaNAY9ELx4xkYeBDZkdGpgwM7uZmTUhK1/hxYPGePVz Pk3D AHBS ppFLVne6uMOZMG8/7dnJ 6xubW/n ws7u3 5B8 CoqaNEUWzQiEeqHRKNnElsGGY4 mOFRIQcW+H4Zua3HlFpFsl7M4kxEGQo2YBRYqz08FT KiKHHH Fkl 25nBXiZ+REmSo94p 3X5EE4HSUE607 hebIKUKMMox2mhm2iMCR2TIXYslUSgD L5wVP3zCp9dxApW9K4c/X3REqE1hMR2k5BzEg ezPxP6+TmMFVkDIZJwYlXSwaJNw1kT 73u0zhdTwiSWEKmZ demIKEKNzahgQ/CXX14lzYuyXylX7i5L essjjycwCmcgw9VqME 1KEBFAQ8wyu8Ocp5cd6dj0V zslmjuEPnM8 w4iQZw==</la exi >
|7i
<la exi sha1_base64="wdwwa0yWWcdalpzBqNx4Ie2T KQ=">AAAB/nicbVDLSsNAFJ3UV62 qLhyM1gEoRASkeqy6MZlB uAJoTJdNIOnTyYuRFLKPg blwo4 b cO OG2z0NYDFw7n3Mu99wSp4Aps+9so ayu W+UNy b2zu7e+b+QVslmaSsRRORyG5AFBM8Zi3gIFg3lYxEgWCdYHQz9TsPTCqexPcwTpkXkUHMQ04JaMk3j1zFBxHxa6TmAnuE GhRa+KbVduyZ8DLxClIFRVo+uaX209oF EYqCBK9Rw7BS8nEjgVbFJxM8VSQkdkwHqaxiRiys n50/wqVb6OEyk hjwTP09kZNIqXEU6M6IwFA elPxP6+XQXjl5TxOM2AxnS8KM4EhwdMscJ9LRkGMNSFUcn0 pkMiCQWdWEWH4Cy+ Eza55ZT +p3F9XGdRFHGR2jE3SGHHSJGugWNVELUZSjZ/SK3own48V4Nz7m SWjmDlE 2B8/gCio5VL</la exi >
+a+ h.c.
<la exi sha1_base64="+ 9iF30Bz5sBOqeTb9dJ6 0wDaQ=">AAACAHicbVDLSsNAFJ3UV62 qAsXboJFEAohKVJdF 24 GA 0MQwmU7aoZMHMzdiCdn4K25cKOLWz3Dn3zh s9DWAxcO59zL 4CWcSLO bK62s q1 lDc W9s7u3 6/kFHxqkg E1iHouejyXlLKJ YMBpLxEUhz6nXX98P W7D1RIFkd3MEmoG+JhxAJGMCjJ048cyYYh9m 4 l5zgD5CNjKJmX 61TK GYxlYhekigq0PP3LGcQkDWkEhGMp+7aVgJ hAYxwmlecVNIEkzEe0 6iEQ6pdLPZA7lxqpSBEcRCVQTGTP09keFQyknoq84Qw0guelPxP6+ QnDpZixKUqARmS8KUm5AbEzTMAZMUAJ8oggmgqlbDTLCAhNQmVVUCPbiy8ukUz h m4Pa82 4o4yugYnaAzZKML1EQ3qIXaiKAcPaNX9KY9aS/au/Yxby1pxcwh+gP 8w VGJX </la exi >
+a2+ h.c.
Figu e 2. Sequen ial eme gence o cohe en quan um non-
Gaussian s a es, C= 0.7by wo sequen ial linea and non-
linea 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 Fig. 3.
ene gy eigenbasis. We will ypically ake his o be he
g ound s a e o HΩ, which can be app oached by cool-
ing. Fo a qubi subsys em we ha e HΩ=Ω
2σz, wi h σz
a 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 inciple 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 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 gen-
e 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 [Eq. (2)] wi h di e en k.
A. 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) ab-
so 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 ≪1we 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 se-
quen ial me hod p o ides an al e na i e and immedia ely
3
La ge Symme y B eakSmall Symme y B eakIni ial Symme ic Fock S a e
<la exi sha1_base64="u4Zh +AI6mB48L7osQJCx 3QFHo=">AAAB8HicbVBNSwMxEM3W 1q/qh69BI gadl pXoRil48V A 0i4lm2bb0CS7JLNCK 0VXjwo4 W 481/Y9 uQVs DDzem2FmXpgIbsDz p3c2 G5lZ+u7Czu7d/UDw8apo41ZQ1aCxi3Q6JYYI 1gAOg UTzYgMBWuFo9uZ33pi2 BYPcA4YYEkA8UjTglY6bELJL2uuJVy 1jyXG8O E 8jJRQhnq +NX xzSVTAEVxJiO7yUQTIgGTgWbF qpYQmhIzJgHUsVkcwEk/nBU3xmlT6OYm1LAZ6 ycmRBozlqH lASGZ mbi 95nRSiq2DCVZICU3SxKEoFhhjP sd9 hkFMbaEUM3 ZgOiSYUbEYFG4K//PIqaZZd +pW7y9K Zssjjw6Qa oHPnoE XQHaqjBqJIom 0i 4c7bw4787HojXnZDPH6A+czx9waY+I</la exi >
⌧=3.32
<la exi sha1_base64="AjHU8kZdPVAPJ/eaTWlQ+KcSlj0=">AAAB8HicbVBNSwMxEJ2 X7V+VT16CRbB07I W UgFL14 GA/pF1KNs22oUl2SbJCW o HhQxKs/x5 /x Tdg1Y DDzem2FmXphwpo3n TmFpeWV1bXiemljc2 7p7y719Rxqgh kJjHqh1iTTmT GGY4bSdKIpFyGk HN1M/dYjVZ F8 6MExoIPJAsYgQbKz10DU6 PP y FeueK43A/pL/JxUIEe9V/7s9mOSCioN4Vj ju8lJsiwMoxwOil1U00TTEZ4QDuWSiyoD LZwRN0ZJU+imJlSxo0U39OZFhoPRah7RTYDPWiNxX/8zqpiS6CjMkkNVSS+aIo5cjEaPo96jNFieFjSzBRzN6KyBA TIzNqGRD8Bd /kuaJ65 da 3p5XadR5HEQ7gEI7Bh3OowS3UoQEEBDzBC7w6ynl23pz3eW ByW 24Recj295 o+O</la exi >
⌧=0.95
<la exi sha1_base64="MXX2EM7as BBAO/+Vw65sUO+psA=">AAAB7XicbVBNS8NAEJ34We X1aOXxSJ4KolI9SIU XisYD+gDWWz3bR N9mwOxFK6H/w4kER /4 b/4b 20O2 pg4PHeDDPzgkQKg6777ays q1 bBa2i s7u3 7pYPDplGpZ zBlFS6HVDDpYh5AwVK3k40p1EgeSsY3U791hPXRqj4AccJ9yM6iEUoGEU Nb I02u3Vyq7FXcGsky8nJQhR71X+u 2FUsjHiOT1JiO5yboZ1SjYJJPi 3U8ISyER3wjqUxjbjxs9m1E3JqlT4JlbYVI5mp ycyGhkzjgLbGVEcmkV K 7ndVIM /xMxEmKPGbzRWEqCSoy Z30heYM5dgSy Sw xI2pJoy AEVbQje4s LpHle8aqV6 1FuXaTx1GAYziBM/DgEmpwB3VoAINHeIZXeHOU8+K8Ox/z1hUnnzmCP3A+ wAXYY7U</la exi >
⌧=0
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.
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 ab-
so 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 illu-
mina ing he ion a he k− h sideband. The e o e imple-
men a ion o he swi ching p o ocol equi es only ha
wo such sideband a e independen ly a ailable and con-
ollable in he same se up [7].
In Fig. 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 decou-
ple 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 num-
be 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)pn(n−1) cos g(1)√n (5)
β( ) = −sin g(2)pn(n+ 1) sin g(1)√n (6)
γ( ) = −icos g(2)pn(n+ 1) sin g(1)√n (7)
δ( ) = −isin g(2)pn(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 su-
pe 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 Fig. 2. The esul is a maximum qubi co-
he ence o C ≈ ln 2 while he Wigne unc ion loses o-
a ional symme y. Eq. (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
4
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 Eq. (4) is
indeed bounded by he wo dimensional subspace. How-
e e , cohe ence only mildly inc eases when epea ing he
swi ching p ocedu e. Fu he mo e a ying he indi id-
ual 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) de-
c eases he anges o imes o which he maximum co-
he 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.
B. Long- ime Eme gence o Cohe ence
Fig. 3shows ypical examples o he oscilla o a se -
e al s ages a e he compound in e ac ion simul ane-
ously in ol ing V(1) and V(2). The ini ial s a e is al-
ways 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 dom-
ina 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 den-
si y ma ix cohe ences (o -diagonal elemen s o he den-
si 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 isi-
ble in Fig. 3, and he Wigne unc ion app oaches a com-
ple 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 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 co-
he ence. Examining he densi y ma ix cohe ences in
Fig. 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
Ene gy ise ia composi e abso p ion
Cohe ence ise ia composi e abso p ion
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) = 0.1o 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.
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 Fig. 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 ac-
companied 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
5
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.
Fig. 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. displace-
men 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.
III. DISCUSSION
A. Weak Coupling Regimes, 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 o-
lu ion alongside he compound in e ac ion and ex e nal
dephasing p ocesses. Thus a , o simplici y, hese dis-
cussions 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 i-
misa ion. The e o e we compa e he maximum cohe ence
when he ee mo ion is ele an wi h he maximum co-
he ence ob ained om ou example in Fig. 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
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 co-
he 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 de-
cohe 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.
B. Ex ension o F 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 inco-
he 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 oscil-
la 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 ]=0o (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 pas-
si 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 co-
he 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
6
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 non-
linea 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 sim-
ple example o Eq. 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.
IV. CONCLUSION
Clea ly, non-Gaussian quan um supe posi ions in os-
cilla 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 co-
he 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.
The necessi y o a mu ually ac i e ans o ma ion im-
plies, h ough conse a ion o ene gy, ha ano he phys-
ical 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 ech-
nology, 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 dynam-
ics can be de i ed which allows his sou ce/sink o en-
e gy o be ully ex e nalised. The e ec i e Hamil o-
nian 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 en-
ginee 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 ap-
plied 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 h-
ou he abo e men ioned ools ypically used in supe -
conduc ing ci cui s [25,30], apped ions [31], op ome-
chanical 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 co-
he 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.
V. ACKNOWLEDGEMENTS
The au ho s acknowledge unding om P ojec No.
21-13265X o he Czech Science Founda ion. This wo k
has ecei ed unding om he Eu opean Union’s 2020
esea ch and inno a ion p og amme (CSA–Coo dina ion
and Suppo Ac ion, H2020-WIDESPREAD-2020-5) un-
de G an Ag eemen No. 951737 (NONGAUSS).
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Plo s - Cohe ence o quan um non-Gaussian s a es
ia nonlinea abso p ion o quan a (a ailable a :
h ps://zenodo.o g/ eco ds/15392092)
Appendix A: Sequen ial Me hod
We expand on he sequen ial scheme p oposed in Sec-
ion II A. The sho ime app oxima ion using he i s
8
0.0 0.5 1.0 1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
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.
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 Fig. 3. In Fig. 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.
Appendix B: Cohe ence Dynamics
The cohe ence o he nonlinea abso p ion has a com-
plex ime e olu ion and dependence on he ini ial s a e
and coupling s eng hs, displayed in Fig. 6. In he main
ex we ha e selec ed g(2)
g(1) = 0.1as i p oduces e y la ge
cohe ence. The oscilla o y beha iou is los , bu he mag-
ni 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 Fig. 7. Al hough some highe alues
o nde ia e om his, he inc ease in cohe ence com-
pa ed o his choice o coupling s eng hs is negligible, as
can be seen by compa ing Fig. 7and Fig. 6. A e ix-
ing 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. Fig. 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. Fig. 8shows he Wigne unc ion
and densi y ma ix o he maximum cohe ence o n= 7
and g(2)
g(1) = 0.1wi 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.
No ably, he la ge alues o cohe ence do no simply
occu close o he ul as ong coupling egime, as demon-
s a ed in Fig. 9. Below sa u a ion he e a e la ge egions
whe e he maximum achie ed cohe ence goes beyond ha
o ul as ong coupling, and in addi ion do no ely on
esonance condi ions. Beyond sa u a ion howe e , hese
egions become ex emely small and ul as ong coupling
is mos e icien .
Decohe ence na u ally opposes he eme gence o co-
he ence om he combina ion o linea and nonlinea ab-
so p ion. We illus a e his in Fig. 10. Wi h dephasing
on he oscilla o a a a e 10% ha o g(1), quan um
non-Gaussian cohe ence eme ges, as seen in he nega i e
egions o he cen al Wigne unc ion, bu is hen sup-
p essed by he decohe ence.
Replacing he highly nonclassical Fock s a es wi h di-
agonal s a es possessing he mal noise in he Fock ba-
sis does no elimina e he eme gence o cohe ence, al-
hough as expec ed he amoun o cohe ence and is-
ibili y o quan um non-Gaussian ea u es educes. We
show he mal s a es wi h mean phonon numbe ¯n= 7 in
Fig. 11. S ikingly, despi e ha ing much lowe cohe ence
han he p e ious example gene a ed unde decohe ence,
he quan um non-Gaussian ea u es a e mo e obus o
ini ial he mal noise in he Fock basis. These ea u es
a e inc eased when he he mal s a e is eplaced by phase
andomised cohe en s a es (Fig. 12), which a e also di-
agonal in he Fock basis bu possess Poissonian noise.
In an al e na i e model he abso be can be subs i-
u ed by an oscilla o mode awi h equency Ω, gene -
a ing he Mul i-Wa e (MW) mixe in e ac ion
V(k)
MW =g(k)a†bk+a(b†)k.(C1)
Again, hese in e ac ions p ese e he o al exci a ion
numbe NMW =ka†a+b†bwi h a simila equency
condi ion o he qubi abso be sys em. The amoun o
cohe ence p oduced by hei combina ion is signi ican ly
educed compa ed o he qubi abso be , compa ed wi h
simila pa ame e s. Howe e he Wigne unc ions, while
9
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.
10-2 10-1 100101102
G
0
1
2
3
4
5
C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0 3 6 9 12 15
7n
0
:/2
:
3:/2
2:
=max
Figu e 7. Top: The maximum cohe ence as a unc ion o
he coupling a io G=g(2)
g(1) a in e ac ion ime τ=π. As
he ini ial Fock occupa ion ninc eases he coupling a io a
which he maximum cohe ence occu s con e ges o G= 0.1
up o n= 10. Fo highe n, luc a ions ake he Gaway
om his alue, bu he maximum does no change much as
can be seen in Fig 6, igh panel. Bo om: The in e ac ion
imes τa which he maximum cohe ence is ex ac ed o
G= 0.1. These gene ally occu a ound τ=π, al hough some
luc ua ions occu , pa icula ly a n= 1. Howe e hese a e
small, as can again be seen in Fig 6.
quali a i ely di e en , e ain hei complexi y and quan-
um non-Gaussian ea u es, again despi e he impu i y
induced by he pa ial ace. The densi y ma ices on
he o he hand show ha qui e di e en supe posi ions
esul . The e is a much g ea e con ibu ion om he
classical g ound s a e and ewe supe posi ions be ween
Fock s a es wi h la ge numbe di e ences. Fig 13 shows
a eplica o he example s a es o Fig. 3 o he MW in-
e ac ion.
Appendix D: Cohe ence F om Globally Ac i e
T ans o ma ions
The us a ion o he commu a ion ela ions means
ha he uni a y e olu ion gene a ed by he Hamil onians
Hω+HΩ+Vdesc ibes a globally ac i e ans o ma ion
in which he sys em gains ene gy om he in e ac ion.
In o de o conse a ion o ene gy o hold, he in e ac-
ion mus be an app oxima ion o a la ge sys em om
which his ex a ene gy is d awn. Le us gi e a simple
example o such a phenomenon. Conspicuously absen
om ou discussion o hese ea u es is he possibili y o
sel -in e ac ions o he oscilla o which a e ac i e, such
as single-mode squeezing. Squeezing changes he ene gy
o he oscilla o , and his ene gy mus come om and go
somewhe e. In ac he squeezing sel -in e ac ion Hamil-
onian is s ic ly an app oxima ion o a la ge wo mode
sys em which can ac i ely media e an exchange o en-
e gy ia he downcon e sion o single pho ons in o pai s.
Tha is, he single-mode squeeze is a ilinea in e ac-
ion in which he pump mode does no deple e and can
be ea ed semi-classically [42]. In ac le us dwell u -
he on his example; he ilinea in e ac ion alone does
no p oduce cohe ence in he a ge oscilla o (c ucially,
using ou assump ions on p epa a ion!), ye he semi-
classical app oxima ion o his in e ac ion gene a es he
single-mode squeezing in e ac ion which does p oduce co-
he ence. The ilinea in e ac ion is globally passi e, ye
he single-mode squeeze is ac i e. This is because he
ilinea in e ac ion ep esen s an in e ac ion deple ing
he ex e nal cohe en d i e, ope a ing in a ully quan um
mechanical way and is ully ene gy conse ing among he
eal oscilla o subsys ems. In o de o see he single-mode
squeeze , one has o linea ise he ilinea in e ac ion by
d i ing he pump mode s ongly i.e. he e is an app oxi-
ma ely undeple ed use o classical cohe en s a es, which
allows he p ocess o gain ene gy and p oduce cohe ence.
In he same way, he nonlinea abso p ion sys em can
be seen as an incomple e desc ip ion o a la ge sys em.
A candida e in e ac ion Hamil onian o comple e he de-
sc ip ion in ol es he addi ion o an ex a oscilla o mode
a, so ha we ha e
Vcompl =g(1) σ+ba +σ−b†a†+g(2) σ+b2+σ−b†2
(D1)
Now he in e ac ion in ol es a hi d mode wi h ee
