Dissipative Pairing Interactions Quantum Instabilities Topological Light and Volume-Law Entanglement Andrew Pocklington1 2Yu-Xin Wang1and A. A. Clerk1

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Dissipative Pairing Interactions: Quantum Instabilities, Topological Light, and
Volume-Law Entanglement
Andrew Pocklington,1, 2 Yu-Xin Wang,1and A. A. Clerk1
1Pritzker School of Molecular Engineering, University of Chicago,
5640 South Ellis Avenue, Chicago, Illinois 60637, USA
2Department of Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA
We analyze an unusual class of bosonic dynamical instabilities that arise from dissipative (or non-
Hermitian) pairing interactions. We show that, surprisingly, a completely stable dissipative pairing
interaction can be combined with simple hopping or beam-splitter interactions (also stable) to gen-
erate instabilities. Further, we find that the dissipative steady state in such a situation remains
completely pure up until the instability threshold (in clear distinction from standard parametric
instabilities). These pairing-induced instabilities also exhibit an extremely pronounced sensitivity
to wave function localization. This provides a simple yet powerful method for selectively populating
and entangling edge modes of photonic (or more general bosonic) lattices having a topological band
structure. The underlying dissipative pairing interaction is experimentally resource-friendly, requir-
ing the addition of a single additional localized interaction to an existing lattice, and is compatible
with a number of existing platforms, including superconducting circuits.
Introduction.—Hamiltonian bosonic pairing interac-
tions (where excitations are coherently created or de-
stroyed in pairs) arise in many settings, and underpin a
vast range of phenomena. In the context of quantum op-
tics and information, they are known as parametric am-
plifier interactions, and are a basic resource for generat-
ing squeezing and entanglement [1,2]; they also form the
basis of quantum limited amplifiers [3,4]. In condensed
matter settings, bosonic pairing underlies the theory of
antiferromagnetic spin waves, interacting Bose conden-
sates, and can also be used to realize novel topological
band structures [5,6].
Given the importance of bosonic pairing, it is interest-
ing to explore the basics of purely dissipative (or non-
Hermitian) bosonic pairing. Non-Hermitian dynamics
have garnered attention in a wide range of fields, from
condensed matter [7–9] to optics [10–12] to classical dy-
namical systems [13–15]. In this Letter, we provide a
comprehensive analysis of dissipative bosonic pairing in
a fully quantum setting, showing it possesses a number
of surprising and potentially useful features. We focus on
minimal, experimentally realizable models, where bosons
(e.g. photons) hop on a lattice, in the presence of a sin-
gle dissipative pairing interaction. Remarkably, we find
that while the dissipative pairing interaction on its own
yields fully stable dynamics, when combined with simple
lattice hopping (which is also stable), one can have dy-
namical instability. Further, close to such an instability,
the quantum steady state is perfectly pure, with a se-
lected subset of modes having high densities and strong
squeezing and/or entanglement correlations. The com-
plete state purity up until the instability threshold is a
clear distinction from more standard instabilities associ-
ated with Hermitian pairing terms. Dissipative pairing
is also distinct from the well-studied situation where a
system is driven with squeezed noise; in particular, driv-
ing a quadratic, particle-conserving system with squeezed
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b)a)
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
 
 
 
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FIG. 1. Stability diagram for a minimal three-mode bosonic
system (see inset) with loss on mode ˆa(rate κ), gain on mode
ˆc(rate η2κ), and tunnel couplings J1, J2[cf. Eq. (3)]. In
the absence of dissipative pairing, the system is dynamically
unstable above the dashed line. Adding dissipative pairing
iηκ(ˆaˆc+ H.c.) shifts the onset of instability to the solid line,
see Eq. (4). Remarkably, this boundary is independent of
J/κη, where J=pJ2
1+J2
2. The dissipative steady state
remains pure (with a high density) as one approaches insta-
bility, see main text. Red lines in each plot are the same
cut of parameter space, J/κη = 1 and J1/J2= 0.75. Solid
lines show hopping, dashed line shows the dissipative pairing
interaction.
noise can never generate instability, whereas this readily
occurs with dissipative pairing.
Dissipative pairing becomes even more interesting
when combined with topological band structures. We
find that our new pairing instabilities are highly suscepti-
ble to wave function localization of the underlying lattice
Hamiltonian. Hence, if the lattice supports exponentially
localized topological edge modes, we are able to selectiv-
ity excite and entangle them. Such topological systems
remain a cornerstone of condensed matter physics [16–
18] and photonics [19–21], and selectively exciting edge
modes has been the subject of a flurry of recent proposals
[22–27]. These are motivated by applications including
topological lasing [24,27–30] and topological amplifica-
arXiv:2210.09252v2 [quant-ph] 23 Mar 2023
2
tion and squeezing [25,31–33]. However, these proposals
often require complicated momentum and/or energy se-
lectivity [25–27], as well as control over the entire lattice,
[25–27,31]. Here, we are able to get edge-mode selectiv-
ity almost for free, using a single quasilocal dissipative
interaction.
Minimal model.—We start with a three-mode system
(bosonic annihilation operators ˆa, ˆ
b, ˆc) that exhibits much
of the surprising physics of interest. The key ingredient
will be a dissipative pairing interaction between ˆaand ˆc,
that is an interaction generating dynamics of the form
∂thˆai=−λhˆc†iand ∂thˆc†i=λ∗hˆai. Because of the rela-
tive sign here, this dynamics cannot be obtained from a
Hermitian pairing interaction. Instead, it would seem to
correspond to a non-Hermitian effective Hamiltonian:
ˆ
Hpairing =−i(λˆa†ˆc†+ H.c.).(1)
To obtain this Markovian dissipative dynamics in a
fully quantum setting, this dissipative interaction must
necessarily be accompanied by noise as well as local
damping and antidamping [8,34]. The resulting descrip-
tion has the form of a Lindblad master equation [35,36].
Using a minimal noise realization of the interaction, and
letting ˆρdenote the system density matrix, we obtain
˙
ˆρ=ˆ
Lˆρˆ
L†−(ˆ
L†ˆ
L
2,ˆρ)≡ D[ˆ
L]ˆρ, ˆ
L=√κˆa+η√κˆc†.
(2)
This purely dissipative evolution generates local damp-
ing on ˆawith strength κ, local antidamping on ˆcwith
strength η2κ, and a dissipative interaction of the form of
Eq. (1) with λ=ηκ/2. We take η < 1 (i.e. more local
damping than antidamping), which ensures dynamical
stability (i.e., no tendancy for exponential growth) [37].
The dissipation in Eq. (2) is reminiscent of the dy-
namics generated by driving modes ˆa, ˆcwith broadband
two-mode squeezed (TMS) noise [38]. There are, how-
ever, crucial differences. Driving with TMS noise always
generates two dissipators; to make Eq. (2) equivalent to
injected TMS nosie, we would thus have to add the addi-
tional dissipator D[√κˆc+η√κˆa†]. This complementary
dissipator would completely cancel the effective dissipa-
tive interaction between aand cgenerated by D[ˆ
L], leav-
ing only driving with correlated noise. There would thus
be no interaction from the dissipation in the equations of
motion between hˆa(t)iand hˆc†(t)i. In contrast, we will
show that in Eq. (2), the direct dissipative interaction
between modes ˆaand ˆcplays a crucial role.
To see explicitly that dissipative pairing is distinct
from input TMS noise, we will add coherent hopping
interactions to our system, and consider the evolution
of average values. The hopping is described by ˆ
H=
J1ˆa†ˆ
b+J2ˆ
b†ˆc+ H.c., with the evolution now given by
∂tˆρ=−i[ˆ
H,ˆρ] + D[ˆ
L]ˆρ. Because of linearity, the equa-
tions of motion for averages of mode operators are in-
sensitive to noise, and only influenced by interactions
(coherent and dissipative). For our system, a symme-
try argument [37] lets us reduce the dynamics of these
averages to the closed linear dynamics of the quadra-
tures ~v = (xa, pb, xc), where hˆai= (xa+ipa)/√2, etc;
the orthogonal quadratures (pa, xb, pc) have an analogous
closed evolution. We find ∂t~v =−iD~v, where the dy-
namical matrix D=DJ+Dκcan be interpreted as an
effective 3 ×3 Hamiltonian matrix, and
DJ=
0iJ10
−iJ10−iJ2
0iJ20
, Dκ=κ
2
−i0−iη
0 0 0
iη 0iη2
.
(3)
The off-diagonal ±iη κ
2terms in Dκare the dissipative
interaction, which surprisingly adds a Hermitian contri-
bution at the level of the dynamical matrix. This mir-
rors the fact that had we started with a nondissipative
Hermitian pairing interaction, we would generate a non-
Hermitian dynamical matrix [39]. Note that the hopping
dynamics on its own generates stable dynamics, as does
the dissipative dynamics on its own. More formally, both
the matrices DJand Dκhave no eigenvalues with pos-
itive imaginary part and hence are dynamically stable
(in the Lyapunov sense [40] that there is no tendency for
exponential growth).
We now come to our first surprise: while each part
of our dynamics (hopping, dissipation) is stable individ-
ually, combining them can lead to instability. We find
that for the full dynamics, whenever J16=J2, there will
be a critical value of ηbeyond which we have exponential
growth. Specifically, one can show [37] that the dynami-
cal matrix in Eq. (3) will be unstable if
η > min (|J1/J2|,|J2/J1|).(4)
We stress that this phenomenon is distinct from recently
studied “dissipation-induced instabilities” [41], where the
purely dissipative dynamics is already unstable on its
own. Again, in our case the system is always stable in
the dissipation-only limit J1=J2= 0.
The instability threshold Eq. (4) can be understood
from a simple perturbative argument that is formally
valid only when κJ1, J2[akin to a Fermi’s golden
rule (FGR) calculation]. If we define |ψii(i= 1,2,3) to
be the (nondegenerate) eigenvectors of DJ, and treat Dκ
as a small perturbation on top of this, then to first order
|ψiihas a relaxation rate
Γi=−Imhψi|Dκ|ψii.(5)
If an eigenmode has more amplitude on ˆcthan ˆa, there
will be a value of η < 1 at which Eq. (5) is negative.
This corresponds exactly to the condition in Eq. (4), and
is easy to understand intuitively (i.e. the eigenmode sees
more antidamping than damping). Surprisingly, this sim-
ple FGR argument turns out to be exact to all orders in
3
κ: Eq. (4) is not perturbative [37]. We stress that this
is a nonobvious phenonmenon. For example, consider
a modified model where we eliminate dissipative pairing
by replacing D[ˆ
L]→ D[√κˆa] + D[√κηˆc†] in our master
equation. We are left with just incoherent gain and loss.
In this case, the instability threshold would depend sen-
sitively on the value of κ, with the FGR prediction only
valid for κ→0, see Fig. 1.
We thus see that even at the semiclassical level, the
dissipative pairing interaction yields surprises: instabil-
ity from the combination of two individually stable dy-
namical processes, with a threshold that is independent
of the overall dissipation scale. Note that the above phe-
nomena could alternatively be described (in a squeezed
frame) as the interplay of asymmetric loss and Hermitian
pairing interacting (see [37] for details and application to
two-mode models).
Extension to quantum lattices.—We now explore dissi-
pative pairing in general multimode lattice systems, fo-
cusing on the possibility of nontrivial dissipative steady
states. Consider an N-site bosonic lattice, with annihi-
lation operators ˆaifor each site. The coherent dynamics
corresponds to a quadratic, number conserving Hamilto-
nian ˆ
H=Pij Hij ˆa†
iˆaj. The only constraint we impose
is that Hpossesses an involutory chiral sublattice sym-
metry U, such that UHU†=−H; our simple three-site
model also had this symmetry. Chiral symmetry ensures
that for every eigenmode of Hwith nonzero energy, there
is a different eigenmode with an opposite energy.
We now add a single dissipative pairing interaction to
the lattice, between two arbitrary sites 0,1. Motivated by
our three-mode example, we take 0,1 to be on the same
sublattice (as defined by the chiral symmetry). The full
dynamics on the lattice is given by [42]
∂tˆρ=−i[ˆ
H,ˆρ] + D[ˆ
L]ˆρ, ˆ
L/√κ= ˆa0+ηˆa†
1.(6)
Our goal is to understand instabilities and steady states
of this setup. Note that previous work studied chiral-
symmetric bosonic lattices driven by single-mode squeez-
ing [44]. Such systems are completely distinct from our
setup: they do not have any dissipative pairing inter-
action, never exhibit dynamical instability, and (unlike
what we describe below) always yield steady states with
aspatially uniform average density.
We start by diagonalizing ˆ
H. Using chiral symmetry,
we can write ˆ
H=Pα≥0α(ˆ
d†
αˆ
dα−ˆ
d†
−αˆ
d−α). Eigenmode
annihilation operators are given in terms of real space
wave functions by ˆ
d±α=Piψ±α[i]ˆai.ˆ
His invariant
under two-mode squeezing transformations that mix a
pair of ±αmodes [37]: for arbitrary rα, φα∈R, if we
take
ˆ
β±α≡cosh(rα)ˆ
d±α+eiφαsinh(rα)ˆ
d†
∓α,(7)
then ˆ
H=Pαα(ˆ
β†
αˆ
βα−ˆ
β†
−αˆ
β−α).
We would like to find a set of rα, φαsuch that
ˆ
L=√κX
α
Nα(ˆ
βα+ˆ
β−α).(8)
If this is possible, the system dynamics are stable, and
we will have a unique steady state (vacuum of the ˆ
β±α
operators). Achieving Eq. (8) requires for each α > 0
[37]:
tanh rα=η
ψα[1]
ψα[0]∗
, φα= arg ψα[1]
ψα[0]∗.(9)
with |Nα|2=|ψα[0]|2(1 − |tanh rα|2).
We now make a crucial observation: Eq. (9) only has a
solution if η < (|ψα[0]∗/ψα[1]| ≡ ηα). If this condition is
violated for a particular α, then the dynamics is unstable:
in this case, we are forced to write ˆ
Lin terms of a Bogoli-
ubov raising operator in the (α, −α) sector, implying that
the dissipation looks like antidamping in this sector. At
a heuristic level, for η > ηα, the αmodes see more gain
than loss. Overall stability requires η < min ηα≡ηc, a
condition that is independent of the dissipation strength
κ. We thus have a generalization and rigorous justifica-
tion of the surprising FGR-like instability condition in
Eqs. (4) and (5) we found for the three-mode model.
Our arguments above imply that as long as η < ηc,
we are dynamically stable and have a pure steady state,
where each (α, −α) pair is in a two-mode squeezed vac-
uum with a squeezing parameter given by Eq. (9). This
will in general be a highly entangled state. Further, as
η→ηcfrom below, the squeezing parameter of the criti-
cal modes is diverging, meaning that we will have a pure
state where a small subset of modes contribute to a di-
verging photon number. Note this is very distinct from
just incoherent gain and loss, which never has a pure
steady state. This behavior is also completely distinct
from standard parametric instabilities, where the steady
state becomes extremely impure as one approaches in-
stability [45,46]. The mode selectivity leads to a highly
nonuniform density that can be exploited for applica-
tions, as we now discuss.
Dissipative pairing and topological edge states.— The
physics discussed above is particularly striking when ap-
plied to chiral hopping Hamiltonians ˆ
Hthat have topo-
logical band structures. There are many such models,
as chiral symmetry is a key part of the standard classi-
fication of topological band structures [47]. As our dis-
sipative interaction always pairs opposite energy modes,
edge modes will only be paired with edge modes, bulk
modes only with bulk modes. Moreover, it is easy to
ensure that the correlated steady-state photon density is
concentrated on the edges. Edge-mode wave functions
are exponentially damped in the bulk, so Eq. (9) tells us
for an edge state α
tanh rα=η
ψα[1]
ψα[0]∝ηe(d0−d1)/ζL,(10)
4
Dissipative Pairing
ത
1ത
0
ത
0ത
1
Dissipative Pairing
ത
1ത
0
FIG. 2. Steady state correlation functions for a 99-site
SSH chain with δ=−0.65. There is a single jump oper-
ator of the form of Eq. (6) with 0 = 4 and 1 = 0, and
η= 0.999ηc∼0.045. The squeezing correlation functions
show a pure, single-mode squeezed state exponentially local-
ized to the edge. Inset: Schematic of the dissipatively stabi-
lized SSH chain. A single jump operator generates a dissipa-
tive pairing interaction, selectively exciting the edge mode.
where d1,0is the distance from 1 and 0 to the edge, re-
spectively, with ζLthe localization length scale of the
edge modes. If d0−d1>0 (i.e. the gain site closer to the
edge than the loss site), we obtain a superexponential en-
hancement in the squeezing parameter of the edge modes.
This yields large populations and squeezing on the edge
(while still having a pure state), see Figs. 2and 3.
For large enough systems, the bulk modes will be
nearly translationally invariant, implying they will have
tanh rα=η. Thus, by spreading the two sites out over a
few localization lengths ζL, a weak pump rate η1
can set tanh rα∼1 for only the edge modes. Here,
the total number of excitations in the bulk would be
very small, hˆnαi=O(η2), whereas the number of ex-
citations in the edge mode, as one approaches instabil-
ity, will be superexponentially enhanced and scales like:
hˆnαi=O([1 −ηe(d0−d1)/ζL]−1).
The upshot is that by using a single dissipative pair-
ing interaction, we can selectively populate, squeeze and
entangle edge modes of a topological bosonic band struc-
ture. Such states could be useful for applications in topo-
logical photonics [20], and are reminiscent of topological
lasing states [24,27] (which typically require complex
schemes to only pump the edge states). We analyze this
physics more carefully below for two prototypical topo-
logical hopping models (see Fig. 3).
SSH chain.—A paradigmatic topological model is the
Su–Schrieffer–Heeger (SSH) chain [48,49], see Fig. 2in-
set. This is a linear, 1D lattice with staggered hopping
strengths, given by the Hamiltonian
ˆ
H=−J
N−1
X
i=1
(1 + (−1)iδ)ˆa†
iˆai+1 + H.c.(11)
Such a model has been realized with bosons in a variety
of experiments (e.g. [28–30,50]). The topological regime
of ˆ
Hadmits one (two) protected edge modes if there are
an odd (even) number of lattice sites, with a localization
length ζL= (1 + δ)/(1 −δ). As α→ −1, ζL→0, and
the edge modes become infinitely localized.
FIG. 3. (a) A 24 ×24 site Hofstadter lattice, which has uni-
form hopping and a quarter flux per plaquette Φ = 1
4Φ0.
There is a single dissipator of the form of Eq. (6), with
1 = (11,23) and 0 = (12,20), and with η= 0.999ηc∼0.0007.
The color corresponds to local steady state photon number,
which is exponentially localized to the edges of the lattice.
(b) The same system, now showing steady state squeezing
correlations between the randomly chosen edge site (18,23)
and the rest of the lattice. Every edge site has exponentially
enhanced squeezing with every other edge site on the same
sublattice.
We consider for simplicity an odd number of lat-
tice sites (see [37] for even N). This yields a single
zero-energy edge mode, localized on a single sublattice.
Hence, if we place the pairing dissipator on the correct
sublattice, we can selectively excite just the edge mode
into a single-mode squeezed vacuum with a superexpo-
nentially enhanced squeezing parameter. The dissipative
steady state for such a situation is plotted in Fig. 2. We
thus have a resource-friendly approach for creating topo-
logically protected, bright nonclassical squeezed light, us-
ing a SSH chain with a single, quasilocal, linear dissipa-
tor. One could imagine using the stabilized photons by
weakly coupling the edge lattice site to an output waveg-
uide, see [37] for more details. Note that topological fea-
tures of the SSH chain are protected against disorder in
the hopping coefficients up to the bulk gap 4|δ|J. We
find that the qualitative nature of the dissipative steady
state is also protected against hopping disorder over a
similar scale (see [37]).
Hofstadter lattice.— 2D topological systems admit ex-
tended boundaries, allowing one to more easily study en-
tanglement properties. Motivated by this, we consider
a finite, quarter-flux Hofstadter lattice [51]. This corre-
sponds to a square lattice with a quarter magnetic flux
quanta per plaquette (see Fig. 3) giving the Hamiltonian:
ˆ
H=X
m,n
ˆa†
m,nˆam+1,n +eiπm/2ˆa†
m,nam,n+1 + H.c., (12)
which has been realized experimentally in Refs. [52–54].
This Hamiltonian supports exponentially localized
modes which propagate chirally around the edge [52,53,
55,56]. The fact that they are extended around the full
edge is critical for generating long-range entanglement.
With the same prescription of adding a dissipative in-
5
teraction of the form of Eq. (6) with 1 on the edge and 0
in the bulk, the steady state solution has exponentially
localized edge photon density, with nearly all-to-all edge
correlations, Fig. 3. For a fixed η, these sites will obey a
volume-law scaling in entanglement entropy, [37], where
maximally separated edge sites are now highly entangled,
Fig. 3. Having all edge sites lie on the same topological
boundary is crucial for this to occur [37].
In the limit that η→ηc, the steady state will be dom-
inated by the topological edge modes approaching insta-
bility. Treating the edge as a ring, we can label these by
their momenta k; the steady state has all momenta kand
k+πin a TMS vacuum. Close enough to instability, a
single momentum will dominate, generating uniform edge
photon densities, see Fig. 3(a), and a “checkerboard” of
correlations, see Fig. 3(b). The checkerboard is a result
of the chiral symmetry, which admits only correlations
within a sublattice. The values of the correlations and
densities can be understood directly from Eq. (10), where
hˆni,j i ∼ sinh(rk)2and hˆai,j ˆai0,j0i ∼ sinh(rk) cosh(rk)
are superexponentially enhanced compared to the bulk
modes. This gives an arbitrary amount of entanglement
between any two edge sites on the same sublattice as
η→ηc. This also means that for a relatively weak di-
mensionless pumping (η < 10−3in Fig. 3), the steady
state can still have a large number of photons (O(102) in
Fig. 3), that is completely independent of the strength of
the dissipation κcompared to the Hamiltonian.
Implementation.— The basic master equation is natu-
rally suited for any circuit- or cavity-QED experimental
platform that can generate tunable couplings, along with
an engineered lossy mode. Quantum systems that have
been able to successfully create topological photonic or
phononic lattices span superconducting circuits [50,54],
micropillar polariton cavities [28], photonic cavities [57],
photonic crystals [55], ring resonators [29,30,52,53,58],
and optomechanics [59,60]. In order to generate the
requisite jump operator in Eq. (6), one can couple the
dissipation sites to an auxiliary bosonic mode ˆ
bwith the
interaction
ˆ
HI=gˆ
b†(ˆa0+ηˆa†
1)+H.c.(13)
In the limit that the auxiliary mode ˆ
bis very lossy with
a loss rate κg, this gives the desired jump operator,
with an effective strength Γ = 4g2/κ, [61]. This allows
one to easily engineer the desired reservoir with few ad-
ditional resources.
Conclusions.— We have demonstrated that dissipative
pairing interactions lead to a previously unexplored class
of instabilities in bosonic systems, where stable Hamil-
tonians and stable dissipation combine to give unstable
dynamics. We have shown that these instabilities are
incredibly sensitive to topological boundaries, providing
a new mechanism to selectively excite topological edge
modes without needing any momentum or frequency se-
lectivity. Moreover, the steady state of the dynamics
remains pure all the way up to the instability point, al-
lowing one to populate the edge with an arbitrary number
of zero-temperature excitations. Our ideas are compat-
ible with a variety of different experimental platforms,
and require few resources to implement.
This work is supported by the Air Force Office of Sci-
entific Research under Grant No. FA9550-19-1-0362, and
was partially supported by the University of Chicago Ma-
terials Research Science and Engineering Center, which is
funded by the National Science Foundation under Grant
No. DMR-1420709. A.C. also acknowledges support from
the Simons Foundation through a Simons Investigator
Award (Grant No. 669487, A.C.).
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摘要：

DissipativePairingInteractions:QuantumInstabilities,TopologicalLight,andVolume-LawEntanglementAndrewPocklington,1,2Yu-XinWang,1andA.A.Clerk11PritzkerSchoolofMolecularEngineering,UniversityofChicago,5640SouthEllisAvenue,Chicago,Illinois60637,USA2DepartmentofPhysics,UniversityofChicago,5640SouthEllisA...

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Dissipative Pairing Interactions Quantum Instabilities Topological Light and Volume-Law Entanglement Andrew Pocklington1 2Yu-Xin Wang1and A. A. Clerk1.pdf

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