Topological aspects in nonlinear optical frequency conversion Stefano Longhi1 2 1Dipartimento di Fisica Politecnico di Milano Piazza L. da Vinci 32 I-20133 Milano Italy

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Topological aspects in nonlinear optical frequency conversion

Stefano Longhi1, 2

1Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133 Milano, Italy

2IFISC (UIB-CSIC), Instituto de Fisica Interdisciplinar y Sistemas Complejos - Palma de Mallorca, Spain

(Dated: October 24, 2022)

Nonlinear optical frequency conversion, observed more than half a century ago, is a corner stone

in modern applications of nonlinear and quantum optics. It is well known that frequency conver-

sion processes are constrained by conservation laws, such as momentum conservation that requires

phase matching conditions for eﬃcient conversion. However, conservation laws alone could not fully

capture the features of nonlinear frequency conversion. Here it is shown that topology can provide

additional constraints in nonlinear multi-frequency conversion processes. Unlike conservation laws,

a topological constraint concerns with the conserved properties under continuous deformation, and

can be regarded as a new indispensable degree of freedom to describe multi-frequency processes. We

illustrate such a paradigm by considering sum frequency generation under a multi-frequency pump

wave, showing that, akin topological phases in topological insulators, topological phase transitions

can be observed in the frequency conversion process both at classical and quantum level.

I. INTODUCTION

Since the ﬁrst observation of optical harmonics more

than half a century ago [1], frequency conversion and

wave mixing processes in nonlinear optical media [2–5]

have enabled the manipulation and control of the elec-

tromagnetic radiation to a great extent, with a variety of

applications ranging from coherent harmonic generation

[4–8] to ultrafast optics and nonlinear spectroscopy

[9–11], quantum optics [12–18], nonlinear imaging and

biological microscopy [19,20], to mention a few.

Modern nonlinear optics has borrowed many concepts

from quantum mechanics and condensed-matter physics,

and in return, enriched the variety of theoretical and

experimental platforms where quantum phenomena can

be studied (see e.g. [21–31] and references therein).

Prominent examples include the geometric (Berry)

phase accompanying nonlinear frequency mixing [22],

adiabatic processes in frequency conversion [21,27,28],

and the design of novel photonic structures which

combine topological phases of light with appreciable

nonlinear response [22], thus extending to the nonlinear

realm the recent developments in the area of topological

photonics [32–36]. Recently, it has been suggested that

various nonlinear optical eﬀects can be described in a

uniﬁed fashion by topological quantities involving the

Berry connection and Berry curvature [37].

Frequency conversion processes in nonlinear χ(2) media,

such as sum/diﬀerence frequency generation and para-

metric down-conversion, are constrained by conservation

laws: energy, ﬂux, momentum and angular momentum

of photons should be conserved during the nonlinear

interaction [2,3,38,40]. Such conservation laws are

expressed by well-known conditions, such as the Manley-

Rowe relations and the phase matching requirement for

momentum conservation. However, conservation laws

alone could not fully capture the properties of nonlinear

frequency conversion. In this work we unravel that,

akin to topological phases in condensed matter physics

[41–43], topology can provide additional constraints to

nonlinear multi-frequency conversion processes, which

can undergo topological phase transitions. Unlike

conservation laws, topology concerns with the conserved

properties under continuous deformation, and can be

regarded as a new indispensable degree of freedom to

describe nonlinear frequency conversion processes.

To unveil the topological aspects underlying fre-

quency conversion, let us consider the process of sum

frequency generation (SFG), where two input photons

at frequencies ω1(signal wave) and ω2(pump wave) an-

nihilate while, simultaneously, one photon at frequency

ω3=ω1+ω2(SFG wave) is created under perfect

phase matching in the nonlinear crystal. The process is

quite simple when we deal with single-frequency ﬁelds,

while topological features emerge when we consider

multi-frequency waves. Let us assume that we inject

one signal photon at frequency ω1and a stream of N2

and N0

2pump photons at slightly diﬀerent frequencies

ω2and ω0

2=ω2+ Ω, respectively [Fig.1(a)]. Clearly,

the signal photon can annihilate with one pump photon

of either frequency ω2or ω0

2, so that the frequency

of the SFG photon can be either ω3=ω1+ω2or

ω0

3=ω1+ω0

2=ω3+ Ω with probabilities N2/(N2+N0

and N0

2/(N2+N0

2), respectively. In repeated measure-

ments, on average the frequency of the SFG photon is

thus hω3i=ω3+νΩ, with ν=N0

2/(N2+N0

2). Clearly, ν

is not quantized, i.e. it not an integer number, and could

be any real number depending on the values of N2and

2. However, this result holds for a short interaction

length z: further interaction in the nonlinear crystal

makes it possible the backward process, i.e. the newly

generated SFG photon can annihilate and generate a

pair of signal and pump photons. Energy conservation

imposes that the frequency of the created signal photon

should belong to the set {ω1, ω1−Ω, ω1+ Ω}. Such a

newly created signal photon can then annihilate with one

pump photon to generate a SFG photon at a frequency

that must belong to the set {ω3−Ω, ω3, ω3+ Ω, ω3+ 2Ω}

for energy conservation. This reasoning can be iterated

arXiv:2210.11526v1 [physics.optics] 20 Oct 2022

FIG. 1. Multi-frequency SFG. (a) A weak monochro-

matic signal wave at frequency ω1interacts with a strong

bi-chromatic pump wave, at frequencies ω2and ω0

2=ω2+ Ω,

in a nonlinear crystal to generate a SFG wave. (b) The signal

and SFG photons describe in tandem a quantum walk in a

synthetic binary lattice in frequency space. The probability

distribution of the photon frequency, depicted on a pseudo-

color map, broadens as the interaction length zincreases. The

topology of the synthetic lattice provides a constraint on the

mean frequency of signal and SFG photons.

and the frequency of both signal and SFG photons

basically undergo a diﬀusion process in frequency space.

Hence, as the interaction length zin the nonlinear crystal

increases, we have an evolving probability distribution

for the frequency of the created SFG photon. Energy

conservation requires that such a frequency should

belong to the set ω3+nΩ (ninteger), but does not

pose any constraint about the mean value hω3iof such

a distribution, which in principle could take any value

ω3+νΩ with νa real number. Here topology comes into

play: as we show in this work, in the multi-frequency

conversion process the signal and SFG photons describe

in tandem a quantum walk on a topological lattice

in synthetic (frequency) space [Fig.1(b)], resulting

in the quantization of νfor long interaction lengths.

Speciﬁcally, the integer νturns out to be a topological

invariant (winding number) associated to the synthetic

lattice and determined by the multi-frequency properties

of the injected strong pump wave. This is the main

message of this work, which is developed and presented

with the due mathematical details in the next sections.

II. TOPOLOGICAL SIGNATURE IN

SUM-FREQUENCY GENERATION WITH A

MULTI-FREQUENCY PERIODIC PUMP WAVE

A. Classical analysis

The quantization of νcan be readily proved in the

framework of a classical analysis of three-wave frequency

mixing in a non-linear χ(2) crystal. In the plane-wave

approximation, the electric ﬁeld propagating along the

longitudinal zdirection of the crystal can be written as

E(z, t) = 1

2(3

l=1 r2~ωl

0c0nl

ψlexp(−iωlt+iklz) + c.c.),

where ω1,ω2and ω3are the carrier frequencies of signal,

pump and SFG waves, respectively, kl= (ωl/c0)nlare

the wave numbers and nl=n(ωl) the (linear) refractive

indices. Under perfect phase matching, the three coupled

equations governing the evolution of the ﬁeld envelopes

ψl(z, t) read (see e.g. [2,3,15,27,38,39])

i∂

∂z +1

vg1,2

∂

∂t ψ1,2=−σψ3ψ∗

2,1(1)

i∂

∂z +1

vg3

∂

∂t ψ3=−σψ1ψ2,(2)

where σ≡[de/(n1n2n3)]p2~k1k2k3/0,deis the ef-

fective nonlinear interaction coeﬃcient, and vgl =

1/(dk/dω)ωlis the group velocity at carrier frequency

ωl. In the above equations, the ﬁeld envelopes have been

normalized such that |ψl|2is the photon ﬂux of the e.m.

wave at frequency ωl. As usual in problems of sum and

diﬀerence frequency generation [2,21,27,44], we assume

that the crystal is excited by a strong pump ﬁeld, not nec-

essarily monochromatic, and by a monochromatic weak

signal at frequency ω1. In the undepleted pump approx-

imation and after letting ξ=zand η=t−z/vg3, one

has ψ2(ξ, η)'ψ2(ξ= 0, η), and Eqs.(1,2) reduce to the

linear two-level equations

i∂ψ1

∂ξ =i1

vg2−1

vg1∂ψ1

∂η +h(η)ψ3(3)

i∂ψ3

∂ξ =i1

vg2−1

vg3∂ψ3

∂η +h∗(η)ψ1(4)

where h(η)≡ −σψ∗

2(ξ= 0, η) describes the temporal

shape of the injected strong pump wave. As shown in Ap-

pendix A, for a suﬃciently spectrally-narrow pump wave

the group velocity mismatch terms can be neglected, so

that Eqs.(3,4) can be readily integrated with the initial

condition ψ1(ξ= 0, η) = 1 and ψ3(ξ= 0, η) = 0, yielding

ψ1(ξ, η) = cos[∆(η)ξ]

ψ3(ξ, η) = −isin[∆(η)ξ] exp[−iϕ(η)],

where we have set h(η)≡∆(η) exp[iϕ(η)], i.e. ∆(η)

and ϕ(η) are the amplitude and phase of the normal-

ized pump wave. Let us now assume that h(η) is peri-

odic with period T= 2π/Ω, i.e. that the pump wave

carries a stream of photons at frequencies ω2+nΩ, and

let us set k= Ωη. Correspondingly, the signal and SFG

wave ψ1,3(ξ, k) are periodic with respect to kwith pe-

riod 2πand can be thus written as a Fourier series,

ψ1,3(ξ, k) = Pl(al, bl)(ξ) exp(−ilk) with ξ-dependent

amplitudes al(ξ), bl(ξ). At the propagation distance ξ,

the mean of the frequency of the signal wave, given by

hω1i=ω1+PllΩ|al(ξ)|2/Pl|al(ξ)|2, reads hω1i=ω1,

whereas the mean frequency of the SFG wave, given by

hω3i=ω3+PllΩ|bl(ξ)|2/Pl|bl(ξ)|2, can be written as

hω3i=ω3+νΩ, where we have set (technical details are

given in Appendix A)

ν=Rπ

−πdk sin2[∆(k)ξ]∂ϕ

∂k

Rπ

−πdk sin2[∆(k)ξ](5)

If we assume that ∆(k)6= 0 for any k, i.e. that the pump

wave is non-vanishing at any time instant, for long inter-

action lengths ξwe can set sin2[∆(k)ξ]'1/2 in Eq.(5),

yielding ν'(1/2π)Rπ

−πdk(dϕ/dk)≡ν∞. This relation

clearly shows that the index νis quantized and equals

the phase spanned by the pump wave in one oscillation

cycle, normalized to 2π. For example, for an injected

bichromatic pump at frequencies ω2and ω0

2=ω2+ Ω,

h(k) = h0+h1exp(ik) and thus ν= 0 for |h0|>|h1|and

ν= 1 for |h0|<|h1|, the case |h0|=|h1|corresponding

to a topological phase transition.

To illustrate the quantization of νin a realistic setting,

let us consider SFG in a periodically-poled lithium nio-

bate (PPLN) crystal with a strong pump at the wave-

length λ2= 810 nm and a weak signal at λ1= 1.55 µm.

The SFG wave corresponds to λ3= 532 nm. We as-

sume extraordinary wave propagation, with a nonlinear

coeﬃcient d33 '27 pm/V. Phase matching is realized

by a ﬁrst-order QPM grating (7.38 µm period), so that

de= (2/π)d33 [45]. Figure 2 shows the behavior of

the index νversus propagation distance zin the crys-

tal for a bichromatic pump wave with a frequency oﬀset

Ω=2π×1 GHz and with two diﬀerent values of the ra-

tio h1/h0=p(I1/I0) between the two harmonic pump

amplitudes. The simulations take into account group ve-

locity mismatch, as calculated using Sellmeier equations

for n(ω) [46]. The ﬁgure clearly illustrates the asymp-

totic quantization of νfor long interaction lengths and

the topological phase transition as the ratio of pump in-

tensities I1/I0varies from below to above one.

B. Quantum analysis

The quantization of the index νpredicted by the clas-

sical analysis can be at best captured in the second-

quantization framework of SFG [44,47–51]. Here, the

signal and SFG photons undergo in tandem a quantum

walk on a synthetic lattice with nontrivial topology in fre-

quency space, the index νcorresponding to a topological

invariant of the lattice. The second-quantization analysis

shows that the topological origin of ν-quantization holds

for an arbitrary non-classical state of the injected signal

wave, i.e. not necessarily for classical (coherent) states.

As in the classical analysis, we assume a multi-frequency

pump with frequencies ω2+nΩ, centered at around the

carrier ω2,and neglect group-velocity mismatch eﬀects.

The second-quantization Hamiltonian of the photon ﬁeld

FIG. 2. Quantization of index ν.(a,b) Behavior of the

index νversus propagation distance in the process of SFG in

a PPLN crystal. The pump wave is bichromatic with pump

intensities I0and I1at frequencies ωpand ωp+ Ω. In (a)

I0= 800 MW/cm2,I1= 400 MW/cm2; in (b) I0= 400

MW/cm2,I1= 800 MW/cm2. The insets show the behavior

of h(k) = h0+h1exp(ik) in complex plane, parametrized

in the scaled time k=ηΩ. Parameter values are given in

the text. (c) Synthetic SSH lattice in frequency space along

which the signal and SFG photons undergo a quantum walk

in tandem.

then reads [44,47]

H=ˆ

H0+ˆ

HI,

where

H0=X

~(ω1+nΩ)ˆa†

nˆan+X

~(ω3+nΩ)ˆ

b†

nˆ

bn+

~(ω2+nΩ)ˆc†

nˆcn

is the Hamiltonian of the free ﬁeld,

HI=−~σvgX

n,l

(ˆ

bn+lˆa†

lˆc†

n+H.c.)

is the interaction Hamiltonian, ˆan,ˆ

bnand ˆcnare the

bosonic annihilation operators of photon modes at fre-

quencies ω1+nΩ, ω3+nΩ and ω2+nΩ, respectively. As-

suming a strong and classical pump wave, the operators

ˆcncan be considered as c-numbers [44], and the Heisen-

berg equations of motion of the destruction operators ˆan,

bn, after the transformation ˆan→ˆanexp[−i(ω1+nΩ)t],

bn→ˆ

bnexp[−i(ω3+nΩ)t], read (see Appendix B for

details)

idˆan

dt =−σvgX

C∗

lˆ

bn+l, idˆ

dt =−σvgX

Clˆan−l(6)

where Cn=hcniand the interaction time tis related

to the interaction length ξby the relation t=ξ/vg.

Equation (6) indicates that the signal and SFG photons

undergo in tandem a continuous-time quantum walk on

the sublattices A and B of a one-dimensional (1D) lat-

tice with chiral symmetry and long-range hopping am-

plitudes σvgC∗

l, which provides an extension of the fa-

mous Su-Schrieﬀer-Heeger (SSH) 1D topological insula-

tor [41,52]. Note the the c-numbers Clare basically the

Fourier amplitudes of the classical strong pump wave-

form, namely ψ2(k) = PlClexp(−ikl), with k= Ωη.

After letting ˆ

ψ1(k, t) = Pnˆanexp(−ikn) and ˆ

ψ3(k, t) =

Pnˆ

bnexp −(ikn), the evolution equations for the oper-

ators ˆ

ψ1,3(k, t) read i(d/dt)( ˆ

ψ1,ˆ

ψ3)T=vgH(k)( ˆ

ψ1,ˆ

ψ3)T

with matrix Hamiltonian

H(k) = 0h(k)

h∗(k) 0 

= ∆(k){cos[ϕ(k)]σx−sin[ϕ(k)]σy}(7)

where we have set

h(k) = −σψ∗

2(k)≡∆(k) exp[iϕ(k)]

and σx,y are the Pauli matrices. Note that the Heisen-

berg equations for the ˆ

ψ13 operators are analogous to the

classical ones [Eqs.(3) and (4)] with vg1=vg2=vg3=vg

after the substitution t→z/vgand considering ˆ

ψ1,3(k, t)

as c-numbers.

Let us assume that the crystal is excited with a

monochromatic signal ﬁeld at frequency ω1in an arbi-

trary quantum state, given by a superposition of Fock

states |ψ(0)i=P∞

l=1(αl/√l!)ˆa†l

0|0iwith arbitrary am-

plitudes αland Pl|αl|2= 1. Note that excitation

with a single-photon Fock state corresponds to αl=

δl,1, whereas excitation with a classical ﬁeld (a coher-

ent state) corresponds to a Poisson distribution αl=

αlexp(−|α|2/2)/√l!, with α=ψ1(0). After a propa-

gation distance ξ=vgt, the mean value of the frequency

of the signal and SFG photon ﬁelds can be readily calcu-

lated and read (details are given in Appendix B)

hω1i=ω1,hω3i=ω3+νΩ,

where the value of νis the same as the one obtained

from the classical analysis [Eq.(5)], regardless of the ini-

tial state |ψ(0)iof the signal photon ﬁeld.

C. Frequency conversion and winding number

The main result, that unravels the topological as-

pects in the SFG process, is that for long interaction

lengths ξthe index νconverges to the topological invari-

ant (winding number) ν∞of the 1D gapped topologi-

cal insulator. For example, if we assume a bichromatic

pump as in the simulations of Fig.2, corresponding to

h(k) = h0+h1exp(ik), the signal and SFG photons un-

dergo a quantum walk on a synthetic SSH lattice in fre-

quency space with alternating hopping amplitudes h0and

h1[see Fig.2(c)], the two sublattices A and B correspond-

ing to the various frequency components ω1,3+nΩ of the

two ﬁelds. The topological invariant of a 1D gapped topo-

logical insulator with chiral symmetry is provided by the

Zak phase γ±of the two lattice bands, given by [41]

γ±=iZπ

−π

dkhu±|∂

∂k u±i=1

2Zπ

−π

dk ∂ϕ

∂k =πν∞

where

u±=1

√21

±exp[−iϕ(k)] 

are the two eigenstates of the Bloch Hamiltonian H(k)

[Eq.(7)] corresponding to the eigen-energies ±|h(k)|, and

ν∞=1

2πZπ

−π

dk ∂ϕ

∂k

is the winding number. Note that the Zak phase in the

two bands takes and same value, related to the winding

number ν∞, and that ν∞is the asymptotic value of ν(ξ)

[Eq.(5)] as ξ→ ∞. The quantization of νas ξ→ ∞,

such as the one observed in Fig.2(a,b), can be explained

in terms of the asymptotic quantization of the mean

displacement that the signal and SFG photons undergo

in the tandem quantum walk in the synthetic frequency

space. In fact, as shown in previous works [53–58]

for a gapped 1D topological insulator such a mean

displacement is asymptotically quantized and equals the

winding number ν∞of the topological lattice. According

to the bulk-boundary correspondence [41,52,59],

|ν∞|measures the number of topologically-protected

zero-energy edge states, and the quantum walk provides

a bulk probing method to measure |ν|[53].

III. TOPOLOGICAL SIGNATURES UNDER A

MULTIFREQUENCY APERIODIC PUMP

The previous analysis can be extended to the case

where the envelope ψ2(η) of the strong pump wave is

aperiodic in time and given by the superposition of N

mutually-incommensurate frequencies Ω1, Ω2,..., ΩN. In

this case, the signal and SFG photons undergo a quantum

walk on a high-dimensional synthetic lattice in frequency

space [60], which can display nontrivial topological fea-

tures.

Let us consider the simplest case of N= 2 incommen-

surate frequencies Ω1and Ω2, and let k1= Ω1ηand

k2= Ω2η. The temporal pump waveform ψ2(η) can be

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TopologicalaspectsinnonlinearopticalfrequencyconversionStefanoLonghi1,21DipartimentodiFisica,PolitecnicodiMilano,PiazzaL.daVinci32,I-20133Milano,Italy2IFISC(UIB-CSIC),InstitutodeFisicaInterdisciplinarySistemasComplejos-PalmadeMallorca,Spain(Dated:October24,2022)Nonlinearopticalfrequencyconversion,ob...

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Topological aspects in nonlinear optical frequency conversion Stefano Longhi1 2 1Dipartimento di Fisica Politecnico di Milano Piazza L. da Vinci 32 I-20133 Milano Italy

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