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Received 25 Nov 2015 | Accepted 20 Jan 2016 | Published 2 Mar 2016

There is enormous interest in engineering topological photonic systems. Despite intense activity, most works on topological photonic states (and more generally bosonic states) amount in the end to replicating a well-known fermionic single-particle Hamiltonian. Here we show how the squeezing of light can lead to the formation of qualitatively new kinds of topological states. Such states are characterized by non-trivial Chern numbers, and exhibit protected edge modes, which give rise to chiral elastic and inelastic photon transport. These topological bosonic states are not equivalent to their fermionic (topological superconductor) counterparts and, in addition, cannot be mapped by a local transformation onto topological states found in particle-conserving models. They thus represent a new type of topological system. We study this physics in detail in the case of a kagome lattice model, and discuss possible realizations using nonlinear photonic crystals or superconducting circuits.

DOI: 10.1038/ncomms10779 OPEN

Topological phase transitions and chiral inelastic transport induced by the squeezing of light

Vittorio Peano1, Martin Houde2, Christian Brendel1, Florian Marquardt1,3 & Aashish A. Clerk2

1 Institute for Theoretical Physics, University of Erlangen-Nrnberg, Staudtstr. 7, 91058 Erlangen, Germany. 2 Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8. 3 Max Planck Institute for the Science of Light, Gnther-Scharowsky-Stra e 1/Bau 24, 91058 Erlangen,

Germany. Correspondence and requests for materials should be addressed to V.P. (email: mailto:[email protected]

Web End [email protected] ).

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Waves are not only ubiquitous in physics, but the behaviour of linear waves is also known to be very generic, with many features that are independent of

the specic physical realization. This has traditionally allowed us to transfer insights gained in one system (for example, sound waves) to other systems (for example, matter waves). That strategy has even been successful for more advanced concepts in the eld of wave transport. One important recent example of this kind is the physics of topological wave transport, where waves can propagate along the boundaries of a sample, in a one-way chiral manner that is robust against disorder scattering. While rst discovered for electron waves, this phenomenon has by now also been explored for a variety of other waves in a diverse set of systems, including cold atoms1, photonic systems2 and more recently phononic systems39.

In the case of topological wave transport, the connection between waves in different physical implementations can actually be so close that the calculations turn out to be the same. In particular, if we are dealing with matter waves moving in a periodic potential, the results do not depend on whether they are bosons or fermions, as long as interactions do not matter. The single-particle wave equation to be solved happens to be exactly the same. This has allowed to envision and realize photonic analogues of quantum-Hall effect1018, the spin Hall effect1922, Floquet topological insulators23,24 and even Majorana-like modes25. More generally, the well-known classication of electronic band structures based on the dimensionality and certain generalized symmetries26 directly applies to photonic systems provided that the particle number is conserved. As we now discuss, this simple correspondence will fail in the presence of squeezing.

Consider the most general quadratic Hamiltonian describing photons in a periodic potential in the presence of parametric driving:

H X

k;n

o0^ay

j ^aj J X

hj;j0i

^ j ^aj0 2

(we set 1). Here we denote by ^aj the photon annihilation

operator associated with lattice site j, where the vector site index has the form j (j1,j2,s). j1,j2AZ labels a particular unit cell of the

lattice, while the index s A,B,C labels the element of the sublattice. hj,j0i indicates the sum over nearest neighbours, and J is the (real

valued) nearest-neighbour hopping rate; o0 plays the role of an onsite energy. As there are no phases associated with the hopping terms, this Hamiltonian is time-reversal symmetric and topologically trivial. We chose the kagome lattice because it is directly realizable both in quantum optomechanics5 and in arrays of super-conducting cavity arrays13,16; it is also the simplest model where purely local parametric driving can result in a topological phase.

We next introduce quadratic squeezing terms to this Hamiltonian that preserve the translational symmetry of the lattice and that are no more non-local than our original,

en k

^by

k;n^bk;n X

k;n;n0

lnn0 k

k;n^by k;n0 h:c:

: 1

^by

The rst term describes a non-interacting photonic band structure, where ^bk;n annihilates a photon with quasimomentum k in the n-th band. The remaining two-mode squeezing terms are induced by parametric driving and do not conserve the excitation number. As we discuss below, they can be controllably realized in a number of different photonic settings. While supercially similar to pairing terms in a superconductor, these two-mode squeezing terms have a profoundly different effect in a bosonic system, as there is no limit to the occupancy of a particular single-particle state. They can give rise to highly entangled ground states, and even to instabilities.

Given these differences, it is natural to ask how anomalous pairing terms can directly lead to topological phases of light. In this work, we study the topological properties of two-dimensional photonic systems described by Equation (1), in the case where the underlying particle-conserving band structure has no topological structure, and where the parametric driving terms do not make the system unstable. We show that the introduction of particle non-conserving terms can break time-reversal symmetry (TRS) in a manner that is distinct from having introduced a synthetic gauge eld, and can lead to the formation of bands having a nontrivial pattern of (suitably dened) quantized Chern numbers. This in turn leads to the formation of protected chiral edge modes: unlike the particle-conserving case, these modes can mediate a protected inelastic (but still coherent) scattering mechanism along the edge (that is, a probe eld injected into the edge of the sample will travel along the edge, but emerge at a different frequency). In general, the topological phases we nd here are distinct both from those obtained in the particle-conserving case, and from those found in topological super-conductors. We also discuss possible realizations of this model using a nonlinear photonic crystal or superconducting microwave circuits. Finally, we discuss the formal analogies and crucial differences between the topological phases of light investigated here and those recently proposed for other kinds of Bogoliubov quasiparicles2731 (see Discussion section).

ResultsKagome lattice model. For concreteness, we start with a system of bosons on a kagome lattice (Fig.1),

H0 X

B C

[afii9840]offei[p10]AC

[afii9840]onei[p10]B

Figure 1 | Setup gure. (a) An array of nonlinear cavities forming a kagome lattice. (b) Photons hop between nearest-neighbour sites with rate J. Each cavity is driven parametrically leading to the creation of photon pairs on the same lattice site (rate non) and on nearest-neighbour sites (rate noff). A spatial pattern of the driving phase is imprinted on the parametric interactions, breaking the time-reversal symmetry (but preserving the C3 rotational symmetry).

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nearest-neighbour hopping Hamiltonian:

2 non X

eifs^

h:c: 3

Such terms generically arise from having a nonlinear interaction with a driven auxiliary pump mode (which can be treated classically) on each site, see for example, ref. 32. As we discuss below, the variation in phases in ^

HL from site to site could be achieved by a corresponding variation of the driving phase of the pump. Note that we are working in a rotating frame where this interaction is time independent, and thus o0 should be interpreted as the detuning between the parametric driving and the true onsite (cavity) frequency ocav (that is, o0 ocav oL/2,

where the parametric driving is at a frequency oL). The parametric driving can cause the system to become unstable; we will thus require that the onsite energy (that is, parametric drive detuning) o0 be sufciently large that each parametric driving term is non-resonant enough to ensure stability. If one keeps o0 xed, this means that the parametric driving amplitudes non, noff will be limited to some fraction of o0 (the particular value

of which depends on J, Supplementary Note 1).

For a generic choice of phases in the parametric driving Hamiltonian of Equation (3), it is no longer possible to nd a gauge where ^

j ^ay

noff X

hj;j0i

eifss0 ^

j ^ay

jk; n0i

h X

u n;k s

0;k s

v n;k s

0;k s

dn;n

0 : 6

Having identied the appropriate Berry connection for a band eigenstate, the Chern number for a band n is then dened in the usual manner:

^ HL is purely real when expressed in terms of real-space annihilation operators: hence, even though the hopping Hamiltonian ^

H0 corresponds to strictly zero ux, the parametric driving can itself break TRS. In what follows, we will focus for simplicity on situations where time reversal and particle conservation are the only symmetries broken by the parametric driving: they will maintain the inversion and C3 rotational

symmetry of the kagome lattice. We will also make a global gauge

transformation so that noff is purely real, while non |non|eij

= An

In this case, the only possible choices for the f phases have the form (fA,fB,fC) (fAB,fBC,fCA) (0,d,2d) with d 2pmn/3,

where mn is an integer and is the vorticity of the parametric driving phases. We stress that these phases (and hence the sign of the TRS breaking) are determined by the phases of the pump modes used to generate the parametric interaction.

Gap opening and non-trivial topology. ^

H0 is the standard tight-binding kagome Hamiltonian for zero magnetic eld, and does not have band gaps: the upper and middle bands touch at the symmetry point C (0,0), whereas the middle and

lower bands touch at the symmetry points K (2p/3,0) and

K0 (p/3,p/(3)1/2) where they form Dirac cones (Fig. 2a).

Turning on the pairing terms, the Hamiltonian ^

^z: 7

The denition in Eq. (5) agrees with that presented in ref. 27 and (in one-dimension) ref. 29; standard arguments27 show that the Cn are integers with the usual properties. We note that, as for superconductors, breaking the U(1) (particle-conservation) symmetry remains compatible with a rst-quantized picture after doubling the number of bands. The additional hole bands are connected to the standard particle bands by a particlehole symmetry; see Methods section. In bosonic systems, the requirement of stability generally implies that particle and hole bands can not touch; this is true for our system. Thus, the sum of the Chern numbers over the particle bands (with E40) must be zero, and there cannot be any edge states with energies below the lowest particle bulk band (or in particular, at zero energy); Supplementary Note 1.

In the special case where we only have onsite parametric driving (that is, noff 0,nona0), the Chern numbers can be

calculated analytically (Supplementary Note 3). They are uniquely xed by the pump vorticity. If mn 0, we have TRS and the band

structure is gapless, while for mn 1, C (81, 0, 1). This

set of topological phases also occurs in a particle-number conserving model on the kagome lattice with a staggered magnetic eld, that is, the OghushiMurakamiNagaosa (OMN) model of the anomalous quantum-Hall effect33,34.

In the general case, where we include offsite parametric driving, entirely new phases appear. We have computed the Chern numbers here numerically, using the approach of ref. 35. In Fig. 3a, we show the topological phase diagram of our system, where J/o0 and mn are held xed, while the parametric drive strengths non,noff are varied. Different colours correspond to different triplets C (C1,C2,C3) of the band Chern numbers, with

grey and dark-grey corresponding to the two phases already present in the OMN model. Strikingly, a nite off-diagonal coupling noff generates a large variety of phases which are not present in the OMN model, including phases having bands with |Cn|41. The border between different topological phases represent topological phase transitions, and correspond to parameter values where a pair of bands touch at a particular symmetry

^ HL can be diagonalized in the standard manner as ^

H P

n;k En k

^ bn;k , where the ^

bn;k are canonical bosonic annihilation operators determined by a Bogoliubov transformation of the form (see Methods section):

n;k

sA;B;C

un;k s

^ay

k;s

vn;k s ^a k;s: 4

Here ^ak;s are the annihilation operators in quasimomentum space, and n 1,2,3 is a band index; we count the bands by

increasing energy. The photonic single-particle spectral function now shows resonances at both positive and negative frequencies, En[k], corresponding to particle- and hole-type bands, Fig. 2d.

Because of the TRS breaking induced by the squeezing terms, the band structure described by En[k] now exhibits gaps, Fig. 2b;

furthermore, for a nite sized system, one also nds edge modes in the gap, Fig. 2d.

The above behaviour suggests that the parametric terms have induced a non-trivial topological structure in the wavefunctions

of the band eigenstates. To quantify this, we rst need to properly identify the Berry phase associated with a bosonic band eigenstate in the presence of particle non-conserving terms. For each k, the Bloch Hamiltonian ^

Hk corresponds to the Hamiltonian of a multi-mode parametric amplier. Unlike the particle-conserving case, the ground state of such a Hamiltonian is a multi-mode squeezed state with non-zero photon number; it can thus have a non-trivial Berrys phase associated with it when k is varied, Supplementary Note 2. The Berry phase of interest for us will be the difference of this ground state Berry phase and that associated with a single quasiparticle excitation. One nds that the resulting Berry connection takes the form

An i k; nj ^

sz=k jk; ni

h : 5 Here the six vector of Bogoliubov coefcients |k, ni (un,k[A],

un,k[B], un,k[C], vn,k[A], vn,k[B], vn,k[C]) plays the role of a singe-particle wavefunction, and ^

sz acts in the particle-hole space, associating 1 to the u components and 1 to the v

components, see Methods section. These effective wavefunctions obey the symplectic normalization condition

k; nj ^

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a b c d

[afii9853]0

Particles

Holes

[afii9840]off

[afii9853]0

Frequency

[afii9853]0

[afii9843][afii9843]

kx kx kxa

Figure 2 | Topological Band structure. (a,b) 3D plots of the bulk band structure. The hexagonal Brillouin zone is also shown. (a) In the absence of parametric driving, neighbouring bands touch at the rotational symmetry points K, K0 and C. (b) The parametric driving opens a gap between subsequent bands. For the chosen parameters, there is a global band gap between the second and third band. (d) Hole and particle bands, Em[kx], in a strip geometry (sketched in c). The line intensity is proportional to the weight of the corresponding resonance in the photon spectral function, Supplementary Note 1. The edge states localized on the right (left) edge, plotted in green (blue), have positive (negative) velocity. Parameters: Hopping rate J 0.02o0 (o0 is the onsite frequency); (b,d), the parametric couplings are non 0.085o0 and noff 0.22o0.

[afii9807]

a b

0.2 2[afii9843] 2[afii9843]

[afii9843][afii9843]

0.2

0.1

[afii9807]

[afii9840] off /[afii9853] 0

[afii9840]2 / ( J [afii9853] )

~~ ~

0 0.2

[afii9840]on /[afii9853]0

Flux [afii9818]

[2,1,1] [1,4,3] [1,2,1] [2,1,3] [4,3,1] [1,0,1]

Other

Unstable

Figure 3 | Symplectic Topological phase diagrams. (a)Topological phase diagram for the parametrically driven kagome lattice model. The y (x) axis corresponds to the strength of the onsite parametric drive non (offsite parametric drive noff), and different colours correspond to different triplets

C (C1,C2,C3) of Chern numbers for the three bands of the model. Note that only the grey and dark-grey phases are found in the particle-conserving version

of our model with a staggered eld. We have xed the hopping rate J/o0 0.02, and the vorticity of the pump mn 1. (b) Same phase diagram, but now

plotted in terms of the effective ux F and effective parametric drive experienced by a quasiparticles.

, and the annihilation operators ^aj are given by a local Bogoliubov (squeezing) transformation ^aj eif

point; we discuss this further below. Via a standard bulk-boundary correspondence (Supplementary Note 4), the band Chern numbers for a particular phase determine the number of protected edge states that will be present in a system with a boundary; as usual, the number of edge states in a particular bandgap is obtained by summing the Chern numbers of lower-lying bands. We discuss these edge states in greater detail in a following subsection. Finally, the black regions in the phase diagram indicate regimes of instability, which occur when the parametric driving strength becomes too strong.

Dressed-state picture. To gain further insight into the structure of the topological phases found above, it is useful to work in a dressed-state basis that eliminates the local parametric driving terms from our Hamiltonian. We thus rst diagonalize the purely local terms in the Hamiltonian; for each lattice site j we have

Hj o0^ay

j ^aj

2 noneif

e ijn=2coshraj eif

eijn sinhr^awj, where the squeez-

ing factor r is

: 9

On a physical level, the local parametric driving terms attempt to drive each site into a squeezed vacuum state with squeeze parameter r; the ^aj quasiparticles correspond to excitations above this reference state. Note that we have included an overall phase factor in the denition of the ^aj, which will simplify the nal form of the full Hamiltonian.

In this new basis of local quasiparticles, our full Hamiltonian takes the form

H X

14 ln

non

h:c:h i

^ay

j ^aj : 8

Here

o20 n2on

^ay

j ^aj X

hj;li

jl^ay

j ^al

hj;li

^ay

j ^ay

h:c:

A: 10

j ^ay

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The transformation has mixed the hopping terms with the non-local parametric terms: The effective counter-clockwise hopping matrix element is

~Jjl Jeid e3id=2 2J cos

sinh2r noff sinh 2r cos

2 jn

;

and the magnitude of the effective non-local parametric driving is

j j j noffe id=2j

2noffcos d=2 jn

sinh2 r

J sinh 2r cos d=2 j :

Note that the phase of can be eliminated by a global gauge transformation, and hence it plays no role; we thus take to be real in what follows.

Our model takes on a much simpler form in the new basis: the onsite parametric driving is gone, and the non-local parametric driving is real. Most crucially, the effective hoppings can now have spatially varying phases, which depend both on the vorticity of the parametric driving in ^

HL (through d), and the magnitude of the onsite squeezing (through r). In this transformed basis, the effective hopping phases are the only route to breaking TRS. Our model has thus been mapped onto the standard OMN model for the anomalous quantum-Hall effect, with an additional (purely real) nearest-neighbour two-mode squeezing interaction. In the regime where the parametric interactions between the ^a quasiparticles are negligible (Supplementary Note 3), the complex phases correspond in the usual manner to a synthetic gauge eld (that is, the effective ux F piercing a triangular plaquette would be F 3arg

~J). In other words, the squeezing creates a synthetic gauge eld for Bogoliubov quasiparticles. However, in the presence of substantial parametric interaction between ^a quasiparticles, the parameter F can not be interpreted anymore as a ux: a ux of 2p can not be eliminated by a gauge transformation because the complex phases reappear in the parametric terms. In that case, only a periodicity of 6p in F is retained, since that corresponds to having trivial hopping phases of 2p.

Understanding the topological structure of this transformed Hamiltonian is completely sufcient for our purposes: one can easily show that the Chern number of a band is invariant under any local Bogoliubov transformation, hence the Chern numbers obtained from the transformed Hamiltonian in Equation (8) will coincide exactly with those obtained from the original Hamiltonian in Equation (3). We thus see that the topological structure of our system is controlled completely by only three dimensionless parameters: the ux F (associated with the hopping phases), the ratio j=

~J j, and the ratio= j

~J j.

The topological phase diagram for the effective model is shown in Fig. 3b. Again, one sees that as soon as the effective non-local parametric drive is non-zero, topological phases distinct from the standard (particle-conserving) OMN model are possible. The sign of the parametric pump vorticity mn determines the sign of the effective ux F, c.f. Equation (11). As such, the right half of Fig. 3b (corresponding to F 4 0) is a deformed version of the phase diagram of the original model for pump vorticity mn 1, as

plotted in Fig. 3a. Changing the sign of mn (and hence F) simply ips the sign of all Chern numbers, Supplementary Note 3.

Our effective model provides a more direct means for understanding the boundaries between different topological phases. Most of these are associated with the crossing of bands at one or more high-symmetry points in the Brillouin zone; this allows an analytic calculation of the phase boundary (Supplementary Note 3). Perhaps most striking in Fig. 3b is the horizontal boundary (labelled M), occurring at a nite value of

the effective offsite parametric drive,

~J~o p

. This boundary is set by the closing of a band gap at the M points; as these points are associated with the decoupling of one sublattice from the other two, this boundary is insensitive to the ux F. Similarly, the vertical line labelled T denotes a line where the system has TRS,

and all bands cross at the symmetry points K, K0 and C. The case of zero pump vorticity mn 0 (not shown) is also interesting.

Here the effective ux F depends on the strength of the parametric drivings, but is always constrained to be 0 or 3p. This implies that the effective Hamiltonian has TRS, even though the original Hamiltonian may not (that is, if Im noffa0, the original

Hamiltonian does not have TRS). For mn 0, the parametric

Quasimomentum

a b

[afii9853]L[afii9853]P

[afii9853]P

0.8

0 Elastic

c d

Frequency ([afii9853] P [afii9853] L/2)/[afii9853] 0

Inelastic

[afii9853]P

[afii9853]L/2 [afii9853]L/2

[afii9853]cav [afii9853]cav

0.9

0.8

0.9

[afii9853]P

[afii9853]L[afii9853]P

K M K

Trans-mission

Figure 4 | Topologically protected transport in a nite system. (a) A probe beam at frequency op inside the bulk band gap is focused on a site (marked in yellow) at the edge of a nite sample. The probability map of the light transmitted inelastically at frequency oL op (where oL is the frequency

of the drive tone applied to the auxiliary pump modes) clearly shows that the transport is chiral. (b) The elastic and inelastic transmission probability to a pair of sites along the edges (indicated in red in a) is plotted in blue and green, respectively. A cut through the bulk bands is shown to the left. (c,d) Sketch of the relevant scattering processes and energy scales. The inelastic (elastic) transmission has a larger rate when the light is injected in the hole (particle) band gap. Parameters: Hopping rate J 0.02o0 (o0 is the onsite frequency), parametric couplings non 0.4o0 and noff 0.02o0, optical decay rate

k 0.001o0. (a) op oL/2 0.95o0.

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drivings do not open any band gap and the Chern numbers are not well dened.

Edge states and transport. Despite their modied denition, the Chern numbers associated with our Bogoliubov bands still guarantee the existence of protected chiral edge modes in a system with boundaries via a standard bulk-boundary correspondence, see Supplementary Note 4. These states can be used to transport photons by exciting them with an auxiliary probe laser beam, which is focused on an edge site and at the correct frequency. The lack of particle-number conservation manifests itself directly in the properties of the edge states: along with the standard elastic transmission they can also mediate inelastic scattering processes. In terms of the original lab frame, light injected at a frequency op can emerge on the edge at frequency oL

op, where oL is the frequency of the laser parametrically

driving the system. This is analogous to the idler output of a parametric amplier. Here both signal and idler have a topologically protected chirality.

Shown in Fig. 4 are the results of a linear response calculation describing such an experiment, applied to a nite system with corners. We incorporate a nite photon decay rate k in the standard inputoutput formalism, see Methods section. Narrow-band probe light inside a topological band gap is applied to a site on the edge, and the resulting inelastic transmission probabilities to each site on the lattice are plotted, Fig. 4a. One clearly sees that the probe light is transmitted in a unidirectional way along the edge of the sample, and is even able to turn the corner without signicant backscatter. The corresponding elastic transmission (not shown) is also chiral and shows the same spatial dependence. In Fig. 4b we show the elastic and inelastic transmissions to the sites indicated in red (rescaled by the overall transmission, 1 R where R is the reection probability at the

injection site) as a function of the probe frequency op. By

scanning the laser probe frequency, one can separately address particle and hole band gaps. The relative intensity of the inelastic scattering component is highly enhanced when the probe beam is inside a hole band gap, see also the sketches in Fig. 4c,d. When the parametric interaction between the ^a quasiparticles is negligible, the ratio of elastic and inelastic transmissions depends only on the squeezing factor r, (c.f. Equation (9)), see Methods section.

Physical realization. Systems of this type could be implemented in photonic crystal coupled cavity arrays36 fabricated from nonlinear optical w2 materials3739. The array of optical modes participating in the transport would be supplemented by pump modes (resonant with the pump laser at twice the frequency). One type of pump mode could be engineered to be spatially co-localized with the transport modes (non processes), while others could be located in-between (noff). The required periodic phase pattern of the pump laser can be implemented using spatial light modulators or a suitable superposition of several laser beams impinging on the plane of the crystal. One method for realizing the required kagome lattice of defect cavities was discussed in ref. 5. Optomechanical systems offer another route towards generating optical squeezing terms40,41, via the mechanically induced Kerr interaction, and this could be exploited to create an optomechanical array with a photon Hamiltonian of the type discussed here. Alternatively, these systems can be driven by two laser beams to create phononic squeezing terms42. A fourth alternative consists in superconducting microwave circuits of coupled resonators, where Josephson junctions can be embedded to introduce w2 and higher order nonlinearities, as demonstrated in refs 43,44. Kagome lattices of superconducting resonators have recently been implemented45.

DiscussionBefore concluding, it is worthwhile to discuss the connections between our work and other recent studies. A Hamiltonian of the general form of Equation (1) arises naturally in the mean-eld description of a Bose-condensed phase. In this setting, the anomalous pairing terms describe the interactions with the condensate treated at the mean-eld level. A few recent studies have proposed to take advantage of these interactions to selectively populate topological edge states28,30 or, closer to our study, to induce novel topological phases. These include a study of a magnonic crystal27, as well as general BoseEinstein condensates in one-dimension29 and in two-dimensions31.

There are some crucial differences between the above studies and our work. In our case, Equation (1) describes the real particles of our system, not quasiparticles dened above some background. This difference is not just a question of semantics: in our case, topological effects can directly be seen by detecting photons, whereas in refs 29,31, one would need to isolate the contribution of a small number of Bogoliubov quasiparticles sitting on a much larger background of condensed particles. In addition, in our work the pairing terms in Equation (1) are achieved by driving the system, implying that negative and positive frequencies are clearly physically distinguished (that is, they are dened relative to a non-zero pump frequency). This is at the heart of the topologically protected inelastic scattering mechanism we describe, and is something that is not present in previous studies.

Our work opens the door to a number of interesting new directions. On the more practical side, one could attempt to exploit the unique edge states in our system to facilitate directional, quantum-limited amplication. On the more fundamental level, one could use insights from the corresponding disorder problem46 and attempt to develop a full characterization of particle non-conserving bosonic topological states that are described by quadratic Hamiltonians. This would then be a counterpart to the classication already developed for fermionic systems26.

Methods

Bogoliubov transformation and rst-quantized picture. We nd the normal mode decompositions leading to the band structures in Fig. 2 and the topological phase diagrams in Fig. 3 by introducing a rst-quantized picture. Since the relevant Hamiltonians do not conserve the excitation number, this is only possible after doubling the degrees of freedom. This is achieved by grouping all annihilation operators with quasimomentum k and the creation operators with quasimomentum k in the 2N vector of operators

Ck ^ak1; . . . ; ^akN; ^aw k1; . . . ; ^aw kN

(where N is the unit cell dimension), and by casting the second quantized Hamiltonian ^

H in the form

Xk^

Cyk ^h k ^

Ck: 13

The 2N 2N hermitian matrix

^ plays the role of a single-particle Hamiltonian

and is referred to as the Bogoliubov de Gennes Hamiltonian. By denition of the normal modes ^

H P

k;n En k

^ bn;k, we have

^ bwn;k En k

bwn;k.

By plugging into the above equation the Bogoliubov ansatz Equation (4) one immediately nds

h k

^ kn

j i En k

jkni: 14

Likewise, from ^

H; bn; k

En k

bn; k one nds

^ K^

knj i En k ^

knj i : 15

Here K denotes the complex conjugation and the matrix ^

sx exchanges the us and

the vs Bogoliubov coefcients,

0 1N

1N 0

^ :

is formed by the set of 2N

eigenenergies En[k] (belonging to the particle bands) and En[ k] (belonging

Thus, the spectrum of the 2N matrix ^

sz^h k

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to the hole bands). Vice versa, to calculate the eigenenergies En[k] and En[ k]

and the vector of Bogoliubov coefcients in Equation (4), we have to solve the eigenvalue problem

sz ^h k

^ m

j i lm m

j i: 16 The solutions we are interested in should also display the symplectic orthonormality relations Equation (6).

We note in passing that so far we have implicitly assumed that the normal mode decomposition is possible. However, this is not always the case. When the matrix ^

sz ^h k

has any complex eigenvalue, the Hamiltonian is unstable. Moreover,

at the border of the stable and unstable parameter regions, the matrix ^

sz^h k

is not

diagonalizable. The Supplementary Note 1 contains a stability analysis of our specic model.

In the stable regime of interest here the matrix ^

sz^h k

is diagonalizable and all

its eigenvalues are real. In this case, its eigenvectors |mi can be chosen to be

mutually ^

sz orthogonal. In addition, there are exactly N positive (negative) norm eigenvectors. Thus, it is always possible to enforce the symplectic orthonormality relations Equation (6) by identifying the (appropriately normalized) positive and negative norm solutions with |kni and K^

knj i, respectively. The corresponding

eigenvalues are then to be identied with En[k] (particle band structure) and

En[ k] (hole band structure), respectively

Particlehole symmetry. The Bogoliubov de Gennes Hamiltonian has the generalized symmetry Cw

^h k

where the charge conjugation

operator C is anti-unitary and C2 12N. Thus, our system represents the bosonic

analogue of a superconductor in the Class D of the standard topological classication. This is a simple consequence of the doubling of the degrees of freedom in the single-particle picture. It simply reects that the set of ladder operators ^

bwn;k and ^

bn; k calculated from

^h k

^ are the adjoint of the

set of operators ^

bn;k and ^

bwn; k calculated from

_ H; ^aj

k^aj=2 k

p ^

a inj: 17

In practice, we consider an array of detuned parametric ampliers with intensity decay rate k and add to the standard description of each parametric amplier the inter-cell coherent coupling described in the main text. The last term describes the inuence of the input eld ^a inj injected by an additional probe drive including also the environment vacuum uctuations. The eld ^a outj leaking out of each cavity at site j is given by the inputoutput relations

a outj ^a inj k

p ^

aj: 18

The above formulas give an accurate description of a photonic system where the intrinsic losses during injection and inside the system are negligible. Intrinsic photon absorption can be incorporated by adding another decay channel to the equation for the light eld. It reduces the propagation length but does not change qualitatively the dynamics.

In Fig. 3, we show the probabilities TE(o, l, j) and TI(o, l, j) that a photon injected on site j with frequency oin o oL/2 is transmitted elastically

(at frequency o oL/2) or inelastically (at frequency oL/2 o) to site l where it is

detected. From the Kubo formula and the inputoutput relations we nd

TE o; l; j dlj ik

E o; l; j

2; 19

TI o; l; j k2

I o; l; j

HereE=I o; l; j are the elastic and inelastic components of the Greens function

in frequency space,

E o; l; j i Z

dtY t

^al t

h iE

; ^ay

j 0

eiot; 21

I o; l; j i Z

dtY t

^ayl t

; ^ay

j 0

h iE

eiot: 22

In a N site array with single-particle eigenstates |ni (un[1],y,un[N],

vn[1],y,vn[N])T, the Greens functions read

GE o; l; j X

un l

ik=2

v n l

vn j

o E n

ik=2

u n j

o E n

; 23

vn l

GI o; l; j X

u n j

o E n

: 24

We note that for a probe eld inside the bandwidth of the particle (hole) sector but far detuned from the hole (particle) sector, only the rst (second) term of the

summand in Equations (23) and (24) is resonant. Thus, as expected, the inelastic scattering is comparatively larger when the probe eld is in the hole band gap.

It is easy to estimate quantitatively the relative intensities of elastically and inelastically transmitted light when the parametric interaction of the ^a Bogoliubov quasiparticles is small (the regime where F can be interpreted as a synthetic gauge eld experienced by the Bogoliubov quasiparticles). In this case, it is straightforward to show that |vn[j]/un[j]|Etanhr independent of the eigenstate n and the site j. By putting together Equations (20, 23, 24) and neglecting the off-resonant terms we nd that for o;44 j

~J j; k; j o j,

TI o; l; j tanh r

2TE o; l; j

TI o; l; j

coth r

2TE o; l; j

These analytical formulas agree quantitatively with the numerical results shown in Fig. 4b (note that in Fig. 4b the transmission at the output sites is rescaled by the overall transmission,

Pl a jTI(o, l, j) TE(o, l, j)).

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10779

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Acknowledgements

V.P., C.B., and F.M. acknowledge support by an ERC Starting Grant OPTOMECH, by the DARPA project ORCHID, and by the European Marie-Curie ITN network cQOM. M.H. and A.A.C. acknowledge support from NSERC.

Author contributions

V.P., A.A.C. and F.M. contributed to the conceptual development of the project and interpretation of results. Calculations and simulations were done by V.P., M.H. and C.B.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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How to cite this article: Peano, V. et al. Topological phase transitions and chiral inelastic transport induced by the squeezing of light. Nat. Commun. 7:10779doi: 10.1038/ncomms10779 (2016).

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Title

Topological phase transitions and chiral inelastic transport induced by the squeezing of light

Author

Peano, Vittorio; Houde, Martin; Brendel, Christian; Marquardt, Florian; Clerk, Aashish A

Pages

10779

Publication year

2016

Publication date

Mar 2016

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Nature Publishing Group

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20411723

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Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms10779

ProQuest document ID

1769609932

Topological phase transitions and chiral inelastic transport induced by the squeezing of light

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