Qubit-photon corner states in all dimensions
Adrian Feiguin, Juan Jose Garcia-Ripoll, Alejandro Gonzalez-Tudela
QQubit-photon corner states in all dimensions
Adrian Feiguin, Juan Jos´e Garc´ıa-Ripoll, and Alejandro Gonz´alez-Tudela Department of Physics, Northeastern University, Boston, Massachusetts 02115, US Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain (Dated: October 3, 2019)A single quantum emitter coupled to a one-dimensional photon field can perfectly trap a photonwhen placed close to a mirror. This occurs when the interference between the emitted and reflectedlight is completely destructive, leading to photon confinement between the emitter and the mirror. Inhigher dimensions, the spread of the light field in all directions hinders interference and, consequently,photon trapping by a single emitter is considered to be impossible. In this work, we show that isnot the case by proving that a single emitter can indeed trap light in any dimension. We providea constructive recipe based on judiciously coupling an emitter to a photonic crystal-like bath withproperly designed open boundary conditions. The directional propagation of the photons in suchbaths enables perfect destructive interference, forming what we denote as qubit-photon corner states .We characterize these states in all dimensions, showing that they are robust under fluctuations ofthe emitter’s properties, and persist also in the ultrastrong coupling regime.
Introduction.–
The radiation properties of a quantumemitter can change modifying the photonic environmentaround it [1]. A particularly simple example of this con-sists in placing an emitter close to a mirror [2, 3] orto other quantum emitters [4, 5]. These configurationsin free-space already lead to remarkable effects such aslifetime renormalizations or the modification of atomicresonance fluorescence [6–10]. However, they are ulti-mately limited by the reduced solid angle of the emittedlight that the mirrors or emitters can cover. All theseeffects are dramatically enhanced when the emitters cou-ple to one-dimensional (1D) photonic fields such as di-electric [11–16] or microwave [17] waveguides, where, forexample, a single atom can perfectly reflect single pho-tons [18]. These strong interference effects lead to theemergence of bound states in the continuum (BIC) [19]with two emitters [20–27], or a single emitter in frontof a mirror [28–33], in which a single photon becomeslocalized despite being energetically in the middle ofthe continuous spectrum. These BICs, which are en-tangled light-matter states, have experienced a renewedinterest because of their possible applications to real-ize decoherence-free quantum gates [34, 35], or non-reciprocal photon transport [36–38].Among the different configurations, the one using asingle emitter and a mirror [28–33] is especially advanta-geous since the BICs in that case are insensitive to theenergy mismatch between emitters. Exporting this con-figuration to higher-dimensional systems was generallythought not be possible, since the wavepacket diffractionprecludes perfect destructive interference. Here, we showthe contrary by proving that indeed a single quantumemitter can perfectly trap light and create a BIC in anydimension. The key idea is to combine the directionalemission occurring in 2D and 3D photonic crystal-likebaths [39–42], with an adequate design of open boundaryconditions. Then, by placing the emitter close to a cor-ner of the photonic bath, its directional emission and the reflection in the boundary generates a high-dimensionalBIC that we label as qubit-photon corner state . In con-trast to the recently observed topological photon cornerstates [43–50], ours can inherit a strong non-linearityfrom the emitter, and do not require a topologically non-trivial bath. We characterize these states in two and treedimensions using exact numerical techniques to take intoaccount the retardation effects and the corrections in theultra-strong coupling regime, where these states acquirea finite lifetime.
Setup.–
To illustrate the emergence of these states, weuse a d -dimensional photonic lattice composed by N d res-onators with energy ω a , that can tunnel to their nearestneighbour at a rate J . With these assumptions, the bathenergy dispersion ( ω ( k )) then only depends on the pho-tonic lattice geometry, which determines the number ofnearest neighbours resonators ( N nn ). For the emitter,we take a two-level system (qubit) with energy difference∆, that is locally coupled at a position x ∈ R d to thephotonic bath. Thus, the full Hamiltonian reads: H = ∆2 σ z + (cid:88) x ω a a † x a x + J (cid:88) (cid:104) x , y (cid:105) a † x a y + g x σ x ( a x + a † x ) . (1)Notice, that we have kept the full dipole coupling g x between the emitter and the photonic mode. Like thiswe can study situations in which the coupling is com-paratively weak, g (cid:28) ∆ , ω a and the rotating-wave ap-proximation (RWA) is justified, replacing gσ x ( a + a † ) ∼ g ( σ + a + σ − a † ), but also the ultra-strong coupling (USC)regime, which occurs when g/ ∆ ≥ g ∼ W ,where W = 2 N nn J is the photon bandwidth.We are interested in studying the spontaneous emissiondynamics, that is, considering that the emitter is initiallyexcited with no photons in the bath, and then study thetime dynamics governed by e − iHt . In our case, this is a a r X i v : . [ qu a n t - ph ] O c t complicated problem because of the high-dimensional na-ture of the bath and, in the ultra-strong coupling regime,because the number of excitation is not conserved. Thus,before describing the physics, it is worth explaining thetwo complementary approaches we used to study thisproblem. • Polaron Hamiltonian.–
Instead of working with (1)directly, we study the unitarily equivalent polaron Hamil-tonian [51]. This transformed model eliminates much ofthe entanglement between the quantum emitter and thephotonic field, leading to renormalized coupling strengthsand qubit frequencies. For moderate coupling strengthsor finite-bandwidth models, the polaron Hamiltonian hasa single excitation limit that describes the spontaneousemission problem that we want to study H pol , = ˜∆2 σ z (1 + 8 F † F ) + (cid:88) x , y J xy a † x a y (2)+ 2 ˜∆( σ + F + H.c.) + (cid:88) x ω a a † x a x . Within that picture, the emitter interacts with a collec-tive coupling operator F = (cid:80) x f x a x with coupling vector f = { f x } x , and has a renormalized frequency ˜∆. Theseparameters can be obtained solving self-consistently thefollowing equations˜∆ = ∆ e − (cid:80) k | f k | , f = 1 J + ˜∆ g . (3)The single-excitation polaron adopts a RWA stanza andis therefore amenable to analytical treatment, much likeearlier works with regular lattices and point-like interac-tions [39, 40]. As a result, the model supports single-photon solutions | ψ ( t ) (cid:105) = (cid:34)(cid:88) x ψ ( x , t ) a † x + c ( t ) σ + (cid:35) |↓(cid:105) ⊗ | vac (cid:105) , (4)whose photon and qubit components ψ ( x , t ) and c ( t ) fol-low a linear Schr¨odinger equation with H pol , , that canbe evolved in time using different numerical methods. • Chain mapping and DMRG.–
As an additionalbenchmark, we also solve the dynamics of the full spin-boson model of Eq. 1 using a time-dependent versionof the density matrix renormalization group (tDMRG)[52–55]. To simulate large high dimensional bosonicbaths, the non-interacting lattice Hamiltonian is exactlymapped (it is a unitary transformation) onto a 1D chainof free bosons by means of a Lanczos recursion [56–58].The consequent dimensional and entanglement reductionmakes the new Hamiltonian amenable to DMRG simu-lations. Remarkably, with only N bosonic modes in thechain we capture well the dynamics of the emitter. Asshown in [59], this mapping can be combined with thepolaron transformation to reduce the amount of entan-glement, but this was not required for this study. FIG. 1. Formation of a 1D BIC by spontaneous emission on a1D lattice with 400 sites. (a) Pictorical representation (above)and photon number spatial distribution (below) in the BICstate for g = 0 .
1∆ and x = 12 . (b) Total excitation number N excit (5) in the BIC state as a funciton of time. Solid anddashed lines correspond to x = 12-th and x = 11-th site. (c)Qubit and photon component of the bound state, P ↑ and P γ , and probability of creating the bound state P BIC ∼ N ( t ) . (d)Estimated decay rate of the corner state extracted from a fit N excit ( t ) ∼ N ( t ) exp( − γ ( t − t )) after the initial transient aa Values below 10 − are not reliable, due to finite simulationtime. Reminder of 1D BICS.–
Our first set of simulationsrecreates the BICs obtained in a one-dimensional lat-tice with open boundaries and N = 400 sites, takingthe lattice constant as the unit of length. We use anemitter resonant with the middle of the photonic band,∆ = ω a = 2 . J , although this is not strictly needed.We place a quantum emitter at even ( x = 12 , solid) andodd positions ( x = 11 , dashed), excite the emitter, andabruptly switch on the coupling g . When the emitteris placed on an odd site, it decays completely, releasinga propagating photon. However, if the emitter is on aneven site, it can, with some probability, trap a photonbetween the emitter and the end of the lattice, as seenin Fig. 1a. Such states correspond to the BICs that havebeen identified before in one-dimensional systems [20–33], and can be intuitively understood from the inter-ference between the emitted light of the emitter and itsafterimage, as schematically depicted in Fig. 1a. Fig. 1bplots the probability of creating the 1D BIC, defined as N excit = 12 ( σ z + 1) + x (cid:88) x =1 a † x a x = P ↑ + P γ . (5)which contains both a non-neglibible photonic ( P γ ) andqubit ( P ↑ ) component. Note how the emitter in odd sitesdecay (dashed lines), but emitters in even sites have someprobability to excite the BIC, even in the USC regime.As we increase the coupling strength, the BIC transitionsfrom being mostly an excited atom to an equal superpo-sition of both [cf. Fig. 1c]. In the USC regime, the BICstate has a significant fraction of photon component, butit also acquires a finite lifetime [cf. Fig. 1d]. This decaycan be attributed to the renormalization of the qubit en-ergy when g ∼ W , which also changes the emitted photonfrequencies. Thus, the photons will no longer have theexact wavelength that leads to the perfect interference forthe position of the emitter chosen. Finally, note that theresults obtained using the single-photon polaron Hamil-tonian agree very well with our DMRG simulations. Qubit-photon corner states in two-dimensions.–
To ob-tain these phenomena in two dimensions, it is enough toconsider the simpler generalization of the coupled cavityarray to two dimensions, that is, disposing the resonatorsin a square geometry. This model displays an energy dis-persion given by: ω ( k ) = ω a + 2 J (cos( k x ) + cos( k y )) . (6)At the middle of the band, ω ( k ) = ω a , the isofrequen-cies are ”nested”, which means they are straight lines de-fined by k x ± k y = ± , ∓ π . One of the consequences of suchlines is that when an emitter is spectrally tuned to thatenergy, its emission becomes highly directional [39–41].This is what we will harness to induce the perfect trap-ping. As in the 1D case, the intuitive idea (see Fig. 2a)consists in placing the emitter in a position such thatits directional emission is orthogonal to the bath bound-aries, and their afterimages are out-of-phase with respectto the emission from the emitter.Fig. 2b shows a proof-of-principle example of thatmechanism. We have taken a square lattice and removedsites to form a reflective corner in a rhombus with 4 × sites. The quantum emitter is equidistant to its afterim-ages only when placed on the diagonal of the rhombus—positions A to E in the plot—. As in the 1D case, whenwe place the emitter on an odd site, such as B, it failsto acquire the right phase relation and decays releasing aphoton into the lattice. However, for even positions (A, FIG. 2. Formation of a corner state by spontaneous emis-sion on a 2D rombic lattice with 30 sites on each diagonal,for g = 0 .
01∆ and J = 0 . . (a) Pictorical representation ofthe emitter and its afterimages. (b) Locations of the emit-ter in the corner of the photonic lattice (dots), coupling be-tween photonic sites (lines) and distribution of photons (den-sity plot), for a corner state generated by emitter E. (c) Totalexcitation number N excit as a function of time, for differentlocations of the emitter, from A to E . C, D, E) the emitter relaxes to a stationary state withhigh probability [cf. Fig. 2b]. In these states, the photonis trapped in a corner, avoiding the quantum emitter.Fig. 2b shows a density plot of a trapped photon thatis anchored by a quantum emitter at position E. As inthe 1D case, we have a very good agreement betweenDMRG and the single-photon polaron Hamiltonian forthe rhombus. However, since the DMRG is working witha reduced number of modes (up to four per bath site)it allows the simulation of larger lattices—see Fig. 2c,where the DMRG uses 400 sites—, and even moving tohigher dimensional scenarios as we will show next. Qubit-photon corner states in 3D–
In the three-dimensional case there are many different geometries inwhich the resonators can be disposed, but not all of themare suitable for our purposes. Using the intuition devel-oped in Ref. [42], we choose a body-centered-cubic lat-tice in which each resonator is connected to four nearest
FIG. 3. (a) A cube of light in a corner state trapped by aquantum emitter at position D ( x = y = z = 5) on the diago-nal of a BCC photonic lattice. (b) Density of photons on thecorner state, as seen from above. (c) Probability of creatinga corner state for emitters at A, B, C and D (respectively x = y = z ∈ { , , , } ), for J = 0 . , g = 0 . neighbours. This model has an energy dispersion: ω ( k ) = ω a + 2 J (cid:104) cos( k x ) + cos( k y )+ (7)+ cos( k z ) + cos( k x + k y + k z ) (cid:105) , (8)with nested equifrequencies that yield highly collimatedemission in 3 directions. This is especially well-suited toprovide reflection in 3D corners. Other geometries, likethe cubic-simple lattices, also display collimated emis-sion but in more directions [42], such that they are notadequate for the desired goal.In Fig. 3 we provide a proof-of-principle numerical con-firmation of the trapping for a lattice with N =? and g/ ∆ = 0 .
1. Fig. 3a) shows the 3D photon distribution ofa qubit-photon corner state when placed in the positiondenoted by the red dot (D), while in Fig. 3b) we plot anhorizontal cut of this distribution. Finally, in Fig. 3c) weplot the probability of exciting the BIC as a function of time for the positions A-D depicted in Fig. 3b) comparingagain the polaron Hamiltonian (lines) and chain-mappedDMRG (dots). Here again, we see the difference betweenthe positions A, C, and D, where the phase relation withthe afterimages is the right one, compared the B situationwhere the photon is not trapped, and BIC probability isvery small.
Discussion.–
Summing up, in this work we haveshown that a single quantum emitter can trap a pho-ton in any dimension. The emitter must be placed in aphotonic crystal-like medium, with the right separationfrom the reflective boundaries of the medium. Undersuch conditions, the emitter interferes destructively withthe afterimages reflected by the boundaries, generatinga bound-state-in-the-continuum that we denote as qubit-photon corner state.
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