ICAP Summer School 2016

'Chiral' Quantum Optics: UNIVERSITY OF INNSBRUCK A Novel Driven-Dissipative Quantum Many-Body System

Peter Zoller UQUAM • ‘chiral' photonic quantum circuit / network ERC Synergy Grant

unidirectional light-matter couplings chiral appear naturally in fiber nanophotonic devices RESEARCH LETTER

appropriately described as an optomechanical polariton whose deco- the launched power constant. This allows the displaced cavity mode ^a herence time exceeds the period of the Rabi oscillations between light (of frequency D ) to be brought in and out of resonance with the j j ^ and mechanics. Although the ground state has recently been reported mechanical mode b (of frequency Vm). For each detuning point, we 10 with optical cooling , in that work Vc was much smaller than k.In acquire the coherent response of the system to an optical excitation of contrast, the present system achieves a coupling rate exceeding both swept frequency vl 1 Vmod in a first step (Vmod is the frequency dif- the optical and mechanical decoherence rates, thereby satisfying the ference from the coupling laser). These spectra (Fig. 2a) allow us to necessary conditions for full control of the quantum state of a mech- determine all parameters of the model characterizing the optomecha- anical oscillator with optical fields9,12–14,21,22. The experimental setting nical interaction (Supplementary Information). For large detunings is a micro-optomechanical system in the form of a spoke-anchored D . Vm, they essentially feature a Lorentzian response of width k 24 j j toroidal optical microcavity . Such devices exhibit whispering gallery and centre frequency D . The sharp dip at Vmod < Vm originates from j j 27 mode resonances of high quality factor (with a typical cavity decay rate optomechanically induced transparency (OMIT), and for Vm 5 –D, 2 k/2p , 10 MHz) coupled to mechanical radial breathing modes via its width is approximately Vc k. The coupling rate, as derived from a radiation pressure25. The vacuum optomechanical coupling rate fit of the coherent response for a laser power of 0.56 mW, is  g0 5 (vc/R)xZPM can be increased by reducing the radius R of the Vc 5 2p 3 (3.7 6 0.05) MHz (corresponding to an intracavity photon 5 cavity (here vc is the optical cavity resonance frequency and xZPM is number of nc~3|10 ). the zero point motion). However, the larger per photon force Bvc/R is Additionally, for each value of the detuning, the noise spectrum of then usually partially compensated by the increase in the mechanical the homodyne signal is recorded in the absence of any external excita- resonance frequency Vm—and correspondingly smaller zero point tion (Fig. 2b). The observed peak represents the phase fluctuations motion, given by xZPM~ B= 2meff Vm (where meff is the effective imprinted on the transmitted light by the mechanical mode’s thermal oscillator mass). Moreover, smallðÞ structures also generally feature lar- motion. The constant noise background on these spectra is the shot- ger dissipation through clampingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi losses. To compensate these oppos- noise level for the (constant) laser power used throughout the laser ing effects, we use an optimized spoke-anchor design (see Fig. 1 and sweep (see Supplementary Information for details). Importantly, the Supplementary Information) that maintains low clamping losses and a amplitude of the peak is determined by the coupling to and the tem- moderate mechanical resonance frequency while reducing the dimen- perature of the environment, and therefore allows us to extract the sions of the structure. Devices fabricated in this manner (with mechanical decoherence rate. All parameters now having been measured, R 5 15 mm) exhibited coupling rates as high as g0 5 2p 3 3.4 kHz for it is moreover possible to retrieve the mechanical displacement spec- a resonance frequency of 78 MHz and a critically coupled sideband trum (Fig. 2c inset). As can be seen, for detunings close to the sideband, factor Vm/k 5 11. when the (displaced) optical and mechanical modes are degenerate, the To reduce the mechanical decoherence rate c

a b d Ω m

c Quantum Simulation & Quantum Optics Ω c

c κ m Building Quantum Simulators with AMO, [solid state] Optical Mechanical environment environment • k T k T n = B ≈ 0.01 n = B >> 0 etc. 10 μm p m Figure 1 | Optomechanicalnanomechanics microresonators. a, False-colour scanning Pe´rot cavity: quantum-coherent coupling is achieved when the enhanced electron micrograph of a spoke-anchored toroidal resonator 31 mm in diameter coupling rate Vc is comparable to or exceeds the optical and mechanical • cold atoms in optical lattices used for the optomechanical experiments reported in this work. b, Sketch of an decoherence rates k,Cmnm . Owing to the large asymmetry between optical whispering gallery mode in the microresonator (colours indicate optical mechanical and opticalðÞ frequencies, the occupancies of the two environments phase). c, Simulated displacement (exaggerated for clarity) of the fundamental are widely different. • trapped ions radial breathing mode of the structure. d, Equivalent optomechanical Fabry–

• photons … 64 | NATURE | VOL 482 | 2 FEBRUARY 2012 ©2012 Macmillan Publishers Limited. All rights reserved controlled many-body quantum systems

✓dynamics: closed & open systems NV centers ✓preparation & measurement ✓quantum control on level of single quanta

optical fiber optical fiber SC Qubits CQED atoms in optical lattices trapped ions 'Chiral' Quantum Optics 'Chiral' Quantum Optics what it is NOT ;-)

Quantum communication with chiral edge state qubit 2

✓ photonic nanostructure 2D topology ✓ atoms, spin, … qubit 1 Particle in a [synthetic] magnetic field

cyclotron orbit in B-field quantum wave packet

X B~

chiral edge states © L. Fallani “skipping orbits"

'Chiral' Quantum Optics

Quantum communication with chiral edge state qubit 2

✓ photonic nanostructure 2D topology ✓ atoms, spin, … qubit 1 'Chiral' Quantum Optics chiral coupling between light and quantum emitters

Nanophotonic devices: chirality appears naturally …

atoms & nanofibers atoms & CQED quantum dots & photonic nanostructures Nature Review submitted RESEARCH | REPORTS

swaths of differently textured sea-floor fabric to 5. M. S. Steckler, A. B. Watts, Earth Planet. Sci. Lett. 41,1–13 Santos Basin, Brazil and Its Consequences For Petroleum the east and west of the lineament (Fig. 3). The (1978). System Development (Petroleum Geology Conference Series, geometry of the two features suggests that they 6. C. S. Liu, D. T. Sandwell, J. R. Curray, J. Geophys. Res. 87, Geological Society of London, London, 2010), pp. 855–866. 7673–7686 (1982). 19. W. U. Mohriak, P. Szatmari, S. Anjos, Geol. Soc. London Spec. form a pair of an extinct ridge (on the African 7. K. Matthews, R. D. Müller, P. Wessel, J. M. Whittaker, Publ. 363,131–158 (2012). side) and a pseudofault (on the South American J. Geophys. Res. Solid Earth 116,1–28 (2011). side), created by a northward ridge propagation 8. Materials and methods are available as supplementary ACKNOWLEDGMENTS episode between ~100 and 83 Ma. An absolute materials on Science Online. The CryoSat-2 data were provided by the European Space Agency, 9. J. Pindell, L. Kennen, in The Geology and Evolution of the hot spot–based plate reconstruction using the and NASA/Centre National d"Etudes Spatiales provided data Region Between North and South America,K.James, from the Jason-1 altimeter. This research was supported by rotation parameters from O’Neill et al.(10)in- M. A. Lorente, J. Pindell, Eds. (Special Publication, Geological NSF (grant OCE-1128801), the Office of Naval Research (grant dicates that the Cardno hot spot (Fig.RESEARCH 3) may | REPORTSSociety of London, London, 2009), vol. 328, pp. 1–55. N00014-12-1-0111), the National Geospatial Intelligence Agency have been situated not far north of the northern 10. C. O'Neill, R. D. Müller, B. Steinberger, Geochem. Geophys. (grant HM0177-13-1-0008), the Australian Research Council Geosyst. 6,Q04003(2005). tip of the ridge propagator, where it abuts the (grant FL099224), and ConocoPhillips. Version 23 of global grids swaths of differently textured sea-floor fabric to 5. M. S. Steckler, A. B. Watts, Earth Planet. Sci. Lett. 41,1–13 Santos Basin, Brazil and Its Consequences For11. Petroleum R. D. Müller, W. R. Roest, J. Y. Royer, Nature 396,455–459 of the gravity anomalies and VGG can be downloaded from the the east and west of the lineament (Fig. 3). The (1978). Bodo Verde Fracture Zone;System Development this is where(Petroleum the Geology Conference(1998). Series, supplementary materials and also at the following FTP site: ftp:// geometry of the two features suggests that they 6. C. S. Liu, D. T. Sandwell, J. R. Curray,propagatorJ. Geophys. Res.came87, to a halt.Geological These Society observations of London, London, 2010),12. pp. R. 855 Granot,–866. J. Dyment, Y. Gallet, Nat. Geosci. 5,220–223 topex.ucsd.edu/pub/global_grav_1min. The manuscript contents 7673–7686 (1982). 19. W. U. Mohriak, P. Szatmari, S. Anjos, Geol. Soc. London Spec. form a pair of an extinct ridge (on the African conform with the inference that ridges have a (2012). are the opinions of the authors, and the participation of W.H.F.S. 7. K. Matthews, R. D. Müller, P. Wessel, J. M. Whittaker, Publ. 363,131–158 (2012). 13. W. H. F. Smith, D. T. Sandwell, Science 277,1956–1962 side) and a pseudofault (on the South American J. Geophys. Res. Solid Earth 116tendency,1–28 (2011). to propagate toward hot spots/plumes should not be construed as indicating that the contents of the side), created by a northward ridge propagation 8. Materials and methods are available as supplementary ACKNOWLEDGMENTS (1997). paper are a statement of official policy, decision, or position on and that propagation events and resulting spread- 14. J. A. Goff, W. H. F. Smith, K. A. Marks, Oceanography 17,24–37 episode between ~100 and 83 Ma. An absolute materials on Science Online. The CryoSat-2 data were provided by the European Space Agency, behalf of NOAA or the U.S. government. 9. J. Pindell, L. Kennen, in The Geology and Evolution of the (2004). hot spot–based plate reconstruction using the ing asymmetries are frequentlyand NASA/Centre contained National d"Etudes within Spatiales provided data Region Between North and South America,K.James, from the Jason-1 altimeter. This research was supported15. N. K.by Pavlis, S. A. Holmes, S. C. Kenyon, J. K. Factor, rotation parameters from O’Neill et al.(10)in- M. A. Lorente, J. Pindell, Eds. (Specialindividual Publication, spreading Geological corridors bounded by FZs SUPPLEMENTARY MATERIALS NSF (grant OCE-1128801), the Office of Naval ResearchJ. (grant Geophys. Res. 117,B04406(2012). dicates that the Cardno hot spot (Fig. 3) may Society of London, London, 2009), vol. 328, pp. 1–55. N00014-12-1-0111), the National Geospatial Intelligence Agency www.sciencemag.org/content/346/6205/65/suppl/DC1 (11). The existence of major previously unknown 16. M. Seton et al., Earth Sci. Rev. 113,212–270 (2012). have been situated not far north of the northern 10. C. O'Neill, R. D. Müller, B. Steinberger, Geochem. Geophys. (grant HM0177-13-1-0008), the Australian Research Council Supplementary Text Geosyst. 6,Q04003(2005). ridge propagation events will also be relevant for 17. W. Mohriak, M. Nóbrega, M. Odegard, B. Gomes, W. Dickson, (grant FL099224), and ConocoPhillips. Version 23 of global grids Figs. S1 and S2 tip of the ridge propagator, where it abuts the 11. R. D. Müller, W. R. Roest, J. Y. Royer, Nature 396,455–459 Petrol. Geosci. 16,231–245 (2010). interpreting marine magneticof the gravity anomaly anomalies and sequences VGG can be downloaded from the References (20–33) Bodo Verde Fracture Zone; this is where the (1998). supplementary materials and also at the following18. FTP I. site: Scotchman, ftp:// G. Gilchrist, N. Kusznir, A. Roberts, R. Fletcher, in 12. R. Granot, J. Dyment, Y. Gallet,duringNat. Geosci. the5,220 Cretaceous–223 Normal Superchron on propagator came to a halt. These observations topex.ucsd.edu/pub/global_grav_1min. The manuscriptThe contents Breakup of the South Atlantic Ocean: Formation of Failed 2 July 2014; accepted 2 September 2014 conform with the inference that ridges have a (2012). conjugate ridge flanksare (12 the). opinions of the authors, and the participation of W.H.F.S. 13. W. H. F. Smith, D. T. Sandwell, Science 277,1956–1962 Spreading Axes and Blocks of Thinned Continental Crust in the 10.1126/science.1258213 tendency to propagate toward hot spots/plumes should not be construed as indicating that the contents of the (1997). One of the most importantpaper are a statement uses of official this policy, new decision, or position on and that propagation events and resulting spread- 14. J. A. Goff, W. H. F. Smith, K. A. Marks,marineOceanography gravity17,24 field–37 willbehalf be of NOAAto improve or the U.S. government. the esti- ing asymmetries are frequently contained within (2004). 15. N. K. Pavlis, S. A. Holmes, S. C.mates Kenyon, J. of K. sea-floor Factor, depth in the 80% of the oceans individual spreading corridors bounded by FZs SUPPLEMENTARY MATERIALS J. Geophys. Res. 117,B04406(2012). NANOPHOTONICS having no depth soundings.www.sciencemag.org/content/346/6205/65/suppl/DC1 The most accurate (11). The existence of major previously unknown 16. M. Seton et al., Earth Sci. Rev. 113,212–270 (2012). Supplementary Text ridge propagation events will also be relevant for 17. W. Mohriak, M. Nóbrega, M. Odegard,method B. Gomes, of mappingW. Dickson, sea-floor depth uses a mul- Figs. S1 and S2 Petrol. Geosci. 16,231–245 (2010). interpreting marine magnetic anomaly sequences tibeam echosounder mountedReferences ( on20–33 a) large research during the Cretaceous Normal Superchron on 18. I. Scotchman, G. Gilchrist, N. Kusznir, A. Roberts, R. Fletcher, in The Breakup of the South Atlanticvessel. Ocean: Formation However, of Failed even after2 July 2014; 40 accepted years 2 of September mapping 2014 Chiralweek nanophotonic ending waveguide PRL 110, 243603conjugate ridge (2013) flanks (12). PHYSICALSpreading Axes and Blocks REVIEW of Thinned Continental Crust LETTERS in the 10.1126/science.1258213 14 JUNE 2013 One of the most important uses of this new by hundreds of ships, one finds that more than marine gravity field will be to improve the esti- 50% of the ocean floor is more than 10 km away interface based on spin-orbit mates of sea-floor depth in the 80% of the oceans from a depth measurement. Between the sound- having no depth soundings.Coherence The most accurate PropertiesNANOPHOTONICS of Nanofiber-Trappedings, the sea-floor depth is estimated Cesium from Atoms marine method of mapping sea-floor depth uses a mul- gravity measurements from satellite altimetry interaction of light tibeam echosounder mounted on a large research (13). This method works best on sea floor where vessel. However, even after 40 years of mapping Chiral nanophotonic waveguide by hundreds of ships,D. one Reitz, finds that C. more Sayrin, than R. Mitsch, P. Schneeweiss,sediments are thin, and resulting A. Rauschenbeutel in a high correla-* Jan Petersen, Jürgen Volz,* Arno Rauschenbeutel* Vienna50% of Center the ocean for floor Quantum is more than 10 Science km away andinterface Technology, Atominstitut, basedtion between TU on sea-floor Wien, spin-orbit topography Stadionallee and gravity 2, 1020 Vienna, Austria from a depth measurement. Between the sound- Controlling the flow of light with nanophotonic waveguides has the potential of (Received 27 March 2013;anomalies published in the 13 12-km June–to 2013)–160-km wavelength ings, the sea-floor depth is estimated from marine band. The shorter wavelengths are attenuated transforming integrated information processing. Because of the strong transverse gravity measurements from satellite altimetry interaction of light because of Newton’sinversesquarelaw,whereas confinement of the guided photons, their internal spin and their orbital angular (13). ThisWe method experimentally works best on sea study floor where the ground state coherence properties of cesium atoms in a nanofiber-based the longer wavelengths are partially cancelled by momentum get coupled. Using this spin-orbit interaction of light, we break the mirror sedimentstwo-color are thin, dipole resulting trap, in a high localized correla- Jan200 Petersen, nm away Jürgen Volz, from* Arno the Rauschenbeutel fiber surface.* Using microwave radiation tion between sea-floor topography and gravity  the gravity anomalies caused by the isostatic symmetry of the scattering of light with a gold nanoparticle on the surface of a anomaliesto coherently in the 12-km drive–to–160-km the clockwavelength transition,Controlling we the record flow of Ramseytopography light with fringesnanophotonic on the as Moho well waveguides (13 as). spin The has abyssal echo the potential signals hill ofnanophotonic and infer waveguide and realize a chiral waveguide coupler in which the handedness band. The shorter wavelengths are attenuated transforming integratedfabric information created processing. during the Because sea-floor of the spreading strong transverseof the incident light determines the propagation direction in the waveguide. We control a reversible dephasing time of Tconfinementà 0:6 ms of theand guided an irreversiblephotons, their internal dephasing spin and time their orbital of T0 angular3:7 ms. By because of Newton’sinversesquarelaw,whereas 2 2 the directionality of the scattering process and can direct up to 94% of the incoupled the longer wavelengths are partially cancelled by momentum¼ get coupled.process Using this has spin-orbit characteristic interaction wavelengths of light, we of break 2 to¼ the mirror modeling the signals, we find that, for our experimental parameters, Tà and T0 are limited by the finite symmetry of the scattering12 km, of lightso it with is now a gold becoming nanoparticle2 visible2 on in the the surface ver- of alight into a given direction. Our approach allows for the control and manipulation of the gravity anomalies caused by the isostatic RESEARCH | REPORTS topographyinitial temperatureon the Moho (13). of The the abyssal atomic hill nanophotonic ensemble andwaveguide thetical andheating gravity realize rate, gradient a chiral respectively. waveguide (VGG) models, coupler Our especially in which results the representhandednesslight in optical a waveguides and new designs of optical sensors. fabricfundamental created during step the sea-floor towards spreading establishingof the incident nanofiber-based light determineson the traps flanks the propagation for of cold the slower-spreading atoms direction as in the a building waveguide. ridges blockWe control in an process has characteristic wavelengthsswaths of differently of 2 to texttheured directionality sea-floor fabric of to the(5.14 scattering M.). S. Additionally, Steckler, A. process B. Watts, seamountsEarth and Planet. can Sci. direct Lett. between41,1– up13 to 1 and94%Santos 2 of km Basin,the Brazilincoupled and Itshe Consequences development For Petroleum of integrated electronic cir- light are no longer independent physical quan- 12optical km, so it is fiber now becoming quantumthe visible east network. in and the west ver- of thelight lineament into a(Fig. given 3). The direction.(1978). Our approach allows for the control andSystem manipulation Development (Petroleum of Geology Conference Series, geometry of the two features suggests that they tall,6. C. S. which Liu, D. T. were Sandwell, not J. R. Curray, apparentJ. Geophys. in Res. the87, older gravityGeological Society of London,cuits London, laid 2010), the pp. 855 foundations–866. for the informa- tities but are coupled (4, 5). In particular, the spin tical gravity gradient (VGG) models, especially light in optical waveguides7673 and–7686 new (1982). designs of optical sensors. 19. W. U. Mohriak, P. Szatmari, S. Anjos, Geol. Soc. London Spec. on the flanks of the slower-spreadingform a pair ofridges an extinct ridge (on the African models,7. K. Matthews, are R. D. becoming Müller, P. Wessel, J. visible M. Whittaker, in the newPubl. data.363,131–158 (2012). tion age, which fundamentally changed depends on the position in the transverse plane (14DOI:). Additionally,10.1103/PhysRevLett.110.243603 seamounts betweenside) and 1 and a pseudofault 2 km (on thehe South development American of integratedAsJ. CryoSat-2 Geophys.PACS electronic Res. numbers: Solid continues Earth cir-116,1–28light 42.50.Ct, (2011). to are map no 37.10.Gh,longer the ocean independent sur- 37.10.Jk, physical 42.50.Exmodern quan- society. During the past decades, and on the propagation direction of light in the side), created by a northward ridge propagation 8. Materials and methods are available as supplementary ACKNOWLEDGMENTS tall, which were not apparent in the older gravity cuits laid the foundations for the informa- tities but are coupled (4, 5). In particular, the spin episode between ~100 and 83 Ma. An absolute facematerials topography, on Science Online. the noise in the globalThe marine CryoSat-2 data were providedatransitionfromelectronictophotonicin- by the European Space Agency, waveguide—an effect referred to as spin-orbit in- models, are becoming visible in the new data. tion age, which fundamentally9. J. Pindell, L. Kennen, changed in The Geologydepends and Evolution on of the the positionand NASA/Centre in the transverse National d"Etudes plane Spatiales provided data hot spot–based plate reconstruction using the gravity field will decrease. Additional analysis Tformation transfer took place, and nowadays, teraction of light (SOI). This effect holds great As CryoSat-2 continues to map the ocean sur- modern society. DuringRegion the Between past decades,North and South Americaand,K.James, on the propagationfrom direction the Jason-1 altimeter. of light This in research the was supported by rotation parameters from O’Neill et al.(10)in- ofM. the A. Lorente, existing J. Pindell, data, Eds. (Special combined Publication, Geological with thisNSF steady (grant OCE-1128801),nanophotonic the Office of Naval Research circuits (grant and waveguides promise promises for the investigation of a large range of Over theface past topography, years, the noise hybrid in thedicates global quantum that marine the Cardno systems hot spotatransitionfromelectronictophotonicin- (Fig. 3)have may SocietySpecifically, of London, London, 2009), we vol.waveguide 328, experimentally pp. 1–55.—an effectN00014-12-1-0111), referred to characterize as the spin-orbit National Geospatial in- Intelligence and Agency model gravity field will decrease. Additional analysis Tformation transfer tookdecrease place,10. C. O'Neill, and R. in nowadays,D. Müller, noise, B. Steinberger, willteraction enableGeochem. dramatic Geophys. of light (SOI). improve-(grant This HM0177-13-1-0008), effectto holds partially the Australian great replaceResearch Council their electronic counterparts physical phenomena such as the spin-Hall effect have been situated not far north of the northern Geosyst. 6,Q04003(2005). attracted considerableof the existing data, attention combinedtip with of [ the1 this]. ridge steady In propagator, thenanophotonic specific where it abuts circuits case the andments waveguidesthe in ground our promise understanding statepromises coherence of for deep the investigation ocean of(grant the FL099224), tec- clock of a and largeand ConocoPhillips. transition range to enable of Version radically 23 of global cesium grids new functionalities (1–3). (6, 7)andextraordinarymomentumstates(8) 11. R. D. Müller, W. R. Roest, J. Y. Royer, Nature 396,455–459 of the gravity anomalies and VGG can be downloaded from the of light-matterdecrease quantum in noise, will enable interfaces dramaticBodo Verde improve- [ Fracture2–4], Zone;to they partially this combine is replace where the their electronictonic(1998).atoms processes. counterparts stored inphysical the nanofiber-based phenomena suchsupplementary as the materials spin-Hall two-colorThe and strong also effect at the followingconfinement trap FTP realized site: ftp:// of light provided by such and has been observed for freely propagating light ments in our understanding ofpropagator deep ocean came tec- to a halt.and Theseto enable observations radically new12. functionalities R. Granot, J. Dyment, ( Y.1– Gallet,3). Nat.(6 Geosci., 7)andextraordinarymomentumstates(5,220–223 topex.ucsd.edu/pub/global_grav_1min.8) The manuscript contents (2012). waveguides leads to large intensity gradients on fields (9, 10)inthecaseoftotalinternalreflection the advantagestonic processes. of photons forconform transmitting with the inference quantumThe strong that ridges confinement infor- have a ofREFERENCES lightin provided [12]. by AND Remarkably, such NOTESand has been the observed inferredare for the freely opinions propagating coherenceof the authors, and light the participation times of W.H.F.S. extend 13. W. H. F. Smith, D. T. Sandwell, Science 277,1956–1962 should not be construedthe as wavelengthindicating that the contents scale. of Inthe this strongly nonparaxial (11, 12), in plasmonic systems (13 15), and for tendency to propagate towardwaveguides hot spots/plumes leads to large1. intensity(1997). J. T. Wilson, gradientsNature on207,343fields–347 (9 (1965)., 10)inthecaseoftotalinternalreflection – mation andREFERENCES of long-coherence-time AND NOTES systems, such as up to milliseconds even thoughpaper are thea statement experiments of official policy, decision, ortake position onplace and that propagation eventsthe and wavelength resulting scale.spread- In this2.14. strongly J. S. A. Cande, Goff, W. nonparaxial H. J. F. LaBrecque, Smith, K. A. Marks, W.( Haxby,11Oceanography, 12),J. in Geophys.17 plasmonic,24–37 Res.behalf Solid systems of Earth NOAA (or13 the–regime,15 U.S.), government. and for spin and orbital angular momentum of radio frequency waves in metamaterials (16). Re- 1. J. T. Wilson, Nature 207,343–347 (1965). (2004). ing asymmetries are frequentlyregime, contained spin and within orbital angular93,13479 momentum–13492 (1988). of radio frequency waves in metamaterials (16). Re- cently, it has been demonstrated in a cavity- dopant ions2. in S. Cande, crystals, J. LaBrecque, nitrogen W. Haxby, J. Geophys.individual vacancy Res. spreading Solid Earth centers, corridors bounded quantum by FZs 15. N.at K. Pavlis, a distance S. A. Holmes, S. C. where Kenyon, J. K. Factor, the atom-surfaceSUPPLEMENTARY interaction MATERIALS starts to be 93,13479–13492 (1988). 3.J. C. Geophys. Heine, Res. J. Zoethout,117,B04406(2012). R. D. Müller, Solid Earth 4,215–253 Vienna Center for Quantum Science and Technology, TU cently, it has been demonstratedwww.sciencemag.org/content/346/6205/65/suppl/DC1 in a cavity- quantum electrodynamics setup in which SOI 3. C. Heine, J. Zoethout, R. D. Müller, Solid(11). Earth The4,215 existence–253 of majorVienna previously Center for unknown Quantum Science16. M. and Seton Technology,et al., Earth TU Sci. Rev. 113,212–270 (2012). dots, single trapped neutral atoms and ions, and atomic (2013).significant. quantum electrodynamicsSupplementary setup Text inWien which–Atominstitut, SOI Stadionallee 2, 1020 Vienna, Austria. ridge propagation events will also be relevant for 17. W. Mohriak, M. Nóbrega, M. Odegard, B. Gomes, W. Dickson, (2013). Wien–Atominstitut, Stadionallee 2,4. 1020 L. A. Vienna, Lawver, Austria. L. M. Gahagan, I. W. Dalziel, Mem. Natl.Figs. Inst. S1 Polar and S2 *Corresponding author. E-mail: [email protected] (J.V.); arno. fundamentally modifies the coupling between a 4. L. A. Lawver, L. M. Gahagan, I. W. Dalziel,interpretingMem. Natl. Inst. marine Polar magnetic*Corresponding anomaly author. sequences E-mail: [email protected]. Geosci. (J.V.);16,231 arno.–245 (2010). fundamentally modifies the coupling between a ensembles, for storing and processing quantum informa- Res. 53The,214–229 experimental (1998). setup isReferences sketched (20–33) [email protected] in Fig. 1(a) (A.R.)and is single atom and the resonator field (17). Res. 53,214–229 (1998). during the Cretaceous [email protected] Superchron (A.R.) on 18. I. Scotchman, G. Gilchrist, N. Kusznir,single A. Roberts, atom R. Fletcher, and in the resonator field (17). tion and for realizing long-distance quantum communica- Thedescribed Breakup of the South Atlantic in Ocean: detail Formation in of Failed Refs.2 July [2014;12 accepted,19]. 2 September Cesium 2014 atoms are conjugate ridge flanks (12). Spreading Axes and Blocks of Thinned Continental Crust in the 10.1126/science.1258213 tion [5]. In the context of quantumOne of networks the most important [6], uses it of would this new trapped in the evanescent field surrounding the nanofiber SCIENCE sciencemag.org marine gravity field will be to improve the esti- SCIENCE sciencemag.org 3OCTOBER2014• VOL 346 ISSUE 6205 67 3OCTOBER2014• VOL 346 ISSUE 6205 67 be highly desirable to connectmates these of sea-floor matter-based depth in the 80% of storage the oceans waist of a tapered optical fiber. The atoms are located about having no depth soundings. The most accurate NANOPHOTONICS and processing units via opticalmethod fiber of mapping links. sea-floor A depth promising uses a mul- 200 nm above the nanofiber surface in two diametric one- approach towards the realizationtibeam of echosounder such fiber-based mounted on a large quan- research dimensional arrays of potential wells, with at most one vessel. However, even after 40 years of mapping Chiral nanophotonic waveguide tum interfaces consists in couplingby hundreds cold of ships, neutral one finds that atoms more than to atom per trapping site. By using a red-detuned standing 50% of the ocean floor is more than 10 km away photonic crystal fibers [7–9]. Anotherfrom a depth measurement. technique Between with the highsound- interfacewave and a basedblue-detuned on running spin-orbit wave, localization of the ings, the sea-floor depth is estimated from marine potential involves trapping andgravity interfacing measurements from cold satellite atoms altimetry in interactionatoms in the three of (radial, light azimuthal, and axial) directions the evanescent field surrounding(13). This optical method works nanofibers. best on sea floor where By is achieved with trap frequencies of (200, 140, 315) kHz. sediments are thin, resulting in a high correla- Jan Petersen, Jürgen Volz,* Arno Rauschenbeutel* using the optical dipole forcetion exerted between sea-floor by topography a blue- and and gravity a In order to drive transitions between the hyperfine ground anomalies in the 12-km–to–160-km wavelength Controlling the flow of light with nanophotonic waveguides has the potential of red-detuned nanofiber-guided lightband. The field shorter [10 wavelengths,11], two-color are attenuated transformingstates integrated of the informationtrapped processing. atoms, Because we use of the a strong tunable transverse microwave traps have been demonstratedbecause experimentally of Newton’sinversesquarelaw,whereas with laser- confinement(MW) of field the guided at photons, a frequency their internal of spin 9.2 and GHz. their orbital In angular the following, the longer wavelengths are partially cancelled by momentum get coupled. Usingin this spin-orbit interaction of light, we break the mirror cooled cesium atoms [12,13]. the gravity anomalies caused by the isostatic symmetrywe limit of the scattering our study of light to with the a so-called gold nanoparticle clock on the transition surface of a between topography on the Moho (13). The abyssal hill nanophotonic waveguide and realize a chiral waveguide coupler in which the handedness In order to implement quantumfabric created protocols during the sea-floor with spreading atoms of thethe incident states light determinese 6 theS1 propagation=2;F 4 direction;mF in the0 waveguide.and g We control6S1=2; coupled to nanophotonic devices,process good has characteristic coherence wavelengths proper- of 2 to the directionality ofj thei scattering j process and¼ can direct up¼ to 94%i of thej incoupledi  j 12 km, so it is now becoming visible in the ver- lightF into a given3;m direction.F 0 Our. This approachg allows fore thetransition control and manipulation exhibits of only a ties are a prerequisite but cannottical gravity be gradient taken (VGG) for models, granted: especially light in¼ optical waveguides¼ andi new designsj i!j of opticali sensors. on the flanks of the slower-spreading ridges Nanofiber-based two-color dipole trap Various effects, like Johnson noise(14). Additionally, [14] or seamounts patch between potentials 1 and 2 km (a)he development of integrated electronic cir- light are no longer independent (b) physical quan- tall, which were not apparent in the older gravity cuits laid the foundations for the informa- tities but are coupled (4, 5). In particular, the spin [15], may occur and hampermodels, long are coherence becoming visible times in the new [16 data.]. tion age, which fundamentally changed depends on the position in the transverse plane When coupling to optical nearAs fields,CryoSat-2 continues this is to all map the ocean more sur- modern society. During the past decades, and on the propagation direction of light in the out face topography, the noise in the global marine atransitionfromelectronictophotonicin- waveguide—an effect referred to as spin-orbit in- gravity field will decrease. Additional analysis Tformation transfer took place, and nowadays, teraction of light (SOI). This effect holds great critical because of the small atom-surface distance of of the existing data, combined with this steady nanophotonicR. Mitsch, circuits A Rauschenbeutel and waveguides et promise al., Naturepromises Communications for the investigation (2015) of a large range of typically a few hundred nanometers.decrease in noise, This will enable is dramatic more improve- than to partially replace their electronic counterparts physical phenomena such as the spin-Hall effect I. Söllner, P. Lodahl et al., Nature Nanotechnology 10, 775–778 (2015) one order of magnitude closerments to the in our surface understanding than of deep in, ocean e.g., tec- and to enable radically new functionalities (1–3). (6, 7)andextraordinarymomentumstates(8) tonic processes. The strong confinement of light provided by such and has been observed for freely propagating light waveguides leads to large intensity gradients on fields (9, 10)inthecaseoftotalinternalreflection atom chip experiments, whereREFERENCES coherence AND NOTES times on the the wavelength scale. In this strongly nonparaxial (11, 12), in plasmonic systems (13–15), and for order of seconds have been observed1. J. T. Wilson, Nature [17207].,343 Similar–347 (1965). coher- chirality natural / generic feature of photonic nanostructures 2. S. Cande, J. LaBrecque, W. Haxby, J. Geophys. Res. Solid Earth regime,FIG. spin 1 and (color orbital angular online). momentum (a) of Sketchradio frequency of wavesthe in experimental metamaterials (16). Re- setup ence times have been obtained in93,13479 specially–13492 (1988). designed opti- cently, it has been demonstrated in a cavity- 3. C. Heine, J. Zoethout, R. D. Müller, Solid Earth 4,215–253 Vienna Center for Quantum Science and Technology, TU quantum electrodynamics setup in which SOI (2013). Wien–includingAtominstitut, Stadionallee the 2, 1020 tapered Vienna, Austria. optical fiber, the trapping, probe, and cal dipole traps far from surfaces4. L. A. [18 Lawver,]. L. M.Here, Gahagan, I. usingW. Dalziel, Mem. Ramsey Natl. Inst. Polar *Corresponding author. E-mail: [email protected] (J.V.); arno. fundamentally modifies the coupling between a Res. 53,214–229 (1998). [email protected] (A.R.) laser fields, the microwavesingle atom and antenna, the resonator and field (17 the). single- interferometry as well as spin-echo techniques, we mea- photon counter (SPCM). (b) End view of the nanofiber display- sure, to the best of our knowledgeSCIENCE forsciencemag.org the first time, the ing the orientation of the plane of the3OCTOBER2014 quasilinear• VOL 346 polarizations ISSUE 6205 67 of reversible and irreversible dephasing times of atoms that the blue- and red-detuned trapping fields, the atoms, and the are trapped and interfaced with an optical near field. magnetic offset field.

0031-9007=13=110(24)=243603(5) 243603-1 Ó 2013 American Physical Society 2 HE11 Mode: Polarization Properties… with atoms … with quantum dots y a Spin rbit z E-Field O x

nanofiber 2

b c d y a Spin rbit z E-Field O x

+z, σ− or -z, σ+ +z, σ+ or -z, σ− Kz, π nanofiber e f g b c d

10

+z, σ− or -z, σ+ +z, σ+ or -z, σ− Kz, π +z, σ− or -z, σ+ +z, σ+ or -z, σ− Kz, π

e f g FIG. 2. Spin-orbit coupling in optical nanofibers. a When the guided light is quasi linearly polarized along the y- axis, longitudinal polarization components occur. For light traveling in +z-direction, this leads to nearly circular − or + +z, + or -z, − Kz, FIG. 1. Experimental setup. a A single nanoparticle is (+) polarization on the top (bottom) of the fiber, see cir- +z, σ -z, σ σ σ π deposited on a silica nanofiber and illuminated with a laser cular green arrows. For light propagating in z direction, + beam that propagates in the negative x-direction. The polar- and are interchanged. As a consequence, the spin ization of the light can be set with a quarter wave plate. The angular momentum of the light (yellow arrows) is oriented fiber can be rotated around the z-axis by the angle ,which perpendicular to the propagation direction and anti-parallel FIG. 2. Spin-orbit coupling in optical nanofibers. a amounts to changing the azimuthal position of the particle. to the orbital angular momentum (red arrows). b Overlap When the guided light is quasi linearly polarized along the y- + + Here, =90 corresponds to the case where the nanopar- between the HE11,y mode (HE11 ,y mode) and ( ) polar- axis, longitudinal polarization components occur. For light + ticle is on the top of the fiber. The light scattered into the ization. c Overlap between the HE mode (HE mode) traveling in +z-direction, this leads to nearly circular 11,y 11,y + nanofiber is detected using single photon counting modules at + FIG. 1. Experimental setup. a A single nanoparticle is ( ) polarization on the top (bottom) of the fiber, see cir- and () polarization. d Overlap between the HE11± ,y each output port (left and right) of the fiber. b The presence modes and ⇡depositedpolarization. on ae-g silicaSame nanofiber as b-d andbut forilluminated the fiber with a laser cular green arrows. For light propagating in z direction, + of the nanofiber modifies the intensity distribution of the in- beam that propagates in the negative x-direction. The polar- and are interchanged. As a consequence, the spin modes HE11± ,x. The values are calculated for our experimental cident light field. The relative intensity distributions for an parametersization (nanofiber of the diameter: light can 2a be=315nm,opticalwave- set with a quarter wave plate. The angular momentum of the light (yellow arrows) is oriented incident field polarized along y-andz-axis are shown. c,d length: =fiber 532 nm) can and be rotated the fiber around mode the profilez-axis functions by the are angle ,which perpendicular to the propagation direction and anti-parallel Scanning electron microscope images of the nanofiber and 2 normalizedamounts such that to✏± changing(x = y the=0) azimuthal=1. position of the particle. to the orbital angular momentum (red arrows). b Overlap the single nanoparticle used in our experiments. From the | HE,i | + + Here, =90 corresponds to the case where the nanopar- between the HE11,y mode (HE11 ,y mode) and ( ) polar- images, we determine diameters of 2a =(315 3) nm for the ticle is on the top of the fiber. The light scattered into the + ± ization. c Overlap between the HE11,y mode (HE11 ,y mode) fiber and 2r =(90 3) nm for the nanoparticle. nanofiber is detected using single photon counting modules at + ± and () polarization. d Overlap between the HE± We choose HE± and HE± such that their main polar- 11,y each11 output,x port11 (left,y and right) of the fiber. b The presence modes and ⇡ polarization. e-g Same as b-d but for the fiber ization componentof the nanofiber points along modifies the thex-direction intensity distribution ( =0) of the in- modes HE11± ,x. The values are calculated for our experimental frequency of the light and c.c. the complex conjugate. and the y-directioncident light ( field.= 90 The), relative respectively. intensity Figure distributions 2 for an parameters (nanofiber diameter: 2a =315nm,opticalwave- incident field polarized along y-andz-axis are shown. c,d The total power of the light scattered into a given fiber shows the overlap of the profile functions ✏HE± ,x and ✏HE± ,y length: = 532 nm) and the fiber mode profile functions are Scanning electron microscope images of the nanofiber and 2 mode is then given by of the electric part of the fiber modes (see methods) with normalized such that ✏± (x = y =0) =1. the single nanoparticle used in our experiments. From the | HE,i | circular ± =(ie e )/p2 and linear ⇡ = e polariza- images,z we± determiney diameters of 2ax=(315 3) nm for the I d ✏ (r, ) 2 = ↵ ✏ (r, ) 2, (1) tion as a function of the position in the fiber transverse scat ⇤ exc ⇤ fiber and 2r =(90 3) nm for the nanoparticle.± / | · | | E · | plane for the parameters used± in our experiment, where We choose HE11± ,x and HE11± ,y such that their main polar- where (r, ) denotes the position of the scatterer in the we have chosen x as the quantization axis. Here, ex,y,z ization component points along the x-direction ( =0) nanofiber transverse plane. As a consequence, the emis- are the unit vectors along the corresponding axes. From frequency of the light and c.c. the complex conjugate. and the y-direction ( = 90), respectively. Figure 2 sion rate is directly proportional to the overlap between Fig. 2 it is apparent that the local polarization depends The total power of the light scattered into a given fiber shows the overlap of the profile functions ✏± and ✏± the field of the excitation light and the fiber mode at the both on the position in the fiber transverse plane and on HE,x HE,y mode is then given by of the electric part of the fiber modes (see methods) with particle’s position. the direction of propagation of the mode. This is a clear circular ± =(iez ey)/p2 and linear ⇡ = ex polariza- For a single-mode nanofiber, all guided light fields can signature of spin-orbit coupling of the2 nanofiber guided 2 ± I d ✏⇤(r, ) = ↵ ✏⇤(r, ) , (1) tion as a function of the position in the fiber transverse be decomposed into the quasi linearly polarized fiber light. We now considerscat the situation where theexc particle / | · | | E · | plane for the parameters used in our experiment, where eigenmodes[15, 16] HE11± ,x and HE11± ,y,wherethez-axis is located at the top ( = 90) of the nanofiber. At this where (r, ) denotes the position+ of the scatterer in the we have chosen x as the quantization axis. Here, ex,y,z coincides with the nanofiber axis and the sign indicates position, the overlap between the HE mode (HE ± nanofiber transverse plane.11 As,y a consequence,11,y the emis- are the unit vectors along the corresponding axes. From the propagation direction of the light in the fiber ( z). mode) and polarization is maximal (minimal) and ± sion rate is directly proportional to the overlap between Fig. 2 it is apparent that the local polarization depends the field of the excitation light and the fiber mode at the both on the position in the fiber transverse plane and on particle’s position. the direction of propagation of the mode. This is a clear For a single-mode nanofiber, all guided light fields can signature of spin-orbit coupling of the nanofiber guided be decomposed into the quasi linearly polarized fiber light. We now consider the situation where the particle eigenmodes[15, 16] HE11± ,x and HE11± ,y,wherethez-axis is located at the top ( = 90) of the nanofiber. At this + coincides with the nanofiber axis and the sign indicates position, the overlap between the HE mode (HE ± 11,y 11,y the propagation direction of the light in the fiber ( z). mode) and polarization is maximal (minimal) and ± H Pichler T Ramos B Vermersch P Hauke

‘Chiral' Quantum Optics → Many-Body Quantum Physics • Theory … • Review T. Ramos, H. Pichler, A.J. Daley, and PZ, PRL 2014 P. Lodahl, A. Rauschenbeutel, PZ et al., K. Stannigel, P. Rabl, and PZ. NJP (2012) submitted to Nature Reviews 2016 H. Pichler, T. Ramos, A.J. Daley, PZ, PRA, 2015 T. Ramos, B. Vermersch, P. Hauke, H. Pichler, and PZ, PRA 2016 B. Vermersch, T. Ramos, P. Hauke, and PZ, PRA 2016 H. Pichler and PZ, PRL 2016 P.O. Guimond, H. Pichler, A. Rauschenbeutel and PZ, arXiv June 2016 C. Dlaska, B. Vermersch, and PZ, arXiv July 2016 ‘Chiral' Quantum Optics

open fiber boundaries ∞L ∞R left-moving photon right-moving photon

✓ ‘chiral’ atom-light interface: broken left-right symmetry ∞ ∞ L 6= R ‘Chiral' Quantum Optics

open fiber boundaries ∞R right-moving photon

• photons never return / are never reflected • carry away entropy ✓ ‘chiral’ atom-light interface: broken left-right symmetry ∞ 0;∞ L = R ‘chirality' ~ open quantum system ‘Chiral' Photon-Mediated Interactions

open fiber boundaries

✓ ‘chiral’ interactions broken left-right symmetry

atoms only talk to atoms on the right ‘Chiral' Interactions … How to Model?

• interactions mediated by photons - quantum optics we know

left - right symmetric

✓ dipole-dipole interaction H æ°æ+ æ+æ° by integrating out photons ª 1 2 + 1 2 - chiral quantum optics

broken left - right symmetry

✓ unidirectional interaction H æ°æ+ ª 1 2 ? Theory: ‘Cascaded Master equation’ = open quantum system Theory Quantum Optical Master Equation

open fiber boundaries

• We integrate the photons out as ‘quantum reservoir’ in Born-Markov approximation

• Master equation for reduced dynamics: density operator of atoms

i Ω˙ H ,Ω L Ω = ° sys + fl £ § Theory 1. ‘Bidirectional’ Master Equation

≠ ≠

open boundaries

• Master equation: symmetric

driven atoms 1D dipole-dipole

Ω˙ i[H ∞sin(k x x )(æ+æ° æ+æ°),Ω] = ° sys + | 1 ° 2| 1 2 + 2 1 1 2∞ cos(k xi x j )(æi°Ωæ+j {æi+æ°j ,Ω}). + i,j 1,2 | ° | ° 2 X= collective spontaneous emission

“Dicke" master equation for 1D: D E Chang et al 2012 New J. Phys. 14 063003 Theory 2.’Cascaded’ Master Equation

≠

open boundaries

• Master equation: (purely) unidirectional

† † Ω˙ L Ω i(HeffΩ ΩHeff) æΩæ = ¥° ° + Lindblad form • non-Hermitian effective Hamiltonian ∞ Heff H1 H2 i æ+æ° æ+æ° 2æ+æ° = + ° 2 1 1 + 2 2 + 2 1 • quantum jump operator: collective° ¢ C.W. Gardiner, PRL 1993; æ æ° æ° H. Carmichael, PRL 1993 = 1 + 2 • general case:positions N atoms, of the chiral atoms does not matterH. Pichler et al., PRA 2015 Theory of Cascaded Quantum Systems

in 1 out 1 in 2 system 2: out 2 system 1: ´ "driven "source" unidirectional coupling system"

:

• References Cold Atoms: Volume 2 T. Ramos, H. Pichler, A.J. Daley, and PZ, PRL 2014 The Quantum World of Ultra-Cold Atoms and Light Book I: Foundations of Quantum OpticsK. Stannigel, P. Rabl, and PZ. NJP 2012 By (author): Crispin GardinerH. Pichler, (University T. Ramos, of Otago, A.J. New Daley, Zealand PZ,), PR PeterA, 2015 Zoller (University of Innsbruck, Austria) T. Ramos, B. Vermersch, P. Hauke, H. Pichler, and PZ, PRA 2016 Cold Atoms: Volume 4 H. Pichler and PZ, PRL 2016 The Quantum World of Ultra-Cold Atoms and Light Book II: The Physics of Quantum-OpticalN=2: DevicesC.W. Gardiner, PRL 1993; H. Carmichael, PRL 1993 By (author): Crispin Gardiner (University of Otago, New Zealand), Peter Zoller (University of Innsbruck, Austria) J.I. Cirac, P. Zoller, H.J. Kimble, H Mabuchi, PRL 1998 Our Model System: ‘Chiral' Many-Body Quantum Optics

≠

open boundaries drive

✓ ‘chiral’ photon-mediated interactions ✓ laser driving ✓open quantum system

Driven-dissipative quantum many-body system Chiral Photonic Quantum Network chiral = asymmetric coupling of atoms to wave guide Chiral

channel

node

input / output port

• open quantum many body system - driven-dissipative (quantum optics) • why? -quantum info / non-equilibrium cond mat (quantum phases) • how? - physical realization -photons, spin waves, … Markovian Quantum Network Theory

many-body quantum quantum system reservoir fiber drive

Born-Markov Approximation in/out

Many body Quantum Optics • Dynamics: Master equation • Steady state:

validity … Markovian Quantum Network Theory

many-body quantum quantum system reservoir fiber drive

Engineer system-reservoir coupling! in/out

Many body Quantum Optics • Dynamics: Master equation • Steady state: !

validity … pure & (interesting) entangled state (dark state of dissipative dynamics) Dark States: Single Particle

• optical pumping t Ω(t) ⇥ g 1 g 1 ⇧⇧⇧ | + ⌅⇤ + |

pumping into a pure “dark state”

• Optical Bloch Equations quantum jump operator (nonhermitian) Ω˙ i[H,Ω] = † 1 † 1 † ∞Æ cÆΩc c cÆΩ Ω c cÆ + Æ 2 Æ 2 Æ Æ µ ∂ X • steady state as a pure “dark state”

H D E D t | = | Ω(t) ⇥ D D Æ c D 0 ⇧ | ⌅⇤ | Æ| ⇥ = conditions pumping into a pure state 24 Dark States: Many Particle qubits or particles on a lattice

c

quasi-local Lindblad operators

• master equation quantum jump operator (nonhermitian) Ω˙ i[H,Ω] = † 1 † 1 † ∞Æ cÆΩc c cÆΩ Ω c cÆ + Æ 2 Æ 2 Æ Æ µ ∂ X • desired state as “dark state” t H D E D Ω(t) ⇥ D D | = | ⇧ | ⌅⇤ | Æ cÆ D 0 | ⇥ = construct a parent Liouvillian desired state Kraus et al., PRA 2008 25 Examples: Engineered Dissipative Atomic Systems Topology via dissipation BCS-pairing from dissipation 2 At late times, we can then expand the state around BCS with the ansatzreservoir⌅ = q ⌅q,where⌅q contains |the two momentum modes q necessary to describe pair- ± Q ing. Using the projection prescription1 † ⌅q =tr† =q⌅,we ji 2 (ai ai ai 1 ai 1) d-wave pairing then find the equations of motion= + for the+ single+ + pair⌃ den- sity matrices ⌅q in the presence of nonzero mean fields ...... resulting from a couplingi i 1 to other momentum modes, whose values areMajorana dictated+ edge by the modes proximity to the final state.S. Diehl The resultinget al., Nature e⇥ective Phys. Hamiltonian 2012; PRL 2013 is quadratic ...... S. Diehl et al., PRL 2010 andJ. given∞ BudichL c1 byc2 et al., preprintc2i 1 c2i c2N 1 c2N ∞R ¥ ° ° ¥ Diss.i⇥ Quantum Phase2 transitions Entangled States from Dissipation H = n˜(c† c + ⌥ c c† ) (2) e 2 q,⇧ q,⇧ | q| q,⇧ q,⇧ q,⇧ X ˜ 1 i trapped ions system ancilla +qs⇧c (q,⇧)cq,⇧ +h.c. = ⇤qq†,⇧q,⇧, 5 2 2 q,⇧ FIG. 1. (a) Symmetry in the d-wave state, represented by 3 X asingleosite fermion pair exhibiting the characteristic sign with s = 1,s = 1 and dimensionless ”gap function” change under spatial rotations. In a d-wave BCS state, this ⇤ ⌅ ˜ q = ˜ ⌥q, and4 where the diagonal and o⇥ diagonal pair is delocalized over the whole lattice. (b,c) The dissi- 2 ˜ dq ⌃q pative pairing mechanism builds on both (b) Pauli blocking mean fields evaluate ton ˜ = =2 2 | | 2 | | (2⌅) 1+ ⌃q ⇤ and (c) delocalization via phase locking. (b) Illustration of 0.72 on the d-wave state. The e⇥ective Hamiltonian| | is Exp. ions: Blatt et al., Nature '11; Nat Phys '13 R the action of Lindblad operators using Pauli blocking for a diagonalized in the second line, introducing quasiparticle N´eelExp. stateneutral (see atoms: text). DeMarco, (c) The d-wave Oberthaler, state may … be seen as a Lindblad operators delocalization of these pairs away from half filling. [Polzik et al., PRL '11] 1 q,⇧ = (c q,⇧ + s⇧⌥qcq† , ⇧). 2 This convergence to a unique pure state is illustrated Reviews:1+ M ⌥Müllerq et al., Adv Atom Mol Opt Phys, 2012; C Bardyn et al., NJP 2013 in Fig. 2 using numerical simulations for small systems.26 q In Fig. 2a we show convergence to a pure state via the In this basis, the resulting master equation reads t⌅ = entropy of the full density matrix for a small 1D system, iHe ⌅+i⌅He† + q,⇧ ⇤qq,⇧⌅q†,⇧. The linearized Lind- blad operators have analogous properties to quasiparticle and in Fig. 2b. the fidelity of the BCS state for a small operators familiarP from interaction pairing problems: (i) 2D grid as a function of time, computed via the quantum trajectories method. They annihilate the (unique) steady state q,⇧ BCS = | Lindblad operators for D-wave states –Wenowturn 0; (ii) they obey the Dirac algebra q,⇧, † = { q0,⇧0 } to the construction of the Lindblad operators for the d- ⇥q,q0 ⇥⇧,⇧0 and zero otherwise [? ]; and (iii) therefore are related to the original fermions via a canonical transfor- wave BCS state as given in Eq. (1). We will perform this mation. The imaginary spectrum of the e⇥ective Hamil- construction first for an antiferromagnetic N´eel state at tonian features a ”dissipative pairing gap” half filling, and then generalize to the BCS state. Our task can be formulated as finding for a given many body 2 state d a set of (non-hermitian) Lindblad operators j⌥ ⇤q = ⇤ n˜ (1 + ⌥q) ⇤ n.˜ ⇥ so that| it becomes the unique ”dark state”, j d =0 ⌥| The dissipative gap implies an exponential approach to l. Both the N´eel and the BCS state have product form, ⌃ steady state d-wave BCS state for long times. This can d = d† vac . Thus, we note as a su⇤cient dark | m m| be most easily seen in a quantum trajectory represen- state condition [j⌥,d† ] = 0. Q m tation of the master equation, where the time evolu- There are two antiferromagnetic N´eel states at tion of the system is described by a stochastic system half filling N+ = i A ci†+e , ci,† vac , N = | ⇧ x ⇤ ⌅| | wavefunction ⌃(t) undergoing a time evolution with c† c† vac with A a sublattice in a two- i A i+ex, i, | iHeff t | Q non-hermitian Hamiltonian ⌃(t) = e ⌃(0) / ... dimensional⇧ ⌅ bipartite⇤ (square) lattice, which di⇥er by | 2 | Q interrupted with rate ⇤ j⌥ ⌃(t) by quantum jumps an overall spin flip. Introducing “N´eel unit cell op- | a a ⌃(t) j⌥ ⌃(t) / ... so that ⌅(t)= ⌃(t) ⌃(t) stoch. erators” Sˆ = c† ⇧ c†, a = , e⇤ = ex, ey , | ⌅ | ⌥| ⌥ | i,⇤ i+e i We thus see that (i) the BCS state is a ”dark state” of the whose usefulness will become apparent± soon,{± the± state} dissipative dynamics in the sense that j⌥ BCSN =0im- can be written in eight di⇥erent forms, N = | plies that there will never be quantum jump, i.e. the state ˆ M/2 ˆ | ± i A Si,±⇤ vac =( 1) i B Si,⇥ ⇤ vac ,withM the ⇧ | ⇧ | remains in BCSN , and (ii) states near BCSN show a lattice size. We then see that the Lindblad operators | | exponential decay according to the dissipative gap. Note Q a ˆb Q must obey [ji,⇤ , Sj,µ] = 0 for all i, j located on the same that it is in marked contrast to dissipative preparation of sublattice A or B, which is fulfilled for the set a non-interacting BEC state in bosonic systems, where a a an approach polynomial in time is expected [12]. j = c† ⇧ ci,i A or B. (3) i,⇤ i+e ⇧ Dynamics of spins coupled to a chiral waveguide d

≠

drive ∞L ∞R

Special case:

• Distance commensurate with • Equal Rabi frequencies and photon wavelength staggered detunings

For Purely dissipative Dicke model 27 Two-Level Atoms with ‘Chiral' Waveguide Coupling

≠

open boundaries

entropy carried away • Unique, pure steady state: • Quantum Dimers product of pure quantum spin-dimers

singlet / EPR singlet fraction

Note: only for N even • Entanglement by Dissipation 28 N even: cascaded ∞L

• Iterative solution from left to right:

≠

K. Stannigel et al. NJP (2012) 29 N spins? Consider cascaded case first ∞L

• Iterative solution from left to right:

≠

Dark manifold

Dark state pumping No output on the right!

30 N even: cascaded ∞L

• Iterative solution from left to right:

≠

Dark state No input from No output on the left the right!

quantum interference: no light

31 N even: cascaded ∞L

• Iterative solution from left to right:

≠

Dark state Dark state No input from No output on No output on the left the right! the right!

32 N even: cascaded ∞L

• Iterative solution from left to right:

≠

constant “purification speed"

33 N odd: cascaded ∞L

• Iterative solution from left to right:

≠

Last spin cannot pair up, but still dimers are formed

34 N even: Chiral waveguide ∞ ∞ L 6= R

≠

System purifies “as a whole"

35 N odd: chiral waveguide ∞ ∞ L 6= R • Odd number of spins?

≠

• Any unpaired spin destroyed the formed dimers: No dark state!

1

0.8

0.6

0.4

0.2

0 0 50 100 150 200

36 diffdierentfferent classes classes of of multi-partite multi-partite entanglement. entanglement. In In gen- gen- ofof the the underlying underlying mechanisms mechanisms leading leading to to them. them. In In par- par- N N eral,eral, the the Nspinsspins will will arrange arrange themselves themselves in in mNmadjacentadjacent ticular,ticular, we we show show that that the the conditions conditions stated stated in in Sec. Sec.IIII B B multi-mers:multi-mers: areare su sufficientfficient to to cool cool the the system system in in such such dark dark states states and and also we solve analytically the dark states for 2,3 and 4 Nm also we solve analytically the dark states for 2,3 and 4 Nm spins, which illustrate the most important physics. We = M , (15) spins, which illustrate the most important physics. We = Mq q , (15) | |i i | | i i recallrecall that that a apure pure quantum quantum state state isis a a dark dark state state if if it it q=1q=1 | |i i OO is is M wherewhere each each multi-mer multi-mer qMqisis the the superposition superposition (1)(1)annihilatedannihilated by by all all jump jump operators, operators, and and | | i i MqM MqM 2 2 (2) invariant under the coherent part of the dynamics, Mq = a g ⌦ +q bjl S g ⌦ q (2) invariant under the coherent part of the dynamics, | Mi q =|aig ⌦ + b|jl iSjl |jlig ⌦ i.e. an eigenstate of the Hamiltonian. Dynamics |of iTLS| coupledi j>l to |a ichiral| i waveguide i.e. an eigenstate of the Hamiltonian. XXj>l MqM 4 4 In the explicit example of the chiral spin (4)thefirst + cjlrs S S g ⌦ q + ..., (16) In the explicit example of the chiral spin (4)thefirst + cjlrs Sjl jl Srs rs g ⌦ + ..., (16) | i| i| i| i| i| i conditioncondition reads readscLcL ==cRcR =0=0, which, which means means that that j>l>r>sj>l>r>s | |i i | |i i XX thethe systems systems does does not not emit emit photons photons on on both both output output ports ports ≠ of the waveguide (hence the term “dark”). The second presentingpresenting multi-partite multi-partite entanglement. entanglement. Here Here we we just just il- il- of the waveguide (hence the term “dark”). The second lustrate some of the “spin cluster” states that can be dis- conditionsconditions is is fulfilled fulfilled if if(H(Hsyssys++HHL L++HHR)R) ==EE , , lustrate some of the “spin cluster” states that can be dis- | |i i | |i i sipativelysipatively prepared prepared with with these these chiral chiral spin spin networks, networks, but but thatthat is is if if the the state state is is an an eigenstate eigenstate of of the the total total Hamil- Hamil- H inin Sec. Sec.????wewe analysize analysize the the general general conditions conditions in in detail. detail. tonian,tonian, consisting consisting not not only only of of the the system system part part Hsyssysbutbut alsoalso of of the the bath bath induced induced coherent coherent parts partsHHL LandandHHR.R. In In generalgeneral these these two two conditions conditions can can not not be be satisfied satisfied at at the the samesame time, time, inhibiting inhibiting the the existence existence of of a a dark dark state. state. To To 1 1 1 1 understandunderstand why why and and when when they they can can be be satisfied satisfied simulta- simulta- 0.9 0.9 neouslyneously it it is is instructive instructive to to first first consider consider the the simple simple ex- ex- 0.80.8 0.8 0.8 ample of only two spins coupled by a chiral waveguide, purity ample of only two spins coupled by a chiral waveguide, 0.7 0.7 sincesince it it contains contains many many of of the the essential essential features, features, and and will will 0.60.6 0.6 0.6 serveserve as as a abuilding building block block to to understand understand larger larger systems. systems.

0.5 0.5 0.40.4 0 0 50 50 100100150150200200250250 0 0 100100 200200 300300 time time A.A. Two Two spins spins coupled coupled by by a a chiral chiral waveguide waveguide

1State1 of many-body spin system1 1 cools / purifies to a ForForNN=2=2adirectsearchfordarkstatesispos-adirectsearchfordarkstatesispos- pure state of spin dimers,0.9 tetramers0.9 , hexamers, … 0.8 0.8 sible.sible. The37 The dark dark state state condition condition (1) (1) restricts restricts the the dark dark 0.80.8 statesstates to to the the nullspaces nullspaces of ofcLcLandandcRc.R. The The nullspace nullspace 0.6 0.6 ofofcLcLisis spanned spanned by by the the trivial trivial state stateggggandand the the state state 0.70.7 1 1 ik(ikx1(x1x2x) 2) | | i i p p( eg( eg e e gege),) which, which does does not not emit emit photons photons 0.4 0.4 2 2| | ii | | i i 0.60.6 propagatingpropagating to to the the left left because because of of destructive destructive interference interference 0 0 100100 200200 300300 400400 0 0 500500 10001000 15001500 ofof the the left-moving left-moving photons photons emitted emitted by by the the two two spins, spins, an an effeectffect well well known known as as sub-radiance sub-radiance [CITE]. [CITE]. However However this this sub-radiantsub-radiant state state in in general general decays decays by by emitting emitting photons photons FigureFigure 2. 2. Dyanmical Dyanmical purification purification of of chiral chiral networks networks into into dif- dif- propagatingpropagating to to the the right. right. The The nullspace nullspace of ofcRcRisis spanned spanned ferentferent entangled entangled multimer multimer states. states. (a) (a) dimers dimers with with a singlea single 1 1 ik(ikx(x1x x)2) bybyggggandand ( eg( eg ee 1 2 gege).). Therefore, Therefore, in in chiralchiral channel channel (b) (b) quadrumers quadrumers with with a single a single chiral chiral channel channel (c) (c) | | i i p2p2| | ii | | i i octumer with two chiral channels (d) Non-local dimers in a generalgeneral only only the the state stateggggisis annihilated annihilated by by both both jump jump octumer with two chiral channels (d) Non-local dimers in a | | i i singlesingle bidirectional bidirectional channel. channel. Parameters Parameters are... are... operators,operators, leaving leaving no no room room for for a a nontrivial nontrivial dark dark state. state. AnAn exception exception occurs, occurs, if if the the distance distance of of the the two two spins spins is is CommentComment on on Fisher Fisher Sec. Sec.????forfor a way a way of of witnessing witnessing the the anan integer integer multiple multiple of of the the wavelength wavelength of of the the photons, photons, that that is isk xk x1 x x2=2=2n⇡n⇡withwithn n=0=0, 1,,12,,...2,..... Then Then the the two two entanglement.entanglement. | |1 2| | jumpjump operators operators coincide coincidecLcL==cRcR(up(up to to an an irrelevant irrelevant phase),phase), and and the the common common nullspace nullspace is is spanned spanned by by the the two two 1 statesstatesggggandandS S 1 ( eg( eg gege) [)?[?].]. The The so so called called III.III. PURE PURE DARK DARK STEADY STEADY STATES STATES OF OF | | i i | |i⌘i⌘p2p2| | ii| | i i CHIRALCHIRAL SPIN SPIN NETWORKS NETWORKS singletsinglet state stateS Sisis perfectly perfectly sub-radiant sub-radiant with with respect respect to to | |i i both,both, photons photons propagating propagating to to the the right right and and photons photons prop- prop- FromFrom a quantum a quantum optics optics point point of of view, view, the the steady steady states states agatingagating to to the the left. left. On On the the other other hand hand the the triplet triplet states states 1 1 of the driven-dissipative dynamics are pure when they T T p( eg( eg++gege) and) andeeeeareare super-radiant, super-radiant, that that is is of the driven-dissipative dynamics are pure when they | |i⌘i⌘p2 2| | i i | | i i | | i i areare dark dark states states of of the the driven-dissipative driven-dissipative dynamics. dynamics. The The theythey decay decay with with2(2(L+L +R)R[See) [See Fig. Fig.3 3andand Fig. Fig.4d)].4d)]. We We scopescope of of this this section section is tois to analyze analyze in in detail detail the the conditions conditions notenote that that in in the the perfectly perfectly cascaded cascaded setup setup this this condition condition underunder which which the the steady steady states states of of the the chiral chiral spin spin networks networks onon the the distance distance of of the the spins spins is is not not required, required, since since then then areare dark, dark, as as well well as as to to establish establish a physical a physical interpretation interpretation therethere is is only only one one jump jump operator. operator. N even: cascaded ∞L

• Iterative solution from left to right:

≠

constant “purification speed"

38 N odd: cascaded ∞L

• Iterative solution from left to right:

≠

Last spin cannot pair up, but still dimers are formed

39 Other realizations … and more insight?

'Chiral' Couplings & 'Chiral' Networks with … • “photonic" wave-guides • "phononic" • spin waves [quantum spintronics]

• theory beyond Born-Markov using tDMRG techniques

40 • Chirality / Photonic Nanostructures two-level atom chiral = asymmetric coupling of atom to wave guide photon

photonic wave guide (fiber) …

Chiral “Spin Wave Guide" • …

chiral coupling - how? magnon

below: Rydberg atom implementation spin wave guide … ✓ magnon = ‘hard core’ bosons ✓ quantum optics with spins • Chirality / Photonic Nanostructures two-level atom chiral = asymmetric coupling of atom to wave guide photon

photonic nanostructure …

Chiral “Spin Wave Guide" • lattice …

chiral coupling - how? magnon

below: all-cold atom implementation spin wave guide …

• theory beyond Born-Markov: tDMRG techniques for spin chains 'Chiral' Quantum Optics with Spin Waveguides

• spin waveguide dipole-dipole J˜

spin waveguide J dipole-dipole

T. Ramos, B. Vermersch, P. Hauke, H. Pichler, and PZ, PRA 2016, Non-Markovian Chiral Networks B. Vermersch, T. Ramos, P. Hauke, and PZ, PRA 2016, Implementation with Rydberg Atoms and Ion Strings 'Chiral' Couplings with Spin Chains

• spin waveguide

©

map to bosons Strong Chirality = synthetic gauge field ‘Chiral’ exponential decay into the spin waveguide

… chirality

no reflection

45 Dimer formation: system + reservoir dynamics ' = ⇡/6

chiral case ⌦ Dimer formation: system + reservoir dynamics

' = ⇡/6

chiral case ⌦ ⌦ pure quantum dimer ‘electric field in waveguide' chirality magnons magnons darkness due to reservoir quantum interference spins:

real space momentum

system: spins tDMRG + quantum trajectories [beyond Born-Markov]

⌦ dimer dimer dimer

waveguide

single trajectory

sink sink tDMRG + quantum trajectories

• Larger reservoirs: tDMRG Hannes Pichler • Sink: wave function trajectories

single trajectory

sink sink

Review: Quantum trajectories & open many-body quantum systems, AJ Daley - Advances in Physics, 2014 49 'Wiring Up' Quantum Modules: ‘Chiral’ Quantum Circuits with Photons & Spins

(a)

input output

......

(c) ‘chiral' quantum channel 'Wiring Up' Quantum Modules: ‘Chiral’ Quantum Circuits with Photons & Spins

QC1 quantum QCinformation2 flow QC3

‘chiral' quantum channel STANNIGEL, RABL, SØRENSEN, LUKIN, AND ZOLLER PHYSICAL REVIEW A 84,042341(2011) STANNIGEL, RABL, SØRENSEN, LUKIN, AND ZOLLER PHYSICAL REVIEW A 84,042341(2011) by rewriting the last two lines of Eq. (37)asasumoverthe by choosing appropriate time-dependent control pulses in the thermal noise sources: nodes, which leads to a deterministic state transfer protocol. by rewriting the last two lines of Eq. (37)asasumovertheIn our setting,by choosing which is closely appropriate related, we time-dependent can adiabatically control pulses in the 1 n n thermal noise sources:γ N ([["† ,µ],"n] ["† ,[µ,"n]]). nodes, which leads to a deterministic state transfer protocol. 2 m m n + n tune the effective qubit-fiber couplings &i (t)aswellasthebare n qubit frequenciesIn ourω setting,i (t). We proceed which by is first closely deriving related, the ideal we can adiabatically 1 ! n n q This expressionγ isN explicitly([["† ,µ in], Lindblad" ] [ form"† ,[ andµ," the]] jump). pulse shapes needed for the state transfer and then comment 2 m m n n n n tune the effective qubit-fiber couplings &i (t)aswellasthebare operator associatedn with the thermal+ noise originating from on their realization. i qubit frequencies ωq (t). We proceed by first deriving the ideal node n !involves all subsequent qubits according to "n λ in in = pulse shapes needed for the state transfer and then comment This expression[ (ωq )] is∗ σ explicitly−. Here, the in matrices Lindblad form(ω)describe and the jump 1. Pulse shapes for state transfer 2 i!n T11 i T operatorpropagation associated from node withn to the node thermali (see noise Appendix originatingB)and fromWe ignoreon all their imperfections realization. for the moment and base our " λ in n we obtain &i /γ if the qubits are resonant with the node n involves2 T11 ∼ all subsequentop qubits according to "derivationn on the ideal cascaded ME, as given in Eq. (31) λ OM normalin modes. in for=N 2qubits.AlongthelinesofRef.[11] we require 2 i n[ 11 (ωq )]∗ σ#i−. Here, the matrices (ω)describe 1. Pulse shapes for state transfer ! T T that the= system remains in a pure state µ(t) ψ(t) ψ(t) propagation from node n to node i (see Appendix B)and We ignore all imperfections=| for⟩⟨ the moment| and base our " λ in C. State transfern protocol during the entire evolution, which is equivalent to requiring we obtain &i /γ if the qubits are resonant with the Up to2 now,T11 the∼ dynamicsop of the cascaded qubit network has the outputderivation of the photodetector on the indicated ideal cascaded in Fig. 4(a) ME,to be as given in Eq. (31) OM normal modes. exactly zero. It is clear from Eqs. (32)and(33)thatthisis been discussed on general# grounds. We now turn to a specific for N 2qubits.AlongthelinesofRef.[11] we require application, namely, the transfer of a quantum state from one the case ifthat we can the= enforce system the remains so-called dark in a state pure condition state µ(t) ψ(t) ψ(t) (t) ψ(t) 0foralltimes,wherethetimedependenceofthe qubit to the next. ThisC. has State been transfer the topic protocol of our recent work [42] S | ⟩during= the entire evolution, which is equivalent=| to requiring⟩⟨ | and we provide here the details of the protocol. The main jump operator is attributed to the time dependence of &i (t). the output of the photodetector indicated in Fig. 4(a) to be Upproblem to now, in transferring the dynamics a quantum of the state cascadedψ0 qubitα 0 networkβ 1 Under has this constraint the evolution of the system is completely between two nodes of a cascaded network| according⟩ = | ⟩ to+ | ⟩ characterizedexactly by the Schr zero.odinger¨ It is equation clear from∂t ψ Eqs.i( (H32eff)and(33)thatthisis been discussed on general grounds. We now turn to a specific | ⟩ = −i i + H0) ψ ,withHeff given in Eq. (33)andH0 ω˜ (t) σ /2 application, namely, the transfer of a quantum state from one| ⟩ the case if we can enforce= the so-calledi q z dark state condition ψ0 1 0 2 0 1 ψ0 2 (44) containing the (renormalized) qubit level splittings. We further | ⟩ | ⟩ → | ⟩ | ⟩ (t) ψ(t) 0foralltimes,wherethetimedependenceofthe" qubit to the next. This has been the topic of our recent workreintroduce [42] the phases θi that had been absorbed in Sec. III A lies in the possibility that the photon emitted by the first node Sjump| operator⟩ = is attributed to the time dependence of & (t). andcould we provide pass the second here node the details instead of of being the reabsorbed protocol. [see The mainand expand the state vector in terms of three time-dependent i amplitudes u,v ,v as follows: problemFig. 4(a) in]. Ittransferring was first realized a quantum by Cirac et state al. [11]inthecontextψ0 α 0 β 1 Under1 2 this constraint the evolution of the system is completely | ⟩ = | ⟩ + | ⟩ betweenof atomic two cavity nodes QED of a that cascaded such photon network loss can according be avoided to characterizedi, (t) by the Schrodinger¨ equation ∂t ψ i(Heff ψ(t) αu(t) e + 00 (45) | ⟩ = −i i + | ⟩ =H0) ψ ,withi,| (tH)⟩ eff given ini, Eq.(t) i (φ33(t) )andH0 i ω˜ q (t) σz /2 β|[v1⟩(t) e − 10 v2(t) e − + 01 ]. = (a) ψ0 1 0 2 0 1 ψ0 2 (44) containing+ the (renormalized)| ⟩ + qubit| level⟩ splittings. We further | ⟩ | ⟩ → | ⟩ | ⟩ (46) " lies in the possibility that the photon emitted by the first node reintroduce the phases θi that had been absorbed in Sec. III A Here the phaseand expand factors have the been state introduced vector in for terms later of con- three time-dependent could pass the second node instead of being reabsorbed [see 1 t 2 1 venience according to , (t) ds [˜ωq (t) ω˜ q (t)] and Quantum State Transfer in a Spin Chain amplitudes u±,v1,=v2 2ast0 follows:± Fig. 4(a)]. It was first realized by Cirac et al. [11]inthecontextφ(t) arg J21 θ2 θ1. The dark-state condition ψ 0 = { }= − $ S| ⟩ = of atomic cavity QED that such photon loss can be avoidednow reads √&1v1 √&2v2 0andtheSchri, (t) odinger¨ equation ( 0 + ⇥ 1 ) 0 0 ( 0 + ⇥ 1 ) ψ(+t) αu=(t) e + 00 (45) | ⇤i | ⇤i | ⇤j ⇥ | ⇤i | ⇤j | ⇤j amounts to u˙ 0and =| ⟩ = i,| (t)⟩ i, (t) iφ(t) (b) &1 &2 β[v1(t) e − 10 v2(t) e − + 01 ]. (a) v˙1 v1, v˙2 + iδ(t) v2 &|1&2⟩v+1, (47) | ⟩ = − 2 = − 2 − − (46) % & # 2 1 ˙ where δ(t) ω˜ q (t) ω˜ q (t) φ(t)isaneffectivedetuning. We first deriveHere= the the pulses− phase for+δ( factorst) 0, which have is been a generalized introduced for later con- = 1 t 2 1 resonance conditionvenience taking according the varying to phase, (t of) the cascadedds [˜ωq (t) ω˜ q (t)] and ± = 2 t0 ± coupling intoφ(t account.) arg J21 θ2 θ1. The dark-state condition ψ 0 Achieving perfect= state{ transfer}= means− that we find$ time- S| ⟩ = now reads √&1v1 √&2v2 0andtheSchrodinger¨ equation (c) dependent amounts&i (t) such that to u˙ the solution0and+ of Eq. (=47)satisfiesthe dark state condition as well= as the boundary conditions (b) &1 &2 v1(tvi˙)1 v2(tf ) v11,,vv˙21(tf ) v2(ti ) 0i,δ(t) (48)v2 &1&2v1, (47) =|= − 2|= = =− 2 −= − % & where ti and tf are the initial and final times, respectively. # In reality, however, one will have2 to tolerate1 slight˙ violations where δ(t) ω˜ q (t) ω˜ q (t) φ(t)isaneffectivedetuning. of these boundaryWe first conditions derive= the due pulses to− finite for pulse+δ( lengths,t) 0, etc. which is a generalized To find suitable pulses we make use of the following= time- symmetry argumentresonance [11 condition,55]: If a photon taking is emitted the varying by the first phase of the cascaded FIG. 4. (Color online) State transfer in a two-node cascaded qubit, then,coupling upon reversing into the account. direction of time, we would see quantum network. (a) Schematic illustration of time-symmetric aperfectreabsorption.WecanexploitthisbyensuringthattheAchieving perfect state transfer means that we find time- photon wave packet (see text). (b),(c) Exemplary pulses &i (t)given emitted photon shape is invariant under time reversal and use 2 dependent &i (t) such that the solution of Eq. (47)satisfiesthe (c) in Eqs. (50)and(51), respectively, and resulting occupations vi atime-reversedcontrolpulseforthesecondqubit.Asaresult, obtained numerically. In both cases, the parameters have been| | the absorptiondark process state condition in the second as node well is as a time-reversed the boundary conditions 2 2 adjusted such that v1(tf ) < 10− . copy of the emission in the first and may thus be—at least in | | v1(ti ) v2(tf ) 1,v1(tf ) v2(ti ) 0, (48) =| |= = = 042341-10 where ti and tf are the initial and final times, respectively. In reality, however, one will have to tolerate slight violations of these boundary conditions due to finite pulse lengths, etc. To find suitable pulses we make use of the following time- symmetry argument [11,55]: If a photon is emitted by the first FIG. 4. (Color online) State transfer in a two-node cascaded qubit, then, upon reversing the direction of time, we would see quantum network. (a) Schematic illustration of time-symmetric aperfectreabsorption.Wecanexploitthisbyensuringthatthe photon wave packet (see text). (b),(c) Exemplary pulses &i (t)given emitted photon shape is invariant under time reversal and use 2 in Eqs. (50)and(51), respectively, and resulting occupations vi atime-reversedcontrolpulseforthesecondqubit.Asaresult, obtained numerically. In both cases, the parameters have been| | the absorption process in the second node is a time-reversed 2 2 adjusted such that v1(tf ) < 10− . copy of the emission in the first and may thus be—at least in | | 042341-10 Quantum State Transfer in a 140 site Spin Chain (cascaded,no sink)

1st qubit position 2nd qubit position 99.2% fidelity

1st qubit excited 2nd qubit excited None of them is excited Wiring up quantum-gadgets

Time Reversal of a Wave Packet … atom 1

QS

TR

S spin 1 q switch S spin 2 atom 2 1 ‘time reversal' box 20 0.2 0.8 'quantum spin-switch’40 0.6 0.15 / transistor60 0.4 0.1 80 0.2 100 0.05

0see also: Time reversal with photons, Mehmet Fatih120 Yanik and Shanhui Fan, Phys. Rev. Lett. 2004 0 50 100 200 50600 1001200 Time Reversal + Quantum Switch

Bath (real space) Bath (Fourier space)

purities

QS

TR spin populations

S spin 1 q switch S spin 2 1 20 0.2 0.8 40 0.6 0.15 60 0.4 0.1 80 0.2 100 0.05 0 120 0 50 100 200 50600 1001200 Photonic Circuits: Quantum Feedback with Delays • Model 1: two driven atoms with a delay line

≠

So far neglected

• Model 2: driven atom in front of mirror = quantum feedback ≠

H. Pichler, P.Z., PRL 2016 We use tDMRG techniques to solve for the dynamics. AL Grimsmo, PRL Aug 2015 Quantum Stochastic Schrödinger Equation

• Simplest example:interaction TLS part + transmission line (RWA) • 0+⇥ H = i d⇥(⇥) b (⇥)c c b(⇥) int ≠ˆ † † 0 ⇥ ⇥ with c a system operator

outputNote: For simplicity,one we way consider coupling here only a singleinput heat bath. Generalization to many reservoirs represents no conceptual difficulty. • System + bath Hamiltonian Validity of the Model

The development of quantum noise theory is based on The rotating wave approximation with a smooth system-bath coupling; • The Markov, or white noise, approximation. • system frequency Rotating wave approximation:

assume reservoir bandwidth quantum noise operatorsiH t/ iH t/ ! i t c (t) e sys ce sys = ce 0 I ⇥ where ⇥0 is the resonance frequency of the system 6 Thursday, June 11, 2009

8 Interpretation QSSE: ‘conveyor belt'

≠

quantum chicken & egg

time time steps

stroboscopic map:

(quantum) Ito increment Interpretation QSSE

≠

quantum chicken & egg

time entangled state:

atomic wave function for sequence of emission events: quantum trajectory Matrix Product State Representation

• Total state of system & waveguide ‘time bins'

• MPS representation

MPS

vs. coefficients

Bond dimension ~ entanglement

We will use MPS to solve the Quantum Stochastic Schrödinger Equation.

G. Vidal, Phys. Rev. Lett. 91, 147902 (2003). J. Daley, C. Kollath, U. Schollwöck, G. Vidal, J. Stat. Mech. (2004) P04005. S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004) 60 Conclusions

• Chiral Quantum Optics & Quantum Many-Body Physics

≠

dissipative formation of pure quantum dimers • Physical realization with atoms / solid state emitters + photons, spins, …

• Theory: dynamics of chiral quantum networks with t-DMRG techniques / beyond Markov approximation

• 2D? 61