Introduction to Quantum Optics [Theory]

Home , Peter Zoller, Quantum operation

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

quantum control perspective Peter Zoller University of Innsbruck and IQOQI, Austrian Academy of Sciences

Lectures 3 + 4 Outline uibk

Quantum Optical Systems & Control

Lectures 1+2: Isolated / Driven Hamiltonian quantum optical systems

• Basic systems & concepts of quantum optics - an overview • Example / Application: Ion Trap Quantum Computer

Lectures 3+4: Open quantum optical systems [a modern perspective]

• Continuous measurement theory, Quantum Stochastic Schrödinger Equation, master equation & quantum trajectories

• Example 1: Chiral / Cascaded Quantum Optical Systems & Quantum Many-Body Systems • [Example 2: Entanglement by Dissipation] Theory of Quantum Noise: Quantum Optical Systems The Quantum Stochastic Schrödinger Equation (QSSE)

• quantum operations, Kraus operators - formal quantum information theory • QSSE, master equations etc. - quantum Markov processes - quantum optics

drive system bath e | ⟩ laser Ω Γ photon d = i [H, ] + master equation dt L g | ⟩ 3 Quantum Operations

system U environment quantum operation e e | 0ih 0|

operation elements, Kraus operator

Ref.: Nielsen & Chuang, Quantum Information and Quantum Computations • 4 Quantum Operations

state

system U “k” environment (normalized)

probability

5 Quantum noise & quantum optical systems

• decoherence environment state preparation system • “bath” • read out

quantum operations quantum optics counts

time out U system in

ü master equation ü effect of observation on system: “preparation by quantum jumps” Literature uibk

The Quantum World of Ultra-Cold Atoms and Light:

Book I: Foundations of Quantum Optics Book II: The Physics of Quantum-Optical Devices Book III: Ultra-cold Atoms by Crispin W Gardiner and Peter Zoller

Quantum Noise

A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics

by Crispin W Gardiner and Peter Zoller `Standard Model’ of Quantum Optics: System + Environment

environment system “bath”

unspecified

harmonic oscillators

system operator • driven two-level system + spontaneous emission

quantum jump laser operator

photon detector `Standard Model’ of Quantum Optics: System + Environment

environment system “bath”

unspecified

harmonic oscillators

system operator

system frequency

reservoir bandwidth B Examples

• driven two-level system undergoing spontaneous emission

laser

photon detector

• damped cavity mode counts

time out

in photon detector

H ! a†a sys = ﬂ 0 Time evolution of the system + environment

U Schrödinger equation: system + environment Time evolution of the system + environment

Questions:

• We do not observe the environment: reduced density operator

master equation: U üdecoherence üpreparation of the system (e.g. laser cooling, optical pumping)

• We measure the environment: continuous measurement

conditional wave function: U counts ü counting statistics ü effect of observation on system time evolution (e.g. preparation of the (single quantum) system) Integration of the Schrödinger Equation

• technical step: interaction picture with respect to bath:

interaction picture

Schrödinger equation ~

!0 # 1 + i(! ! )t b(t): b(!)e° ° 0 d! = p2º ˆ! # 0° noise operators

† i! (t s) b(t),b (s) ± (t s)e° 0 ° = s ° h i “white noise on scale 1/B” system frequency

rotating frame reservoir bandwidth B • We integrate the Schrödinger equation in small time steps

0 time t

Q.: size of time step? hierarchy of time scales (see below: "coarse graining") 1st time step

0 time t

• First time step: we start with the first interval and expand U(Δt) to second order in Δt

first order in Δt 1st time step

0 time t

• First time step: to first order in Δt

one photon

time no photon

time 1st time step

0 time t

• First time step: to first order in Δt 1st time step

time 1st time step: summary

0 time t

• Summary of first time step: to first order in Δt

operation elements

time

U 1st time step Discussion:

• We do not read the detector: reduced density operator

U(∆t)

no photon one photon

U 1st time step Discussion:

• We read the detector:

U(∆t) “click” quantum jump operator

U “k” 1st time step Discussion:

• We read the detector:

U(∆t) “no click”

decaying norm

U “k” 2nd time step etc.

… 0 time t

• Second and more time steps:

stroboscopic integration

ü Note: remember … commute in different time slots 2nd time step etc.

… 0 time t

• Second and more time steps:

stroboscopic integration Ito Quantum Stochastic Schrödinger Equation & Master Equation

Ito Quantum Stochastic Schrödinger Equation •

(I) dt ⇥(t) = iH dt + ⌃dB†(t)c ⇥(t) ( ⇥(0) = ⇤ vac ) | ⇧ sys | ⇧ | ⇧ | sys⇧⇥| ⇧ with Ito rules ⇤ ⌅

B(t)B†(t) vac = t vac dB(t)dB†(t)=dt | ⇧ | ⇧⇤ Reduced system density operator: ⇥(t):=Tr ⇥(t) ⇥(t) • B| ⇧⌅ | Master Equation: Lindblad form • ⇥˙(t)= i [H , ⇥(t)] sys 1 + 2c⇥(t)c† c†c⇥(t) ⇥(t)c†c 2 ⇥ • Wave function of the system + environment: entangled state

…

click:

no click: • Tracing over the environment we obtain the master equation

U(∆t)

master equation ü Lindblad form ü coarse grained time derivative

d Proof: (t)=Tr ...dB(t) + ... dB†(t)+dB†(t) dB(t) dt B | | | | | | =0 =0 dB(t)dB†(t)=dt 26 Some Simple Examples Example 1: Two-level atom + spontaneous emission Master Equation

• two-level system Hamiltonian

and in the rotating frame laser ∞

• a quantum jump (detection of an emission) prepares the atom in the ground state

probability for click in time interval (t,t+dt]

• master equation (Optical Bloch Equations) Example 2: Two-level atom Evolution conditional to observation

photon count trajectory (single run)

e | ⟩ Ω Γ photodetector g | ⟩ time

Evolution of the atom, given this counting trajectory?

conditional time evolution / wave function Example 2: Two-level atom Evolution conditional to observation

Fig.: typical quantum trajectory (upper state population)

click: ∞t “quantum jump” = effect of detecting a photon on system

sys(t) p sys(t) | i! | i

no click: with Wigner -Weisskopf Hamiltonian iHeff t 1 sys(t) = e sys(0) Heff !eg i ∞ æee ... | i | i = ° 2 + µ ∂ Example 2: Two-level atom Evolution conditional to observation

Fig.: typical quantum trajectory (upper state population)

∞t • Monte Carlo wave function simulation stochastic wavefunction (t) (dim d) | sys reduced density matrix ⇢(t)= (t) (t) h | sys ih sys | ist DMRG + wave function simulation density matrix (t) (dim d d) sys Example 3: Two-level atom Evolution conditional to observation initial state: e | ⟩ photodetector

g atom time | ⟩ (0) = c g + c e | c i g| i e| i

Outcome of experiment: We observe NO photon up to time t

Question: what is the state of the atom conditional to this observation after time t?

iH t/ t/2 e eff ~ (0) c g + c e e Answer: (t) = | c i = g| i e | i | c i ...... k k k k g for t ! | i !1 We learn that the system is in the ground state poor man’s way of creating entanglement

Preparation of 2 atoms in a Bell state via measurement

atom A atom B atom A laser

low efficiency photodetectors

atom B laser

- Weak (short) laser pulse, so that the excitation probability is small. - If no detection, pump back and start again. - If detection, an entangled state is created. atom A atom B Process:

atom A

atom B Engineered Dissipation — Examples uibk

Open Quantum Many-Body Systems

Example 1:

Chiral Quantum Optics

Example 2:

Entanglement by Dissipation [& Ion Experiment] P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

Example 1:

Chiral Quantum Optics

Theory: Cascaded Quantum Systems

chiral photonic channel node • unidirectional couplings appear naturally in nanophotonic devices P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

'Chiral' Quantum Optics & Nanophotonics

Quantum communication What is Chiral Quantum Optics? with chiral edge state qubit 2

✓ photonic nanostructure 2D topology ✓ atoms, spin, … qubit 1

Ref.: topological quantum optics, Perczel, Lukin et al., PRL 2017; Hafezi, Segev, … P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

'Chiral' Quantum Optics & Nanophotonics

chiral coupling between light and quantum emitters symmetry, protected by not topology Nanophotonic devices: chirality appears naturally …

atoms & nanoﬁbers atoms & CQED quantum dots & photonic nanostructures

P. Lodahl, A. Rauschenbeutel, PZ et al., Nature Review 2017 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 week ending PHYSICALThe Breakup of the REVIEW South Atlanticvessel. Ocean: Formation However, LETTERS of Failed even after2 July 2014; 40 accepted years 2 of September mapping 2014 Chiral nanophotonic waveguide PRL 110, 243603conjugate (2013) ridge flanks (12). Spreading Axes and Blocks of Thinned Continental Crust 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 depthCoherence soundings. The most Propertiesaccurate NANOPHOTONICS of Nanofiber-Trappedings, the sea-floor depth is Cesium estimated 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 and thin, resultingA. Rauschenbeutel in a high correla- Jan Petersen, Jürgen Volz,* Arno Rauschenbeutel* Vienna50% Center of the ocean for floor Quantum is more than Science 10 km away andinterface Technology, Atominstitut, basedtion between TU on sea-floor Wien, spin-orbit Stadionalleetopography and 2, gravity 1020 Vienna, Austria from a depth measurement. Between the(Received sound- 27 March 2013;anomalies published in 13 the June 12-km 2013)–to–160-km wavelength Controlling the flow of light with nanophotonic waveguides has the potential of ings, the sea-floor depth is estimated from marine interactionband. of The light shorter wavelengths are attenuated transforming integrated information processing. Because of the strong transverse gravity measurements from satellite altimetry confinement of the guided photons, their internal spin and their orbital angular (13).We This method experimentally works best on seastudy floor thewhere ground state coherencebecause properties of Newton of’sinversesquarelaw,whereas cesium atoms in a nanofiber-based sedimentstwo-color are thin, dipole resulting trap, in a high localized correla- Jan200 Petersen, nm away Jürgen from Volz,the longer* theArno fiber wavelengthsRauschenbeutel surface. are* partially Using cancelled microwave by radiationmomentum get coupled. Using this spin-orbit interaction of light, we break the mirror 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 clock wavelength transition,Controlling we record the flow Ramsey oftopography light with fringes nanophotonic on the as Moho well waveguides as (13). spin The has echo abyssal the potential signals hill of andnanophotonic 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 T2Ãconfinement0:6 ms ofand the guided an irreversible photons, their dephasing internal spin and time their of orbitalT20 angular3:7 ms. By because of Newton’sinversesquarelaw,whereas ¼ ¼ the directionality of the scattering process and can direct up to 94% of the incoupled momentum get coupled.process Using this has spin-orbit characteristic interaction wavelengths of light, we of break 2 to the mirror themodeling longer wavelengths the signals, are partially we cancelled find that, by for our experimental parameters, T2Ã and T20 are limited by the finite symmetry of the scattering12 km, of lightso it with is now a gold becoming nanoparticle visible 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 temperature on the Moho (13 of). The the abyssal atomic hill ensemblenanophotonic and waveguide thetical heating and gravity realize rate, gradient a chiral respectively. waveguide (VGG) models, coupler Our especially in results which the represent handednesslight in optical a waveguides and new designs of optical sensors. fabricfundamental created during step the towards sea-floor spreading establishingof the nanofiber-based incident light determineson the traps flanks the for propagation of cold the atoms slower-spreading direction as in a the building waveguide. ridges block We 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 fiber is now quantumbecomingthe visible network. east 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. onDOI: the flanks10.1103/PhysRevLett.110.243603 of the slower-spreadingform a pair ofridges an extinct ridge (on the African models,7.PACS K. Matthews, are numbers: R. D. becoming Müller, P. Wessel, 42.50.Ct, J. visible M. Whittaker, in 37.10.Gh, the newPubl. 37.10.Jk, data.363,131–158 42.50.Ex (2012). tion age, which fundamentally changed depends on the position in the transverse plane (14). Additionally, seamounts betweenside) and 1 and a pseudofault 2 km (on thehe South development American of integratedAsJ. CryoSat-2 Geophys. electronic Res. Solid continues Earth cir-116,1–28light (2011). to are map no longer the ocean independent sur- physicalmodern 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. have 3) may Specifically,Society of London, London, 2009), we vol.waveguide 328, experimentally pp. 1–55.—an effectN00014-12-1-0111), referred characterize to as the spin-orbit National Geospatial in- Intelligence and Agencymodel 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 [ of1 the]. this ridge In steady propagator, thenanophotonic specific where it abuts case circuits the andments waveguidesthe ground in our promise understanding statepromises coherence of for deep the investigation of ocean(grant the FL099224), tec- clock of a and largeand ConocoPhillips. transition range to enable of Version radically 23of of globalcesium 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 interfaces enable dramaticBodo Verde improve-[2 Fracture– 4], they Zone;to partially this combine is replace where the their electronictonicatoms(1998). processes. counterparts stored inphysical the nanofiber-based phenomena suchsupplementary as the materials two-color spin-HallThe and strong also effect at the trapfollowing confinement 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 of processes. photons for transmittingconform with the inference quantumThe strong that ridges confinement infor- have a ofREFERENCES lightin provided [12]. by Remarkably, AND such NOTESand has been the observed inferredare for the freely opinions coherence propagating of the authors, and light the times participation of extend W.H.F.S. 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 the a statement experiments of official policy, decision, take or position place on 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). dopant ions in crystals, nitrogening vacancy asymmetries are centers, frequentlyregime, contained quantum spin and within orbital angularat93,13479 a momentum distance–13492 (1988). of whereradio frequency the atom-surface waves in metamaterials interaction (16). Re- starts to be cently, it has been demonstrated in a cavity- 2. S. Cande, J. LaBrecque, W. Haxby, J. Geophys.individual Res. spreading Solid Earth corridors bounded by FZs 15. N. K. Pavlis, S. A. Holmes, S. C. Kenyon, J. K. Factor, SUPPLEMENTARY MATERIALS 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 dots, single3. trapped C. Heine, J. Zoethout, neutral R. D. Müller, atomsSolid(11). Earth The4,215 and existence–253 ions, of majorVienna and previously Center atomic for unknown Quantum Science16.significant. M. and Seton Technology,et al., Earth TU Sci. Rev. 113,212–270 (2012). (2013). 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 ensembles, for4. L. A. storing Lawver, L. M. Gahagan, and I. W. processing Dalziel,interpretingMem. Natl. Inst. marine quantum Polar magnetic*Corresponding anomaly informa- author. sequences E-mail: [email protected] Geosci. (J.V.);16 experimental,231 arno.–245 (2010). fundamentally setup modifies is sketched the coupling between in Fig. a 1(a) and is Res. 53,214–229 (1998). References (20–33) [email protected] (A.R.) 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- describedThe Breakup of the South in Atlantic detail Ocean: Formation in of Failed Refs.2 July [12 2014;, accepted19]. 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 would of 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% storage of 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 promising depth 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 atoms that more thanto 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 technique measurement. Between with the high sound- interfacewave and a blue-detuned based on running spin-orbit wave, localization of the ings, the sea-floor depth is estimated from marine potential involves trapping andgravity interfacing measurements cold from 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 Bywhere 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 a topography 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. Thefield shorter [10 wavelengths,11], two-color are attenuated transformingstates of integrated the trapped information atoms, processing. we Because use of a the tunable strong transverse microwave traps have been demonstratedbecause experimentally of Newton’sinversesquarelaw,whereas with laser- confinement(MW) fieldof the guided at a photons, frequency their internal of spin9.2 and GHz. their orbital In the angular 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 our scattering study of lightto the with so-called a 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 with sea-floor atoms spreading ofthe the incident states lighte determines6S the1=2 propagation;F 4;m directionF in0 the waveguide.and g We control6S1=2; coupled to nanophotonic devices,process good has characteristic coherence wavelengths proper- of 2 to the directionality ofj thei scatteringj process¼ and can direct¼ up toi 94% ofj thei incoupled j 12 km, so it is now becoming visible in the ver- lightF into a3 given;mF direction.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 i and 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 hamper longmodels, coherence are 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 this continues is to all map the 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 R. atransitionfromelectronictophotonicin-Mitsch, A Rauschenbeutel et al., Naturewaveguide Communications—an effect referred (2015) 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 I. Söllner, P. Lodahl et al., Nature Nanotechnology 10, 775–778 (2015) of the existing data, combined with this steady nanophotonic circuits and waveguides promise promises for the investigation of a large range of typically a few hundred nanometers.decrease in noise, This will enable is more dramatic improve- than to partially replace their electronic counterparts physical phenomena such as the spin-Hall effect one order of magnitude closer toments 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. TheAtoms strong confinement + Nanophotonics of light provided exp/theory: by such and Kimble-Cirac-Chang, has been observed for freely Lukin-Vuletic-Park, propagating light Orozco, … + solid state 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 1. J. T. Wilson, Nature 207,343–347 (1965). – order of seconds have been observed2. S. Cande, J. LaBrecque,[17]. W. Similar Haxby, J. Geophys. coher- Res. Solid Earth regime,chirality spin and orbital natural angular momentum / generic of radio frequencyfeature waves inof metamaterials photonic (16). Re- nanostructures 93,13479–13492 (1988). FIG. 1 (color online). (a) Sketchcently, it of has the been demonstrated experimental in a cavity- setup ence times have been obtained3. in C. Heine, specially J. Zoethout, R. D. designed Müller, Solid Earth 4,215 opti-–253 Vienna Center for Quantum Science and Technology, TU quantum electrodynamics setup in which SOI (2013). Wienincluding–Atominstitut, Stadionallee the 2, tapered 1020 Vienna, Austria. optical fiber, the trapping, probe, and cal dipole traps far from surfaces4. L.[18 A. Lawver,].Here, L. M. Gahagan, using I. W. 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] laser (A.R.) fields, the microwavesingle atom antenna, and the resonator and field the (17). 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 polarizations 346 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 Chiral Quantum Optics uibk

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

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

ﬁber ∞R right-moving photon

✓ ‘chiral’ atom-light interface: broken left-right symmetry ∞ 0;∞ L = R ‘chirality' ~ open quantum system Chiral Photon-Mediated Interactions uibk

open ﬁber boundaries

✓ ‘chiral’ interactions broken left-right symmetry

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

• 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 - Master Equation uibk

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 £ § 1. Bidirectional Master Equation uibk

≠ ≠

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 2. Cascaded Master Equation uibk

≠

open boundaries

• Master equation: unidirectional

† † Ω˙ L Ω i(HeffΩ ΩH ) æΩæ = ¥° ° eff + 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 thechiral atoms does not matterH. Pichler et al., PRA 2015 P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

Theory Appendix

• QSSE for Cascaded / Chiral Systems • Cascaded Master Equation Cascaded Quantum Systems

Cascaded quantum systems: ﬁrst system drives in a unidirectional coupling a second quantum system

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

• Quantum Stochastic Schrödinger Equation here • Master Equation

N=2 Cascaded Theory: C.W. Gardiner, PRL 1993; H. Carmichael, PRL 1993 Cascaded Quantum Systems

Example:

counts

in 1 out 2 photon counting time

out 1 in 2 ´ unidirectional coupling

system 1: system 2: "source" "driven system" The Model

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

Hamiltonian H H (1) H (2) H H = sys + sys + B + int ! # 0+ H d!!b†(!)b(!) B ˆ = !0 # ° only right running modes Interaction part

† i!/cx † i!/cx H i d!∑ (!) b (!)e° 1 c c b(!)e+ 1 int = 1 1 ° 1 ´ † i!/cx † i!/cx i d!∑ (!)h b (!)e° 2 c c b(!)e+ 2i (x x ) + 2 2 ° 2 2 > 1 ´ h i unidirectional coupling The Model

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

time delay Interaction picture

† † † † H (t) ip∞ b (t)c b(t)c ip∞ b (t ø)c b(t ø)c (ø 0+) int = 1 1 ° 1 + 2 ° 2 ° ° 2 ! h i h i where time ordering / delays reﬂects causality, and

# 1 + ı(! ! )t b(t) d!b (!)e° ° 0 white noise operator = p2º ˆ # ° The Model

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

Stratonovich Quantum Stochastic Schrödinger Equation with time delays

d † † (S) ™(t) i Hsys(1) Hsys(2) p∞1 b (t)c1 b(t)c dt| i = ° + + ° 1 © ° ¢ h † i † p∞ b (t ø)c b(t ø)c ™(t) (ø 0+) + 2 ° 2 ° ° 2 | i ! h io time delay

where time ordering / delays reﬂects causality

1 +⇥ Scaling: p∞i ci ci ı( 0)t ! b(t)= d⌅b (⌅)e ⌅2⇥ ˆ ⇥ Stratonovich Quantum Stochastic Schrödinger Equation with time delays d (S) (t) = i (H (1) + H (2)) +⌅ b†(t)c b(t)c† dt| ⇤ { sys sys 1 1 1 ⇥ + ⌅ b†(t ⇤)c b(t ⇤)c† (t) 2 2 2 | ⇤ ⇥⇤ (⇤ 0+) ⇥

Scaling: ⌅i ci ci ⇥ Integrating the Schrödinger Eq. in 1 out 2 system 1: out 1 in 2 system 2: ´ t>1/ "source" "driven system"

time

First time step: (for time delay ø 0+) !

¢t ¢t ˆ † † ﬁrst system ™(¢t) 1 iHsys(1)¢t c1 b (t)dt c1 b(t)dt | i = ( ° + ˆ0 ° ˆ0 + ¢t ¢t † † second system iHsys(2)¢t c2 b (t °)dt c2 b(t °)dt ° + ˆ0 ° ˆ0 ¢t t2 2 † † † † ( i) dt1 dt2 b(t1)c1 b(t1°)c2 b (t2)c1 b (t2°)c2 ... ™(0) + ° ˆ0 ˆ0 ° ° + + )| i h≥ ¥≥ ¥ i

1 † † 1 † c c1 0 c c1 c c2 vac ¢t ° 2 1 + ° 2 ° 2 2 | i µ ∂ ﬁrst system emits, second absorbs causality & interaction 53 Integrating the Schrödinger Equation t>1/

in 1 out 2 system 1: out 1 in 2 system 2: ´ "source" "driven time system"

quantum First time step: (for time delay ø 0+) c c c ! jump = 1 + 2 ™(¢t) 1ˆ iH ¢t p∞c ¢B †(0) ™(0) | i = ° eff + | i n o effective (non-Hermitian) system Hamiltonian coherent • interaction: 1 1 H H (1) H (2) i c†c i c†c ic†c asymmetric eff = sys + sys ° 1 1 ° 2 2 ° 2 1 2 2 quantum 1 † † 1 † † Hsys(1) Hsys(2) i c c2 c c1 i (c c )(c1 c2) jump = + + 2 1 ° 2 ° 2 1 + 2 + Ω ≥ ¥æ

… and more steps (as before in Lecture 2) 54 Cascaded Systems

Master Equation for Cascaded Quantum Systems Version 1:

2 d 1 † † † Ω i[Hsys,Ω] 2ci Ωci Ωci ci ci ci Ω dt = ° + 2 i 1 ° ° X= n o [c†,c Ω] [Ωc†,c ] asymmetric in 1 and 2 ° 2 1 + 1 2 n o

counts

in 1 out 2 photon counting time

out 1 in 2 ´ unidirectional coupling

system 1: system 2: "source" "driven system" 55 Cascaded Systems Master Equation for Cascaded Quantum Systems Version 2: Lindblad form

d † 1 † † † Ω i HeffΩ ΩH 2cΩc c cΩ Ωc c dt = ° ° eff + 2 ° ° ≥ ¥ ≥ ¥ coherent with jump operator c c1 c2 and interaction ¥ +

1 † † 1 † Heff Hsys i c c2 c c1 i c c = + 2 1 ° 2 ° 2 ≥ ¥ counts

in 1 out 2 photon counting time

out 1 in 2 ´ unidirectional coupling

system 1: system 2: "source" "driven system" 56 P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

End of Theory Appendix Applications of Chiral Atom-Light Interfaces uibk

1. Quantum Information: Chiral Quantum Networks

open ﬁber boundaries

quantum state transfer protocol

Æ g Ø e 0 g g Æ 0 Ø 1 g g 0 Æ g Ø e | i1 + | i1 | ip | i1 ! | i1 | ip + | ip | i1 ! | i1 | ip | i2 + | i2 ° qubit 1 ¢ wavepacket° in waveguide¢ ° qubit 2 ¢

… with chiral coupling in principle perfect state transfer theory: Cirac et al. 1997; Rabl et al. PRX 2017, Vermersch et al., PRL 2017, Gorshkov et al., PRA 2017 exp: Ritter, Rempe et al, Nature 2012; Schoelkopf et al, 2017; Wallraff et al. 2018 Applications of Chiral Atom-Light Interfaces uibk

2. Driven-Dissipative Many-Body Quantum Systems

≠

open boundaries

• Unique, pure steady state:

Entanglement by dissipation / non-equilibrium quantum phases

product of pure quantum spin-dimers/EPR

T. Ramos, H. Pichler, A.J. Daley, B Vermersch, P. Hauke, P.O. Guimond, K. Stannigel, P. Rabl, PZ, PRL 2014, PRA 2015, PRA 2017, PRL 2017 - see also: A Gorkov, D Chang, … Engineering Chiral Coupling (1): nano-photonics spin-orbit coupling in nano-photonics

open ﬁber boundaries ∞L ∞R left-moving photon right-moving photon Engineering Chiral Coupling (2): synthetic gauge ﬁeld

'meta-atom'

master atom dipole-dipole

Φ quantum reservoir ﬂux open ﬁber boundaries ∞R two emitters: constructive / destructive interference

… could be implemented ‘as is’ with superconducting qubits / microwave

Can we do this in ‘free-space’ / no waveguide? ✓ 1D ✓ chiral Simulating open system dynamics with spin waves uibk two system spins with smooth absorbing boundary ' = ⇡/6

chiral case dimer ⌦

magnons magnons reservoir spins:

real space momentum system: spins Engineered Dissipation — Examples uibk

Open Quantum Many-Body Systems

Example 1:

Chiral Quantum Optics

Example 2:

Entanglement by Dissipation [& Ion Experiment] Entanglement by [Engineered] Dissipation

theory:

Reviews

• M. Müller, S. Diehl, G. Pupillo, and P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, Advances in Atomic, Molecular, and Optical Physics (2012)

• C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Imamoglu, P. Zoller, S. Diehl,Topology by dissipation, arXiv:1302.5135

experiments: J. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller & R. Blatt Nature 470, 486 (2011) P. Schindler, M. Müller, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and R. Blatt, Nat. Phys. 9, 1 (2013).

Krauter et al., Polzik & Cirac, PRL 2011 -- atomic ensembles

64 Open System Dynamics [& Decoherence ☹]

• open system dynamics system environ- not ment observed

completely positive maps:

† Ω E (Ω) Ek ΩEk = k X

Kraus operator

quantum control theory: open-loop [vs. closed loop = measurement + feedback] 65 Entanglement from (Engineered) Dissipation

• open system dynamics system √ √ | ⇥ | environ- not ment observed

engineering Kraus operators:

† Ω E (Ω) Ek ΩEk = k ! X √ √ =| ⇥ |

desired (pure) quantum state

“cooling” into a pure state - non-unitary - deterministic 66 Entanglement from (Engineered) Dissipation

• open system dynamics • Markovian system √ √ … | ⇥ | environ- not ment observed

engineering Kraus operators: master equation:

Ω E (Ω) E ΩE † ⇢˙ = i[H, ⇢] = k k k 1 1 X + c ⇢c† c† c ⇢ ⇢ c† c ! ↵ ↵ ↵ 2 ↵ ↵ 2 ↵ ↵ √ √ ↵ =| ⇥ | X ✓ ◆

desired (pure) quantum jump operators quantum state t ⇢(t) !1 “cooling” into a pure state ! | ih | - non-unitary - deterministic pumping into a pure “dark state” 67 Entanglement from (Engineered) Dissipation

• open system dynamics • atomic physics: single particle

optical pumping system √ √ | ⇥ | environ- not ment observed

engineering Kraus operators:

† t Ω E (Ω) Ek ΩEk (t) ⇥ D D = k ⇥ | ⌅ ⇤ | ! X √ √ =| ⇥ | pumping into a pure “dark state”

desired (pure) quantum state

“cooling” into a pure state - non-unitary - deterministic Q.: generalize to entangled states? 68 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 69 Example: Bell state or stabilizer pumping

• concepts • … and an ion trap experiment

70 Bell State Pumping

• Bell States two spins / qubits Z1Z2 1 1 1 = ( 01 + 10 ) + ° | ⇥2 | | + + 1 1 + = ( 00 11 ) + | ⇥ ⇤2 | ⇥| ⇥ X1X2 1 1 = ( 01 10 ) ° | ⇥ ⇤2 | ⇥| ⇥ 1 + = ( 00 + 11 ) Bell states as eigenstates of (commuting) | ⇥2 | | stabilizer operators X1X2 and Z1Z2

71 Bell State Pumping

Goal: Bell state pumping Ω(t) ™⌅ ™⌅ • ⌅ | ⇤⇥ |

+ + + +

2 x 1

z1z2 x quantum jump c1 X1(1 Z1Z2) c2 Z1(1 X1X2) 2-particle operators ☹ operators = + = + 72 Bell State Pumping

quantum circuit ancilla system qubits • (environment)

one pumping cycle (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) ancilla ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

system S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

Z1Z2(p) X1X2(p) mapping eigenvalue controlled dissipation to ancilla “correction”

73 Bell State Pumping: Ion Experiment environment system • two step deterministic pumping

+ + + +

2 x z1z2 1 x

1 p = 1 0.8 0.6 0.4 + Populations 0.2 + 0 mixed 1 2 3 state Pumping cycles

start here

J. Barreiro, M.Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller and R. Blatt, Nature 2011

74 Bell State Pumping: Ion Experiment environment system • master equation limit: probabilistic pumping

z1z2 x1x2 (i) (ii) (iii) (iv) (i) (ii) (iii) (iv) ) ) 2 2 ) ⎥1〉 0 ) ⎥1〉 1

E 2 ⎥ 〉 2 X Z 1 1 X Z 1 1

1 (Z (X

S -1 -1 M 2 M(Z U (p M M(X X ) UZ(p)

one pumping cycle Z1Z2(p) X1X2(p)

pumping probability p 1 p = 0.5 0.8 0.6 0.4 + Ω˙ i[H,Ω] Populations 0.2 = + 2 0 † 1 † 1 † ∞Æ cÆΩc c cÆΩ Ωmixedc cÆ 1 2 3 + Æ 2 Æ 2 Æ Æ 1 µ state ∂Pumping cycles X= 75 Bell State Pumping: Ion Experiment

1/2 step 1/2 step + + + | i | i | i | i | i | i

+ + + | i | i | i | i | i | i F=77(1)%

1 0 | i| i | i| +i F=92(1)%

0 0 | i| i | i| +i F=91(1)%

0 | i| +i F=74(1)% 76 Stabilizer pumping: 1+4 ions x x x x 1 2 3 4 z1z2 1 0 completely 4 2 mixed state 3 z3z4 z2z3

z1z2

GHZ state

0000 + 1111 | | F = 55.8(4)% z2z3

z2z3 x1x2x3x4

J. Barreiro, M. Müller et al., Nature 470, 486 (2011) Summary uibk

Quantum Optical Systems & Control

Lectures 1+2: Isolated / Driven Hamiltonian quantum optical systems

• Basic systems & concepts of quantum optics - an overview • Example / Application: Ion Trap Quantum Computer

Lectures 3+4: Open quantum optical systems [a modern perspective]

• Continuous measurement theory, Quantum Stochastic Schrödinger Equation, master equation & quantum trajectories

• Example 1: Chiral / Cascaded Quantum Optical Systems & Quantum Many-Body Systems • [Example 2: Entanglement by Dissipation]