Supersymmetry in Quantum Optics

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SOURCE (OR PART OF THE FOLLOWING SOURCE):Type articleTitle Supersymmetry in quantum optics and in spin-orbit coupled systems Author(s) M. Tomka, M. Pletyukhov, V. GritsevFaculty FNWI: Institute for Theoretical Physics (ITF) Year 2015

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2/101SCIENTIFIC REPORTS | 5:13097 | DOI: 10.1038/srep13097

www.nature.com/scientificreports

Supersymmetry in quantum optics

and in spin-orbit coupled systemsMichael Tomka1, Mikhail Pletyukhov2 & Vladimir Gritsev3

Light-matter interaction is naturally described by coupled bosonic and fermionic subsystems.

This suggests that a certain Bose-Fermi duality is naturally present in the fundamental quantum

mechanical description of photons interacting with atoms. We reveal submanifolds in parameter

space of a basic light-matter interacting system where this duality is promoted to a supersymmetry

(SUSY) which remains unbroken. We show that SUSY is robust with respect to decoherence and

dissipation. In particular, the stationary density matrix at the supersymmetric lines in parameter

space has a degenerate subspace. The dimension of this subspace is given by the Witten index and

thus is topologically protected. As a consequence, the dissipative dynamics is constrained by a robust

additional conserved quantity which translates information about an initial state into the stationary

state. In addition, we demonstrate that the same SUSY structures are present in condensed matter

systems with spin-orbit couplings of Rashba and Dresselhaus types, and therefore spin-orbit coupled

systems at the SUSY lines should be robust with respect to various types of disorder. Our ndings

suggest that optical and condensed matter systems at the SUSY points can be used for quantum

information technology and can open an avenue for quantum simulation of SUSY eld theories.

Supersymmetry (SUSY) is one o the most beautiul and attractive concepts in physics, since it establishesa duality between bosons and ermions, cures divergency problems and resolves the mass hierarchy inquantum field theory 1. Furthermore, in cosmology, it can serve as an explanation o the dark matteressence2. I SUSY exists in nature, it needs to be a broken symmetry, since in our surrounding environ-ments there exists no phenomenon in which a boson is converted into a ermion. Tereore, in order toobserve its signatures, the common opinion is that we need powerul accelerators. However, recent pro-gress with quantum simulators using synthetic matter (like cold atoms, trapped ions or coupled cavitiessystems) opened a new possibility o realizing supersymmetric systems in a nowadays laboratory. Herewe show that SUSY systems can be engineered in simple and undamental models, either by means osolid state devices or by quantum optical schemes. One implementation we discuss is based on a gener-alized version o the Rabi model o quantum optics, while the other one is based on the two-dimensionalelectron gas in a magnetic filed with the Rashba and Dresselhaus spin-orbit coupling. Further, we revealthat the maniolds in parameter space, where the SUSY is unbroken, are robust with respect to dissi-pation and decoherence. Tis suggests that SUSY systems have an advantage o being used in quantuminormation science.

Te role o spin-orbit coupling is central or a number o current developments in low-dimensionalmaterials, or example the spin Hall effect, the anomalous Hall effect, spintronics3, topological insulatorsand superconductors4,5 and Majorana ermions6. Recently, synthetic gauge fields and spin-orbit couplinghave also been realized in ultracold Bose and Fermi gases with Raman beams7–14. Behind all these devel-opments stands the simple single-particle Rashba and Dresselhaus model. We ound here that spin-orbitcoupled systems can possess SUSY in a broad range o parameters.

1Department of Physics, Boston University, 590 Commonwealth Avenue, 02215 Boston, Massachusetts, USA.2Institute for Theory of Statistical Physics and JARA—Fundamentals of Future Information Technology, RWTH

Aachen, 52056 Aachen, Germany. 3Institute for Theoretical Physics, University of Amsterdam, Science Park 904,

Postbus 94485, 1098 XH Amsterdam, The Netherlands. Correspondence and requests for materials should be

addressed to M.T. (email: [email protected])

Received: 06 March 2015

Accepted: 17 July 2015

Published: 19 August 2015

OPEN

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2SCIENTIFIC REPORTS | 5:13097 | DOI: 10.1038/srep13097

In the field o quantum optics, an even more undamental role is played by the Jaynes-Cummingsand Rabi models. Tese models describe a system o a single bosonic mode coupled to a two-level sys-tem via dipole interaction. Te understanding o the dynamics in these models led to a breakthroughin cavity-QED systems15 and nanophotonics. Te very presence o bosons (light quanta) and ermions(two-level systems) suggests that there is a hidden SUSY in these quantum optical models.

In this paper, we reveal explicitly the presence o a supersymmetric structure in a generalized versiono the Rabi model o quantum optics. Further, we show that the generalized Rabi model can be realizedin a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit coupling subject to a per-pendicular and constant magnetic field. In the next step, the influence o this SUSY on the dissipative

dynamics o the generalized Rabi model is studied. We observe that, due to the supersymmetry the dis-sipative dynamics, governed by the master equation in the dressed state picture, possesses an additionalconserved quantity when the system is supersymmetric. Furthermore, we studied the behavior o thisadditional conserved quantity, i the system slightly deviates rom the supersymmetric submaniold inparameter space.

The model and its realizationsWe consider one o the simplest and most undamental models describing the interaction o a singlemode bosonic field (represented by the canonical operators â, â†) with a single two-level system (describedby the Pauli matrices σ ˆ i, i = ± , z ),

ω σ σ σ σ σ = + ∆

+ ( + ) + ( + ). ( )− + + −ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ† † † g g H a a

2a a a a

1z gR 1 2

Te energies o the bosonic field and the energy splitting o the two-level system are ω and ∆ , respec-tively, while the interaction constants g 1,2 can be arbitrary real numbers.In the realm o quantum optics, the model (1) describes a single mode electromagnetic field interact-

ing with a two-level emitter via dipole interaction and represents a direct generalization o two unda-mental models in quantum optics. Namely, when either g 2 = 0 or g 1 = 0, it is known as the Jaynes-Cummingsmodel16,17, while when g 1 = g 2, it becomes the Rabi model18,19. In these limits, a number o spectral anddynamical properties are known, while it is much less studied or arbitrary g 1 and g 2. In the weak cou-pling regime close to resonance ω ~ ∆ , only the g 1 term is relevant, and the g 2 term scales to zero (thisis called the rotating wave approximation, RWA). On the contrary, when the strong coupling regime isrealized, both co- and counter-rotating terms have to be kept. We emphasize that, when the Rabi modelis derived rom the microscopic principles, then the coupling constants are such that g 1 = g 2. TeJaynes-Cummings model was studied extensively in the literature and can be solved exactly since thetotal number o excitations, σ σ = + + −ˆ ˆ ˆ ˆ ˆ

†N a aex , is a conserved quantity. In contrast, the analytical solu-tion o the Rabi model is still under active discussions 20, despite the long history o the model. Similarly

to the Rabi model, the Hamiltonian (1) commutes with the parity operator π= ( )ˆ ˆiP exp Nex . Further,while the spectrum o the Jaynes-Cummings model is well known, the spectrum o the Rabi model isgiven by a sel-consistent set o equations which can be solved numerically 20. We note that in the limito strong coupling (both g 1,2/ω are large), the spectrum consists o two quasidegenerate harmonic lad-ders21. Both models are o immense experimental interest or circuit- and cavity-QED15 setups, super-conducting qubits, nitrogen-vacancy (NV) centers, etc. Te solid-state devices are able to approach thestrong-coupling regime, where the g 2 term becomes relevant

22–25. In the field o quantum optics, themodel with unequal g 1 and g 2 can be realized using the Λ-type 3- or 4-level transition schemes

26,27, asshown in Fig. 1.

Indeed, consider two non-degenerate ground states |a〉 , |b〉 coupled to the excited state(s), via a quan-tum field â, with couplings

, g a b. In addition, two classical laser fields with Rabi requencies Ω a,b areapplied to the system, driving transitions rom the ground states to the excited state(s). ∆ a,b denote thedetunings between the lasers and the excited state(s). I ∆ Ω ,, , , g a b a b a b is assumed, one can perorman adiabatic elimination o the excited state(s). Te resulting effective Hamiltonian then takes the ormo the generalized Rabi model (= ĤgR ) with an additional Bloch-Siegert shi, λ σ = +ˆ ˆ ˆ ˆ ˆ

†H H a a z e gR (seethe Supplement or an overview o the derivation). Te parameters o the generalized Rabi model arethen given by = Ω /∆, , , , g g a b a b a b1 2 , ( )= /∆ + /∆ / g g 2a a b b

2 2 , ∆ = (Ω /∆ − Ω /∆ )a a b b2 2 and

( )λ = − /∆ − /∆ / g g 2a a a b2 2 . While the couplings

,a b are predefined, the Rabi requencies Ω a,b as well

as the detunings ∆ a,b can be tuned in a wide range and we can thereore consider the model (1) or variable g 1,2. However, the Bloch-Seigert shi can be canceled i we choose /∆ = /∆ g g a a b b

2 2 and weobtain the generalized Rabi model (1). Re. 28 proposes a simulation o the Rabi model with unequal g 1 and g 2 and with an effective Bloch-Siegert shi ( σ ∝ˆ ˆ ˆ

†a a z ) based on the resonant Raman transitions in anatom that interacts with a high finesse optical cavity mode (our-level transition scheme).

Te same model appears in various branches o condensed matter science, where the spin-orbit inter-action plays an important role. In particular, this is the case or a two-dimensional non-interactingelectron system with Rashba and Dresselhaus spin-orbit coupling in a perpendicular magnetic field. Insolid state devices, this can be realized either by the electron gas in quantum wells, in two-dimensional

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topological insulators or in the quantum dots with a parabolic confinement potential. In cold atomicsystems, spin-orbit coupling can be achieved artificially 7–14,29. For the case o a two-dimensional electrongas subject to a perpendicular magnetic field B = B0ez , the spin-orbit coupled Hamiltonian reads

Figure 1. In the field of quantum optics, SUSY appears in a generalized Rabi model which can berealized in cavity-QED systems (a) using the Λ-type 3- or 4-level transition schemes (b). In solid statesystems the two-dimensional electron gas with Rashba and Dresselhaus spin-orbit couplings subject to aperpendicular magnetic field (c) can also be mapped to the Rabi model with unequal couplings o the co-and counter-rotating terms. In (d) we show the energy spectrum o these models as a unction o thecoupling parameter g 1 ~ αR and or αD ~ g 2 = 0.2, the SUSY lines occur when the parameters satisy

∆ω − = g g 12

22 , in terms o Eq. (1). In Re. 29 a possible realization o tunable Rashba and Dresselhaus spin-

orbit coupling with ultracold alkali atoms is proposed (e), where each state is coupled by a two-photonRaman transition, (f ).

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( ) ( )µ σ

ασ σ

ασ σ =

Π + Π+ + Π − Π + Π + Π , ( )

ˆˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ⁎

⁎

m g BH

2

2 22

x y

B z R

x y y x D

x x y y RD

2 2

0

where Π = −ˆ ˆ ˆp y x x eB

c20 , Π = +ˆ ˆ ˆp x y y

eB

c20 are momentum operators in symmetric gauge, αR represents

the Rashba spin-orbit coupling, while αD denotes the Dresselhaus spin-orbit coupling, m* is the effective

electron mass, g * is the gyromagnetic ratio and µB = eħ/(2mec) is the Bohr magneton. A short derivationo the mapping rom the Hamiltonian (2) to (1) is reproduced in the Supplement. Tis establishes an

equivalence between the electronic Rashba and Dresselhaus model with a magnetic field and theJaynes-Cummings-Rabi model rom quantum optics, which we called the generalized Rabi model.Te correspondence =ˆ ˆH HRD gR has the potential to cross-ertilize two areas o research where these

models play a undamental role: condensed matter physics and the field o quantum optics. Tis is illus-trated below by investigating the quench dynamics in both models.

SupersymmetrySupersymmetric filed theories, which were studied intensively during the last 40 years, have supersym-metric quantum mechanics (SUSY QM) as their low-energy limit. Introduced in 70's the SUSY QMbecame a subfield by itsel 30,31 with many applications. Here we are interested in the N = 2 SUSY QM.Tis SUSY QM is characterized by two supercharges Q̂1 and Q̂2 that satisy the algebra

δ , = , , = , , ( ) ˆ ˆ ˆ{ } i jHQ Q 2 1 2 3i j ij

where ˆ H is known as the SUSY Hamiltonian acting on some Hilber spaceHS. From the definition (3),

it immediately ollows that = = ˆ ˆ ˆH Q Q12

2

2. Tis implies that the spectrum o ˆ H is non-negative and that

the supercharges commute with the SUSY Hamiltonian, making them constants o motion. Te super-charges o a SUSY QM system generate transormations between different eigenstates o the SUSYHamiltonian with non-zero eigenenergy. Tis becomes more apparent when one introduces the ollowinglinear combinations o the supercharges

= ( + ), = ( − ), ( )ˆ ˆ ˆ ˆ ˆ ˆ†i iQ

1

2Q Q Q

1

2Q Q

41 2 1 2

which imply = ( ) =ˆ ˆ †

Q Q 02 2 and , = ˆ ˆ ˆ

†H{Q Q } . Te Witten parity operator ≡ , / ,ˆ ˆ ˆ ˆ ˆ

† †W [Q Q ] {Q Q } is

constructed in such a way that it commutes with the SUSY Hamiltonian, anti-commutes with the super-charges and has the eigenvalues ± 1. Te operator Ŵ gives a natural way to decomposes the Hilbert spaceH

S into a positive and negative Witten parity space, = ⊕+ −H H H

S , where = Ψ ∈ ,±H H

{ S suchthat Ψ = ± ΨŴ }. Tis allows us to represent the operators acting on HS by 2 × 2 matrices. Forexample we can write

=

−

, =

, =

,

( )ˆ ˆ

ˆˆ ˆ ˆ

ˆ

†

†W1 0

0 1Q

0 q

0 0Q

0 0

q 0 5

since =Q̂ 02

and , =ˆ ˆ{Q W} 0. Te operator q̂ transorms a state o negative Witten parity into a state withpositive Witten parity and vice versa or ˆ †q . Te SUSY Hamiltonian becomes diagonal in thisrepresentation

= , =

≡

,

( )

+

−

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ

ˆ

† †

†{ }H Q Qqq 0

0 q q

H 0

0 H 6

where ±Ĥ are called the super partner Hamiltonians. Te supercharge Q̂ transorms a negative Wittenparity eigenstate o ˆ H into a positive Witten parity eigenstate with the same positive energy,

Ψ = ( / ) Ψ+ −ˆE1 QE E and vice versa or ˆ †

Q . Te actor / E1 appears due to the normalization condi-tion o the eigenstates. Tereore, the strictly positive eigenenergies o the super partner Hamiltonians

±Ĥ are the same.Te SUSY o a quantum system is called unbroken i the ground state energy o ˆ H is zero (E0 = 0). In

case that the ground state energy is strictly positive (E0 > 0), the SUSY is said to be broken. From thisdefinition it immediately ollows, that or an unbroken SUSY all the ground states are annihilated by allthe supercharges |Ψ 〉 = , ∀ ,,ˆ i jQ 0i j0 , where j enumerates the possible degeneracy o the ground state. Tisis in analogy to supersymmetric field theories, where or an unbroken SUSY the supercharges leave the

vacuum invariant.

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Our findings can be summarized as ollows: 1) SUSY as a symmetry exists in the generalized Rabimodel or a special combination o parameters,

ω − = ∆ , ( ) g g 712

22

when the Bloch-Siegert shi is zero, λ = 0, (in the special case o g 1 = g 2 SUSY exists only or degenerateatomic levels, ∆ = 0, and the Hamiltonian has the orm o a shied harmonic oscillator). Te associatedsupercharges in matrix representation are given by

ω ω

ω ω

=

, =

.

( )

ˆ ˆ ˆ

ˆ

ˆ

g

g Q

0 q

0 0q

a

a8

1

2

At the SUSY line (7) we can write = ( , ) = ,+ − ˆ ˆ ˆ ˆ ˆ †

H diag H H {Q Q }, with

ω σ σ σ σ σ ω = = + ∆

+ + + + + ( + ) , ( )+ − + + −ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ† † † † g g g g cH qq a a

2a a a a 1

9z 2 2 1 1

= = + , ( )−ˆ ˆ ˆ ˆ ˆ† cH q q H 1 10gR

and ( ) ω = + /( )c g g 212

22

, as demonstrated in the Supplement. Te generalized Rabi model is thus parto the supersymmetric system at the SUSY line (7). When λ ≠ 0 the SUSY condition reads

λ ω λ ω λ ω λ ω λ

∆−

( − ) ( + ) = ( − ) − ( + ). ( )

g g 22 111

222

2) On the SUSY line in parameter space the SUSY is unbroken and only the Hamiltonian −Ĥ has adoubly-degenerate ground state with zero eigenenergy. Te eigenenergies o the Hamiltonian +Ĥ arestrictly positive. Apart rom the ground state o −Ĥ with zero eigenenergy, the Hamiltonian +Ĥ has thesame spectrum as −Ĥ i the parameters satisy the SUSY condition (7). Tis implies that the Witten indexis equal two.

Te Witten index W ind is given by the difference between the number o zero eigenmodes n ± o ±Ĥ ,namely ≡ − = −− + − +ˆ ˆW n ndimker H dimker Hind . Tis index is related to the index o the annihi-lation operator q̂, i.e. = = −ˆ ˆ ˆ †W indq dimker q dimker q

ind, and has the property o topological invar-

iance32 according to the Atiyah-Singer index theorem. We show explicitly in the Supplement that thereare two zero eigenmodes o −Ĥ , and zero or +Ĥ , thus W ind = 2. Similarly, to the Rabi case, the Hamiltonian(1) commutes with the parity operator P̂, thereore the two zero-modes are the eigenstates o the parityoperator P̂ and can be written as ν ν Ψ = ( ( ) |↑〉 ± (− ) |↓〉)/,±

− ˆ ˆD 0 D 0 20 , whereν ν ( ) = ( − )ˆ ˆ ˆ†D exp[ a a is a coherent state displacement operator with ν ω = / g g

1 2 and |↑ 〉 , |↓ 〉 are

the eigenstates o σ ˆ z . Te explicit derivation o the supercharges and the zero-modes or the generalizedRabi model are given in the Supplement.

Dissipative dynamicsIn the quantum optical realization o the generalized Rabi model the effects o coupling the system tothe environment are usually accounted or by the master equation in the Lindblad orm. Here, we showthat the SUSY in the generalized Rabi model is stable against couplings to several types o dissipativebaths. Effects o relaxation and decoherence are described by the Lindblad master equation or the den-

sity matrix in the dressed picture33–36: ρ ρ ρ ∂ = − , +ˆ ˆ ˆ ˆ Li Ht dr gR , where the dissipator Ldr should bewritten in terms o the jump operators j k between the exact eigenstates j , k o the Hamiltonian,

ε=ˆ j jH jgR ,

( )∑ ∑=

Φ

+ Γ + Γ ,( )

κ γ

, >L D D j j j k[ ]

12dr

j j

j k k j

jk jk

:

where ( )ρ ρ ρ ρ = − −ˆ ˆ ˆ ˆ ˆ ˆ ̂ ˆ ̂ ˆ ̂† † †D[O] 2O O O O O O12 is a quantum dissipator. Te different terms in Eq. (12)correspond to different sources o decoherence: Te first term γ σ Φ = ( )/φ ˆ j j0 2 j z , describes thediagonal part o the dephasing o the two-level system in the eigenbasis and γ φ(0) is the dephasing ratequantified by the dephasing noise spectral density at zero requency. Te other two terms describe con-tributions rom the oscillator and the two-level system baths. Tey cause transitions between eigenstates

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with the relaxation coefficients ( ) ( )π αΓ = ∆ ∆ ( )d C 2c jk c kj c kj jkc2

2, where d c (∆ k j) is the spectral density o

the bath and αc (∆ k j) is the system-bath coupling strength at the transition requency ∆ kj = εk− ε j. Tetransition coefficients are = +( ) ˆ ˆ †C j kc c jk

c with σ = , −ˆ ˆ ˆc a . Te spectral density is assumed to be con-

stant, while ( )α ∆ ∝ ∆c kj kj2 . Hence, ϕΓ = ω ∆

C c jk

c jkc 2kj , where ϕc = κc, γ c, which are the standard damp-

ing rates o a weak coupling scenario or the bosonic and spin channels o dissipation35.Using the dressed-picture dissipative ormalism we checked that the dynamics preserved the trace

property and that the ground state evolution has no time dependence. In Fig. 2 we illustrate the timeevolution o the mean-photon number when the initial state is taken in the “spin up” state with zerobosonic occupation. Te evolution at the SUSY line exhibits oscillatory behavior, while away rom theSUSY line the dynamics is damped.

Usually a dissipative quantum system has a unique limit or the stationary state density matrix.However, this is not always the case. Here we ound that the stationary solution o the density matrix

equation has a manifold of stationary states at the SUSY line. Namely, the stationary solution o theLindblad equation ρ ρ − ,

=ˆ ˆ ˆL i H 0dr st st gR , has a our-old degenerate zero eigenvalue when γ φ(0)= 0.

Tis maniold o the stationary states density matrices is spanned by the operators | i〉 〈 j|, where i, j = 1, 2label the two degenerate states, and thus the maniold o the stationary states is equivalent to the spaceo unit quaternions, and can be parametrized by the SU(2) group. On the other hand, when γ φ(0) ≠ 0,only the diagonal part o this SU(2) matrix survives and the stationary state is only doubly degenerate.In the Supplement, we demonstrate that the dimension o the space o the stationary density matrices istopologically protected by the Witten index and the supercharge cohomology. As a consequence o thedegenerate stationary subspace there is, in addition to the trace, another conserved quantity commutingwith the Liouvillian − , ⋅

+

ˆ Li H dr gR . Tese conserved quantities are constructed as an overlap between

le and right eigenstates o the Liouvillan, ρ ρ = ( )( )I 0ii . We explicitly show how to find these con-

served quantities in the Supplement. Te conserved quantities can directly be used to calculate the

Figure 2. Dissipative dynamics of the generalized Rabi model: Te time evolution o the mean-photonnumber or the initial state |0〉 |↑ 〉 (zero photons and excited two-level system). Upper panel: evolution orparameters o the model tuned to the SUSY line (7). Te stationary value (dashed line) was computed withthe help o the conserved quantity I 1 and I 2. Lower panel: dissipation ar away rom the SUSY line. In thiscase the stationary state is given by I 1, which corresponds to the trace and gives the ground state expectation

value o the mean-photon number.

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dynamics and possible classification o topologically non-equivalent stationary state density matrices.

An initial state density matrix is mapped to the stationary state subspace, which implies that the initialstate inormation will be partially stored in the compact space o the stationary state maniold. Tisconcept could be very useul or the realization o (partial) decoherence-ree algorithms in the quantuminormation science.

References1. Weinberg, S. Te Quantum Teory of Fields (Cambridge University Press, 2005).2. Feng, J. L. Supersymmetry and cosmology. Annals of Physics 315, 2 (2005).3. Zutić, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. ev. Mod. Phys. 76, 323 (2004).4. Hasan, M. Z. & ane, C. L. Colloquium: opological insulators. ev. Mod. Phys. 82, 3045 (2010).5. Qi, X.-L. & Zhang, S.-C. opological insulators and superconductors. ev. Mod. Phys. 83, 1057 (2011).6. Mouri, V. et al. Signatures o Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices. Science 336,

1003 (2012).7. Lin, Y.-J. et al. Bose-Einstein Condensate in a Uniorm Light-Induced Vector Potential. Phys. ev. Lett. 102, 130401 (2009).8. Lin, Y.-J., Compton, . L., Jimenez-Garcia, . J., Porto, V. & Spielman, I. B. Synthetic magnetic fields or ultracold neutral atoms.

Nature 462, 628 (2009).9. Lin, Y.-J. et al. A synthetic electric orce acting on neutral atoms. Nature Phys. 7, 531 (2011).

10. Lin, Y.-J., Jimenez-Garcia, . & Spielman, I. B. Spin—orbit-coupled Bose—Einstein condensates. Nature 471, 83 (2011).11. Williams, . A. et al. Synthetic Partial Waves in Ultracold Atomic Collisions. Science 335, 314 (2012).12. Cheu, L. W. et al. Spin-Injection Spectroscopy o a Spin-Orbit Coupled Fermi Gas. Phys. ev. Lett. 109, 095302 (2012).13. Campbell, D. L., Juzeliūnas G. & Spielman, I. B. ealistic ashba and Dresselhaus spin-orbit coupling or neutral atoms. Phys.

ev. A 84, 025602 (2011).14. Goldman, N., Juzeliūnas, G., Ohberg, P. & Spielman, I. B. Light-induced gauge fields or ultracold atoms. ep. Prog. Phys. 77,

126401 (2014).15. Haroche, S. & aimond, J. M. Exploring the Quantum: Atoms, Cavities and Photons (Oxord, Oxord University Press, 2006).16. Jaynes, E. . & Cummings, F. W. Comparison o quantum and semiclassical radiation theories with application to the beam

maser. Proc. Inst. Elect. Eng. 51, 89 (1963).17. Cummings, F. W. Stimulated Emission o adiation in a Single Mode. Phys. ev. 140, A1051 (1965).18. abi, I. I. On the Process o Space Quantization. Phys. ev. 49, 324 (1936).19. abi, I. I. Space Quantization in a Gyrating Magnetic Field. Phys. ev. 51, 652 (1937).20. Braa, D. Integrability o the abi model. Phys. ev. Lett. 107, 100401 (2011).21. Irish, E. ., Gea-Banacloche, J., Martin, I. & Schwab, . C. Dynamics o a two-level system strongly coupled to a high-requency

quantum oscillator. Phys. ev. B 72, 195410 (2005).

Figure 3. Top panels: Te spectrum o a one dimensional array o 3 coupled resonators, each describedby the generalized Rabi model, as a unction o the tunneling amplitude J between the resonators, or ω = 1and ∆ = 2. Bottom panels: Energy difference o the lowest two levels δ 21 = E2− E1. On the le panels theparameters are such that each generalized Rabi cavity is on the SUSY line, g 1 = 1.5 and g 2 = 0.5. On the rightpanels the parameters are chosen not to satisy the SUSY condition, g 1 = 1.4 and g 1 = 0.5.


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AcknowledgementsM.. and V.G. acknowledge support o Swiss NSF and the Delta Institute o Teoretical Physics (DIP);M.. is also supported by the ECOS-SSM grant MP1210, Delta Institute or Teoretical Physics andthe International Institute o Physics in Natal; M.P. is supported by the DFG.

Author ContributionsM.., M.P. and V.G. contributed equally to the realization o the manuscript.

Additional InformationSupplementary information accompanies this paper at http://www.nature.com/srep

Competing financial interests: Te authors declare no competing financial interests.

How to cite this article: omka, M. et al. Supersymmetry in quantum optics and in spin-orbit coupledsystems. Sci. Rep. 5, 13097; doi: 10.1038/srep13097 (2015).

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Supersymmetry in Quantum Optics