UvA-DARE ( Digital Academic Repository ) Supersymmetry in quantum optics and in spin-orbit coupled systems

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Scientific RepoRts | 5:13097 | DOi: 10.1038/srep13097 In the field of quantum optics, an even more fundamental role is played by the Jaynes-Cummings and Rabi models. These models describe a system of a single bosonic mode coupled to a two-level system via dipole interaction. The understanding of the dynamics in these models led to a breakthrough in cavity-QED systems 15 and nanophotonics. The very presence of bosons (light quanta) and fermions (two-level systems) suggests that there is a hidden SUSY in these quantum optical models.
In this paper, we reveal explicitly the presence of a supersymmetric structure in a generalized version of the Rabi model of quantum optics. Further, we show that the generalized Rabi model can be realized in a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit coupling subject to a perpendicular and constant magnetic field. In the next step, the influence of this SUSY on the dissipative dynamics of the generalized Rabi model is studied. We observe that, due to the supersymmetry the dissipative dynamics, governed by the master equation in the dressed state picture, possesses an additional conserved quantity when the system is supersymmetric. Furthermore, we studied the behavior of this additional conserved quantity, if the system slightly deviates from the supersymmetric submanifold in parameter space.

The model and its realizations
We consider one of the simplest and most fundamental models describing the interaction of a single mode bosonic field (represented by the canonical operators â, â † ) with a single two-level system (described by the Pauli matrices σ i , i = ± , z), a a a a 1 z gR 1 2 The energies of the bosonic field and the energy splitting of the two-level system are ω and Δ , respectively, while the interaction constants g 1,2 can be arbitrary real numbers.
In the realm of quantum optics, the model (1) describes a single mode electromagnetic field interacting with a two-level emitter via dipole interaction and represents a direct generalization of two fundamental models in quantum optics. Namely, when either g 2 = 0 or g 1 = 0, it is known as the Jaynes-Cummings model 16,17 , while when g 1 = g 2 , it becomes the Rabi model 18,19 . In these limits, a number of spectral and dynamical properties are known, while it is much less studied for arbitrary g 1 and g 2 . In the weak coupling regime close to resonance ω ~ Δ , only the g 1 term is relevant, and the g 2 term scales to zero (this is called the rotating wave approximation, RWA). On the contrary, when the strong coupling regime is realized, both co-and counter-rotating terms have to be kept. We emphasize that, when the Rabi model is derived from the microscopic principles, then the coupling constants are such that g 1 = g 2 . The Jaynes-Cummings model was studied extensively in the literature and can be solved exactly since the total number of excitations, , is a conserved quantity. In contrast, the analytical solution of the Rabi model is still under active discussions 20 , despite the long history of the model. Similarly to the Rabi model, the Hamiltonian (1) commutes with the parity operator π = ( )î P exp N ex . Further, while the spectrum of the Jaynes-Cummings model is well known, the spectrum of the Rabi model is given by a self-consistent set of equations which can be solved numerically 20 . We note that in the limit of strong coupling (both g 1,2 /ω are large), the spectrum consists of two quasidegenerate harmonic ladders 21 . Both models are of immense experimental interest for circuit-and cavity-QED 15 setups, superconducting qubits, nitrogen-vacancy (NV) centers, etc. The solid-state devices are able to approach the strong-coupling regime, where the g 2 term becomes relevant [22][23][24][25] . In the field of quantum optics, the model with unequal g 1 and g 2 can be realized using the Λ-type 3-or 4-level transition schemes 26,27 , as shown in Fig. 1.
Indeed, consider two non-degenerate ground states |a〉 , |b〉 coupled to the excited state(s), via a quantum field â, with couplings ,  g a b . In addition, two classical laser fields with Rabi frequencies Ω a,b are applied to the system, driving transitions from the ground states to the excited state(s).
. While the couplings ,  g a b are predefined, the Rabi frequencies Ω a,b as well as the detunings Δ a,b can be tuned in a wide range and we can therefore consider the model (1) for variable g 1,2 . However, the Bloch-Seigert shift can be canceled if we choose /Δ = /Δ   g g a a b b 2 2 and we obtain the generalized Rabi model (1). Ref. 28 proposes a simulation of the Rabi model with unequal g 1 and g 2 and with an effective Bloch-Siegert shift ( σ ∝ˆˆ † a a z ) based on the resonant Raman transitions in an atom that interacts with a high finesse optical cavity mode (four-level transition scheme).
The same model appears in various branches of condensed matter science, where the spin-orbit interaction plays an important role. In particular, this is the case for a two-dimensional non-interacting electron system with Rashba and Dresselhaus spin-orbit coupling in a perpendicular magnetic field. In solid state devices, this can be realized either by the electron gas in quantum wells, in two-dimensional topological insulators or in the quantum dots with a parabolic confinement potential. In cold atomic systems, spin-orbit coupling can be achieved artificially [7][8][9][10][11][12][13][14]29 . For the case of a two-dimensional electron gas subject to a perpendicular magnetic field B = B 0 e z , the spin-orbit coupled Hamiltonian reads systems the two-dimensional electron gas with Rashba and Dresselhaus spin-orbit couplings subject to a perpendicular magnetic field (c) can also be mapped to the Rabi model with unequal couplings of the coand counter-rotating terms. In (d) we show the energy spectrum of these models as a function of the coupling parameter g 1 ~ α R and for α D ~ g 2 = 0.2, the SUSY lines occur when the parameters satisfy ∆ω − = g g are momentum operators in symmetric gauge, α R represents the Rashba spin-orbit coupling, while α D denotes the Dresselhaus spin-orbit coupling, m * is the effective electron mass, g * is the gyromagnetic ratio and μ B = eħ/(2m e c) is the Bohr magneton. A short derivation of the mapping from the Hamiltonian (2) to (1) is reproduced in the Supplement. This establishes an equivalence between the electronic Rashba and Dresselhaus model with a magnetic field and the Jaynes-Cummings-Rabi model from quantum optics, which we called the generalized Rabi model.
The correspondence =Ĥ H RD gR has the potential to cross-fertilize two areas of research where these models play a fundamental role: condensed matter physics and the field of quantum optics. This is illustrated below by investigating the quench dynamics in both models.

Supersymmetry
Supersymmetric filed theories, which were studied intensively during the last 40 years, have supersymmetric quantum mechanics (SUSY QM) as their low-energy limit. Introduced in 70's the SUSY QM became a subfield by itself 30,31 with many applications. Here we are interested in the N = 2 SUSY QM. This SUSY QM is characterized by two supercharges Q 1 and Q 2 that satisfy the algebra where Ĥ is known as the SUSY Hamiltonian acting on some Hilber space H S . From the definition (3), it immediately follows that = =ˆĤ Q Q . This implies that the spectrum of Ĥ is non-negative and that the supercharges commute with the SUSY Hamiltonian, making them constants of motion. The supercharges of a SUSY QM system generate transformations between different eigenstates of the SUSY Hamiltonian with non-zero eigenenergy. This becomes more apparent when one introduces the following linear combinations of the supercharges and , ={Q W} 0. The operator q transforms a state of negative Witten parity into a state with positive Witten parity and vice versa for ˆ † q . The SUSY Hamiltonian becomes diagonal in this representation The SUSY of a quantum system is called unbroken if the ground state energy of Ĥ is zero (E 0 = 0). In case that the ground state energy is strictly positive (E 0 > 0), the SUSY is said to be broken. From this definition it immediately follows, that for an unbroken SUSY all the ground states are annihilated by all the supercharges |Ψ 〉= , ∀ , Our findings can be summarized as follows: 1) SUSY as a symmetry exists in the generalized Rabi model for a special combination of parameters, when the Bloch-Siegert shift is zero, λ = 0, (in the special case of g 1 = g 2 SUSY exists only for degenerate atomic levels, Δ = 0, and the Hamiltonian has the form of a shifted harmonic oscillator). The associated supercharges in matrix representation are given by At the SUSY line (7) we can write = ( , ) = , , as demonstrated in the Supplement. The generalized Rabi model is thus part of the supersymmetric system at the SUSY line (7). When λ ≠ 0 the SUSY condition reads

Dissipative dynamics
In the quantum optical realization of the generalized Rabi model the effects of coupling the system to the environment are usually accounted for by the master equation in the Lindblad form. Here, we show that the SUSY in the generalized Rabi model is stable against couplings to several types of dissipative baths. Effects of relaxation and decoherence are described by the Lindblad master equation for the density matrix in the dressed picture [33][34][35][36] where the dissipator L dr should be written in terms of the jump operators j k between the exact eigenstates j , k of the Hamiltonian, is a quantum dissipator. The different terms in Eq. (12) correspond to different sources of decoherence: The first term γ σ Φ = ( )/ φĵ j 0 2 j z , describes the diagonal part of the dephasing of the two-level system in the eigenbasis and γ φ (0) is the dephasing rate quantified by the dephasing noise spectral density at zero frequency. The other two terms describe contributions from the oscillator and the two-level system baths. They cause transitions between eigenstates , where d c (Δ k j ) is the spectral density of the bath and α c (Δ k j ) is the system-bath coupling strength at the transition frequency Δ kj = ε k − ε j . The transition coefficients are = + The spectral density is assumed to be con- , where ϕ c = κ c , γ c , which are the standard damping rates of a weak coupling scenario for the bosonic and spin channels of dissipation 35 .
Using the dressed-picture dissipative formalism 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 time evolution of the mean-photon number when the initial state is taken in the "spin up" state with zero bosonic occupation. The evolution at the SUSY line exhibits oscillatory behavior, while away from the SUSY line the dynamics is damped.
Usually a dissipative quantum system has a unique limit for the stationary state density matrix. However, this is not always the case. Here we found that the stationary solution of the density matrix equation has a manifold of stationary states at the SUSY line. Namely, the stationary solution of the Lindblad equation , has a four-fold degenerate zero eigenvalue when γ φ (0) = 0. This manifold of the stationary states density matrices is spanned by the operators |i〉 〈 j|, where i, j = 1, 2 label the two degenerate states, and thus the manifold of the stationary states is equivalent to the space of unit quaternions, and can be parametrized by the SU(2) group. On the other hand, when γ φ (0) ≠ 0, only the diagonal part of this SU(2) matrix survives and the stationary state is only doubly degenerate. In the Supplement, we demonstrate that the dimension of the space of the stationary density matrices is topologically protected by the Witten index and the supercharge cohomology. As a consequence of the degenerate stationary subspace there is, in addition to the trace, another conserved quantity commuting with the Liouvillian −    , ⋅    + L i H . We explicitly show how to find these conserved quantities in the Supplement. The conserved quantities can directly be used to calculate the stationary value of observables for any initial state. The conserved quantities encode certain information about the initial state into the stationary state. This is demonstrated in Fig. 2.
We also investigated the robustness of the SUSY-like dynamics, when we are slightly detuned from the SUSY line. We observe that the additional integral of motion, I 2 , becomes a time-dependent function with an extremely slow decay. Namely, for deviations up to δ ω / ∼ . , g 0 1 1 2 from the SUSY line, the decay can be fitted with an exponential function κ ( ) ∼ (− ) I t t exp 2 with κ ~ 10 −3 for a very long time interval, corresponding to the scale of Fig. 2. This demonstrates a robustness of the SUSY-related dynamical properties even outside of the SUSY line. We attribute this behavior to the topological nature of the stationary states manifold discussed above. From a more general viewpoint this brings an analogy with the classical KAM theory, where the invariant tori stay stable for a long time.

Cross-links: dynamics
Time evolution starting from a given initial state is very natural in the framework of quantum optics. In the Jaynes-Cummings model, when the evolution starts with a coherent state, one observes Rabi oscillations with a frequency Ω = +Δ / g n 4 , where q^p = l 2 (q × p)·e z and l = (ħc/eB 0 ) 1/2 . Therefore, by preparing the condensed matter system with Rashba spin-orbit coupling in the eigenstate |q〉 of the projected density operator ρ = ( ) q iq q 2 q , one should be able to observe collapse and revival of the Rabi oscillations.
Still another example of cross-links between quantum optical models and spin-orbit coupled condensed matter systems could be provided by the Ramsey π-pulse scheme (kicks) applied to the two-level subsystem 40,41 . Following the previous analogy with Jaynes-Cummings model one can suggest a Ramsey spectrometry magnetic field pulse scheme to measure decoherence effects in the Rashba model.

Coupled systems: prospects for quantum simulation of the SUSY field theories
We coupled several (up to three) cavities, each described by the generalized Rabi model and tuned to the SUSY line. We observe a persistent degeneracy of the ground state in a range of the tunneling parameter, see Fig. 3. A number of recent studies suggest that coupled systems of Jaynes-Cummings-or Rabi-cavities undergo the Mott insulator-superfluid transition, and e.g., in the weak tunneling limit the coupled systems can be mapped to an effective XY-model with a magnetic field (similar to 42 ). To include the effect of the tunneling between the different cavities, one should use a degenerate perturbation theory to study the SUSY points. This leads to the XY-model without an effective magnetic field. Starting from two cavities and transforming to the bonding unit-bonding basis, it is easy to show that the doubly degenerate SUSY line will exist in parameter space, although its position is altered by the tunneling rate. We conjecture that, in the continuum limit, coupled generalized Rabi cavities could be described by a continuum SUSY field theory at a specific parameter manifold. We do not exclude that the continuum model could have a critical line in parameter space, where the effective theory is a super-conformal field theory. This issue will be addressed elsewhere. Another possibility to observe SUSY, would be to design a system which is described by is a continuum analogue of Q as introduced before. An experimental implementation of coupled generalized Rabi cavities, using an ensembles of NV centers coupled to superconducting microwave cavities, was recently proposed 43 .

Discussion
Further connections between dissipative dynamics of quantum optical models and spin-orbit coupled systems can be foreseen in view of the finding of 44 : for a vanishing magnetic field and when g 1 = g 2 , there is a SU(2) dynamical symmetry, which leads to non-diffusive spin transport in disordered spin-orbit coupled systems. The SUSY we found here has the same effect on transport for ∆ω − = g g 1 2 2 2 and a nonzero magnetic field Δ .
In 45 it was found that the parity operations of a generalized Dicke model (a many-level extension of the generalized Rabi model) are discrete transformations of the electric to the magnetic field or vice versa. These transformations are best defined in terms of the electric and magnetic coupling constants Ω E,B = g 1 ± g 2 , respectively. By breaking these symmetries separately in the generalized version of the Dicke model, one can establish separate electric and magnetic phases. It is interesting to note that in this picture, our SUSY line (7) is given by Ω E Ω B = ωΔ , and corresponds to the electro-magnetic self-dual line in parameter space, which is invariant under the exchange Ω E ↔ Ω B .
We observe a nontrivial structure of the stationary state density matrix forming a SU(2) manifold. This inspires a profound study of the nontrivial topology of the density matrix encoded in dissipative Scientific RepoRts | 5:13097 | DOi: 10.1038/srep13097 dynamics and possible classification of topologically non-equivalent stationary state density matrices. An initial state density matrix is mapped to the stationary state subspace, which implies that the initial state information will be partially stored in the compact space of the stationary state manifold. This concept could be very useful for the realization of (partial) decoherence-free algorithms in the quantum information science.

Figure 3. Top panels:
The spectrum of a one dimensional array of 3 coupled resonators, each described by the generalized Rabi model, as a function of the tunneling amplitude J between the resonators, for ω = 1 and Δ = 2. Bottom panels: Energy difference of the lowest two levels δ 21 = E 2 − E 1 . On the left panels the parameters are such that each generalized Rabi cavity is on the SUSY line, g 1 = 1.5 and g 2 = 0.5. On the right panels the parameters are chosen not to satisfy the SUSY condition, g 1 = 1.4 and g 1 = 0.5.