A Passive \mathcal{PT}-Symmetric Floquet-Coupler
AA Passive PT -Symmetric Floquet-Coupler Lucas Teuber, Florian Morawetz, and Stefan Scheel ∗ Institut für Physik, Universität Rostock, Albert-Einstein–Straße 23-24, D-18059 Rostock, Germany (Dated: December 23, 2020)Based on a Liouville-space formulation of open systems, we present a solution to the quantummaster equation of two coupled optical waveguides with varying loss. The periodic modulation ofthe Markovian loss of one of them yields a passive PT -symmetric Floquet system that, at resonance,shows a strong reduction of the required loss for the PT symmetry to be broken. We showcase thistransition for a multi-photon state, and we show how to physically implement the modulated losswith reservoir engineering of a set of bath modes. PACS numbers: 11.30.Er, 42.79.Gn, 42.82.-m, 03.65.Yz
Non-Hermitian systems are an extension to the conven-tional Hermitian theory that provides the basis for ourcurrent understanding of quantum physics. As such, theygarnered growing interest in recent years as they promisenew and exciting ideas and applications. Of special in-terest are the so-called parity-time ( PT ) symmetric sys-tems whose non-Hermitian Hamiltonians can still havereal eigenvalues [1]. Depending on the specific systemparameters, a phase transition to a regime with broken PT symmetry can be observed where the spectrum be-comes complex. This transition is marked by an excep-tional point (EP) where both the eigenvalues as well asthe corresponding eigenvectors coalesce [2].Due to this peculiar behaviour, which results from thecomplex extension of the parameter space, many novelconcepts where conceived and implemented. For exam-ple, the square-root dependence on small deviations forthe eigenvalues near the EP are thought to lead to in-creased sensitivity compared to the linear dependenceof Hermitian degeneracies [3]. Also, due to the specifictopology of the non-Hermitian spectrum, a chiral mode-switching can be observed when encircling the EP [4].Experimental tests of these concepts have already beenperformed in classical setups, e.g. in microwave cavities[5], LRC circuits [6] or in classical optics [7–9]. Theseare mostly active two-mode systems with the loss in onemode being balanced by an equal gain in the other. Re-cently, the first quantum experiments have been per-formed on PT symmetric systems which showed the suc-cessful implementation of a PT directional coupler in in-tegrated waveguides [10] and a full quantum-state to-mography of a qubit over the EP [11]. The main dif-ference to classical realisations is that one has to imple-ment PT symmetry passively so as to avoid additionalgain noise that breaks the PT symmetry [12]. However,it was shown that an all-loss passive system can be mod-elled as an active PT -symmetric system plus an overallloss prefactor [13] when postselecting on the subspacewith highest photon number.The need for passive PT systems does have its limitswhen testing the physics at the EP. Experimental imple- ∗ [email protected] mentations with sufficient visibility are hard to achievedue to the strong losses required to reach the EP and theassociated low success probability of postselection [14].As there are still many questions to be answered, for ex-ample, whether in the quantum domain a real increasein sensitivity can be achieved, or whether this is off-setby quantum noise induced by the self-orthogonality ofthe coalescing eigenstates [15, 16], a setup is needed thatallows to test EP physics with reduced losses.In this Letter, we show that this can be achieved by in-troducing periodic modulation into the system. Based onFloquet theory [17] one can show that the PT symmetry-breaking threshold is greatly reduced when the mod-ulation frequency equals the system’s eigenfrequency[18, 19]. We calculate the PT phase diagram of a two-mode waveguide system with modulated loss by solv-ing the associated quantum master equation in Liouvillespace utilising a Wei-Norman expansion [20]. A phasetransition at a much reduced loss rate amplitude is thenillustrated by the occupation of multiphoton Fock states.The required modulated loss can be implemented by acollection of auxiliary waveguides [21] that act as a reser-voir simulating the Markovian loss [22].The system under study is comprised of two waveg-uides with coupling rate κ and a loss rate γ for one ofthem. Both waveguides support a single mode describedby bosonic operators ˆ a i ( i = 1 , ). The evolution of aquantum state along the propagation direction of thelossy waveguides is given by a Lindblad master equation dd z ˆ ρ = − i (cid:104) ˆ H, ˆ ρ (cid:105) + γ (cid:16) a ˆ ρ ˆ a † − ˆ a † ˆ a ˆ ρ − ˆ ρ ˆ a † ˆ a (cid:17) , (1)with the system Hamiltonian ˆ H = κ (ˆ a † ˆ a + ˆ a † ˆ a ) de-scribing a lossless coupler. The modulation of the lossrate γ is assumed to be slow enough to regard the lossprocess as being Markovian with a dissipator of Lindbladform. We will later discuss this limitation in connectionwith our proposed implementation.Equation (1) can be solved in Liouville space [23]. ALiouville space L is defined as the Cartesian product L = H ⊗ H (cid:48) of two Hilbert spaces and amounts to avectorisation of Hilbert space operators. For example,the density operator ˆ ρ becomes a vector | ˆ ρ (cid:105)(cid:105) in Liouville a r X i v : . [ qu a n t - ph ] D ec space. The master equation (1) then reads as dd z | ˆ ρ (cid:105)(cid:105) = L| ˆ ρ (cid:105)(cid:105) , (2)where L is the Liouvillian acting on the original Hilbertspace operators. A right-action of the Liouvillian can bedefined by introducing the superoperators L − k | ˆ A (cid:105)(cid:105) = ˆ a k ˆ A, L + k | ˆ A (cid:105)(cid:105) = ˆ a † k ˆ A, (3) R − k | ˆ A (cid:105)(cid:105) = ˆ A ˆ a † k , R + k | ˆ A (cid:105)(cid:105) = ˆ A ˆ a k , (4)for some Hilbert space operator ˆ A . They inherit com-mutation relations from the bosonic mode operators as [ L − i , L + j ] = [ R − i , R + j ] = δ ij . The Liouvillian thus reads L = − i κ (cid:0) L +1 L − + L +2 L − − R +1 R − − R +2 R − (cid:1) + 2 γL − R − − γ (cid:0) R +1 R − + L +1 L − (cid:1) . (5)Based on this superoperator form, one can employ Liealgebra techniques to obtain the evolution superoperator U ( z ) that propagates the quantum state as | ρ ( z ) (cid:105)(cid:105) = U ( z, | ρ (0) (cid:105)(cid:105) . (6)Here, we employ a Wei-Norman expansion [20] of U ( z ) whose key steps are to first define a Lie algebra { X k } based on the superoperators occuring in Eq. (5) and thento expand U ( z ) as a product of their exponentials, i.e. U ( z ) = n (cid:89) k =1 U k ( z ) = n (cid:89) k =1 exp [ g k ( z ) X k ] . (7)Together with Eqs. (2) and (6), this results in a set ofgenerally nonlinear differential equations for the g k ( z ) .Inspecting the Liouvillian (5), the Lie algebra isspanned by { L − i R − j , R + i R − j , L + i L − j } ( i, j = 1 , ), wherethe additional operators not present in L are added toclose the algebra under commutation. The procedurecan now be reduced to two separate problems becauseany Lie algebra can be separated into a semisimple anda solvable subalgebra [24]. In the Wei-Norman expan-sion, this means that the total evolution can be sepa-rated into U = U S U R . As the solvable algebra alwaysresults in a set of directly integrable linear differentialequations for the expansion functions, this greatly sim-plifies the overall computation. Here, the solvable subal-gebra is comprised of { L − i R − j }⊕{ (cid:80) k L + k L − k , (cid:80) k R + k R − k } ,and the semisimple subalgebra is a direct sum of two sim-ple algebras { L + k L − k − L + k +1 L − k +1 , L + i L − j (cid:54) = i } ⊕ { R + k R − k − R + k +1 R − k +1 , R + i R − j (cid:54) = i } . Due to this separation, we can fur-ther decompose U S = U S U S . Note that each simple al-gebra is isomorphic to the special linear algebra sl (2 , C ) .The evolution superoperators are thus expanded as U S = e f + L +1 L − e f ( L +1 L − − L +2 L − )e f − L +2 L − , (8) U S = e f ∗ + R +1 R − e f ∗ ( R +1 R − − R +2 R − )e f ∗− R +2 R − , (9) U R = e a ( z ) ( L +1 L − + L +2 L − )e a ( z ) ( R +1 R − + R +2 R − ) × e a ( z ) L − R − e a ( z ) L − R − e a ( z ) L − R − e a ( z ) L − R − . (10) Inserting the ansatz for the evolution superoperator U ( z ) into Eqs. (2) and (6) yields the two sets of differentialequations for the functions f i and a i [13, 25].We now assume the Liouvillian (2) to be periodic, L ( z ) = L ( z + T ) , in order to potentially reduce the PT -breaking threshold. According to Floquet theory, theperiodicity carries over to the evolution superoperator U which then obeys U ( z + T ) = U ( z ) U ( T ) [17]. This meansthat knowledge of the one-cycle evolution U ( T ) (the mon-odromy) allows to construct the evolution for arbitrary z .The eigenvalues of the monodromy can be written as e µ n T with µ n being the Floquet exponents whose real partsare the Lyapunov exponents that indicate the stabilityof periodic systems. Based on the Lyapunov exponentsone can decide whether a lossy system is PT -symmetric,or whether that symmetry is broken. A z -independent, PT -symmetric system is expected to have real eigenener-gies. In the PT -broken phase these eigenvalues becomescomplex. In Liouville space, this behaviour is reversedas the imaginary unit from the Schrödinger equation hasbeen absorbed in the Liouvillian. These arguments canbe directly transferred to passive periodic systems mean-ing that, if the Lyapunov exponents only show an over-all loss of the passive system, then PT symmetry is pre-served. In contrast, when the Lyapunov exponents splitfrom the mean losses, PT -symmetry is broken.For the passive Floquet PT coupler we assume the lossin the first waveguide to be a periodic function with pe-riod T = 2 π/ω , and to be of the form γ ( z ) = 2 B exp[ − β (1 − cos ωz )] (cid:112) − B exp[ − β (1 − cos ωz )] . (11)Its maximum and minimum values depend on the param-eters B and β , and we chose the minimum to be γ min ≈ .Its maximum will be denoted by ¯ γ . Inserting the lossrate γ ( z ) and the coupling strength κ into the differen-tial equations for the functions f i and a i , we numericallycompute them up to the period T . Inserted into Eqs. (8)–(10) gives the evolution superoperator U ( T ) , from whichthe Lyapunov exponents are obtained by diagonalisation.As an example, we consider a single photon prop-agating through the two-mode waveguide system. InFig. 1, the resulting PT phase diagram is shown as afunction of the modulation frequency ω and the maxi-mum loss ¯ γ , normalized with respect to the coupling con-stant κ between the waveguides. The PT -broken phaseis shaded in yellow. The diagram shows a clear reduc-tion of the PT -breaking threshold at the resonance fre-quency ω = 2 κ of the lossless coupler, with additionalregions of reduced thresholds for lower modulation fre-quencies. A similar behaviour was also observed in adifferent context [18], thus pointing at a universal be-haviour. Preparing the passive system with a loss modu-lation frequency equal to the resonance frequency mighttherefore enable one to efficiently probe the transitionbetween PT -symmetric and broken phases.Note that the PT phase diagram as calculated in Liou-ville space is identical to the Hilbert space phase diagram . . . ω/κ ¯ γ / κ FIG. 1. PT phase diagram of the passive coupler with mod-ulated loss γ ( z ) from Eq. (11). For Lyapunov exponents thatsplit from the mean loss values, the system is in the PT -broken phase (yellow). calculated from an effective non-Hermitian Hamiltonianof an active two-mode PT system. If this were not true,the passive system would not be viable to simulate the ac-tive system. That this is indeed the case can be deducedfrom the decomposition of the algebra and the subse-quent product form of the evolution superoperators. Thesuperoperators L − i R − j that are responsible for removingphotons, as well as the sum of superoperators responsiblefor the mean loss, are clearly separated from the sl (2 , C ) algebras that describe the underyling active PT coupler.When postselecting on the outcome where no photon islost in transmission, the contributions from L − i R − j canbe dismissed, and the only remaining part is an evolutiongoverned by the effective non-Hermitian Hamiltonian ofthe active PT system plus an overall mean loss. However,this mean loss is now greatly reduced at the resonancefrequency ω = 2 κ compared to the unmodulated casewhere the threshold is ¯ γ = 2 κ . Additionally, because wealready solved for the monodromy, we are also able tocalculate the full quantum evolution over all subspaceswithout the need for postselection.We highlight one example of how the PT phase tran-sition manifests itself by calculating the occupation P ( n, h, z ) of states | n − h, h (cid:105) P ( n, h ; z ) = (cid:104) n − h, h | ˆ ρ ( z ) | n − h, h (cid:105) , (12)over different subspaces with photon numbers n . Startingwith the input state | ψ (cid:105) = ( | , (cid:105) + | , (cid:105) ) / √ , we showin Fig. 2 the evolution of all photon-number subspacesusing a coupling constant κ = 1 . In the left figure, thesystem has a loss amplitude ¯ γ = 0 . κ and a modulationfrequency ω = 1 . κ associated with the PT -symmetricphase (see Fig. 1), whereas in the right figure, the mod-ulation frequency is set to ω = 2 κ associated with the PT -broken phase. This is reflected in the general be-haviour of the occupations P ( n, h, z ) . In the left panel ofFig. 2, there are two oscillating strands that are equally damped by an overall loss. In contrast, the panel on theright shows one strongly damped strand and one with sig-nificantly lower loss. This behaviour is repeated acrossall subspaces except the continuously filled vacuum sub-space (lowest subpanels). Note also that all subspaceswith fewer than 3 photons are only transiently occupied.The qualitative difference of the two evolutions is aclear sign of a PT symmetry breaking where the systemtransitions from a coherent evolution (plus overall lossin the passive scheme) to an evolution that splits intoexponentially decaying and growing modes. The physicalexplanation for this behaviour is that the damping ofthe z -dependent Floquet modes depend on whether ornot they are concentrated in states | n − h, h (cid:105) with morephotons in the lossy waveguide when γ ( z ) is large. Thisis the Floquet analogue of the usual signature of broken PT symmetry of one mode being amplified and the otherone being suppressed.Recall that this PT -symmetry breaking is only initi-ated by a change of the modulation frequency ω , andthat the loss amplitude ¯ γ is held at a low and constantvalue. In the static case, one instead has to change theloss rate to higher values that lead to significantly re-duced visibilities in the measurements. The passive Flo-quet PT coupler is therefore a possible way to probe the PT phase transition without the obstacle of the overallloss. This is especially interesting as the required loss ratemight even be further reduced as seen from the phase di-agram Fig. 1. However, as the range of frequencies, forwhich PT -symmetry is broken, becomes progressivelynarrower with decreasing values of ¯ γ , an experimentalimplementation becomes more challenging.Finally, we present a proposal on how to implementsuch a lossy coupler using auxiliary waveguides. The gen-eral principle is depicted in the lower part of Fig. 3. Thetop pair of waveguides, together with their mutual cou-pling κ , constitute the system under investigation. Thelower waveguide of the pair is additionally coupled to ahomogeneous array of N auxiliary waveguides (the reser-voir) with the coupling κ l , while the coupling inside thereservoir is denoted by κ b . In order to concentrate on theloss implementation, we briefly consider only one activesystem waveguide ( κ = 0 ). In the weak-coupling regimewhere κ l (cid:28) κ b , the population in the system waveguideapproximately shows an exponential decay with rate γ = 2 κ l (cid:112) κ b − κ l (13)after some short initial parabolic decay [26]. For N → ∞ ,the lost population does not return to the system waveg-uide, and hence constitutes a Markovian loss. For finite N , the exponential decay is only a good approximationup to some recurrence time due to reflections at the endof the array, which scales linearly with the array size.However, a sufficient number of auxiliary waveguides iseasily obtainable in experiments [28].A modulated loss can then be implemented by mod-ulating the coupling κ l which, in the evanscent cou- | i| i| i| i | i| i| i | i| i | i κz | i| i| i| i | i| i| i | i| i | i κz FIG. 2. Occupations P ( n, h ; z ) for states | n − h, h (cid:105) with input state | ψ (cid:105) = ( | , (cid:105) + | , (cid:105) ) / √ over all subspaces with photonnumbers n ∈ { , , , } . (Left) PT -symmetric phase ( ¯ γ = 0 . κ , ω = 1 . κ ). (Right) PT -broken phase ( ¯ γ = 0 . κ , ω = 2 κ ). pling of the intregated photonic waveguides has the gen-eral form κ l ( z ) = A exp[ − αd ( z )] , where d ( z ) is the dis-tance between waveguides and A and α are appropri-ate scaling factors. With a modulation function d ( z ) ∝ (1 − cos ( ωz )) , Eq. (13) yields the loss rate in Eq. (11).Note that the modulation has to be sufficiently slow forthe resulting decay to follow an exponential law, i.e. thatit can be described by a rate that yields the correct formof the dissipator of the quantum master equation (1). . κ b z h ˆ n ( z ) i / h ˆ n ( ) i waveguide reservoir ∝ exp( − R γ ( z )d z ) κκ l ( z ) κ b FIG. 3. (Top) Decay of the mean photon number in the sys-tem waveguide by coupling to a modulated reservoir of waveg-uides (blue solid line) and comparison to analytical form ofdecay (red dashed line). (Bottom) Sketch of the modulatedreservoir coupled to system waveguides.
In order to check the validity of our assumptions, wecompared the numerical evaluation of the N + 1 waveg-uide model with the behaviour of a lossy waveguide witha modulated decay rate ( ∝ exp ( − (cid:82) γ ( z )d z ) with γ ( z ) given by Eq. (11). The result is shown in the upper panelin Fig. 3 for ¯ γ = 0 . κ b and ω = κ b . Setting κ = 0 . κ b this corresponds to the example of the PT -broken phasewith ¯ γ = 0 . κ and ω = 2 κ (right panel in Fig. 2). Notethat the initial parabolic decay in the analytical approx-imation (dashed line) was accounted for by appropriatenormalization [26]. The numerical result (solid line) forthe N + 1 waveguide system matches the exponential de-cay with modulated frequency [Eq. (13)] very well. 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