Dynamics of mode entanglement in a system of cavities coupled with a chiral mirror
aa r X i v : . [ qu a n t - ph ] A ug Dynamics of mode entanglement in a system of cavitiescoupled with a chiral mirror
Ali ¨U. C. Hardal ∗ Department of Physics, Ko¸c University, ˙Istanbul, 34450, Turkey andDepartment of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
We investigate the Hermitian and the non-Hermitian dynamics of the mode entanglement in twoidentical optical cavities coupled by a chiral mirror. By employing the non-Hermitian quantumevolution, we calculate the logarithmic negativity measure of entanglement for initially Fock, coher-ent and squeezed states, separately. We verify the non-conservation of mean spin for the initiallycoherent and squeezed states when the coupling is non-reciprocal and report the associated spinnoise for each case. We examine the effects of non-conserved symmetries on the mode correlationsand determine the degree of non-reciprocal coupling to establish robust quantum entanglement.
PACS numbers: 03.65.-w, 03.65.Ud, 42.50.Pq
I. INTRODUCTION
The interest in the systems which exhibits non-Hermitian quantum mechanical interactions [1–5] hasbeen intensified particularly in the last decade. Theyhave been reported in many research fields includingsoliton-plasmon systems [6–8], hybridized metamateri-als [9], coupled microcavities [10], waveguides [11, 12], op-tical lattices [13, 14] and Bose-Einstein condensates [15–17]. PT -symmetric lasers and anti-lasers [18–21], cloak-ing devices [22] and unidirectional invisibility [23, 24] con-stitute some of the intriguing implementations of suchquantum mechanical systems.Along a similar direction, a generic quantum opti-cal model of coupled resonators which exhibits non-Hermiticity has been proposed very recently [25]. In themodel, two independent quantum oscillators are weaklycoupled with a chiral mirror. The dynamical analysis bythe usual Heisenberg approach has revealed the asym-metric photon exchange between the resonators. In ad-dition, the non-conservation of the total photon numberwas reported for the same model by employing the non-Hermitian quantum evolution [6]. The non-conservationof the mean number of photons is an interplay betweenthe quantum coherence and the non-Hermitian dynamics.Here, we aim to reveal whether there is relation betweenthe mean spin and the entanglement dynamics as well.In the present contribution, we consider the modelsystem proposed in Ref. [25]. We investigate the non-Hermitian dynamics of the mode entanglement by themeans of the logarithmic negativity measure [26]. Modeentanglement occurs in the second quantization pic-ture [27, 28] and can be witnessed via the covariancesof the two distinct modes [29, 30]. Furthermore, therelations between mode correlations and the spin noisein coupled cavity systems have been revealed only re-cently [31]. The absence of bipartite mode entanglement ∗ Electronic address: [email protected] due to the lack of nonlinearity in the model system un-der consideration has also been verified [6]. Here, we re-port the existence of genuine mode entanglement in thegeneric model that are robust and controllable via theasymmetry in the coupling of the two modes.In our numerical analysis, we assume that one cav-ity is in its vacuum while the other in a Fock, coherentand squeezed state, separately. The non-conservation ofthe mean spin is verified for the initially coherent andsqueezed states when the coupling between the cavitiesis non-reciprocal. The associated spin noise measured bythe variances of the corresponding spin operators of thecoupled resonators is reported. We find that the modeentanglement is more robust if the system is in a coherentstate and non-Hermitian, though it displays high ampli-tude oscillations in comply with the noise dynamics. We,then, consider an initially single-mode squeezed state tocompensate spin noise and clarify the interference be-tween population, spin and entanglement dynamics.In a recent contribution [31], we investigated a moregeneral set up consists of two nonlinear cavities coupledeither with single- or two-photon exchange interactions.Quantum entanglement and field coherence were investi-gated in the steady state in a comparative manner. Thefocus of the work was to reveal profound relations be-tween coherence, localization (delocalization) of photonsand quantum correlations. Here, we consider a more fun-damental model which exhibits non-Hermitian dynamics.Our motivation is to dynamically investigate the modalentanglement and its response to the broken symmetriesdue to the asymmetric coupling between the cavities.This paper is organized as follows. In Sec. II, webriefly review the model system and the governing non-Hermitian quantum dynamics. In Sec. III, we presentour results and we finally conclude in Sec. IV.
II. THE MODEL SYSTEM AND THENON-HERMITIAN DYNAMICS
We consider two identical optical cavities, A and B ,which are coupled by a chiral mirror. The dynamics ofthe system is governed by the Hamiltonian [25] H = ω ( a † a + b † b ) + g AB ab † + g BA a † b. (1)Here, a and b are the annihilation operators of the cavitymodes, ω is the resonant transition frequency for eachcavity and g AB and g BA denote the coupling strengths.The Hamiltonian (1) can equivalently be written as H = ω N + g AB S + + g BA S − , (2)where we made use of the pseudo-spin operators for thetwo-resonator system S x := 12 ( a † b + b † a ) ,S y := − i a † b − b † a ) , (3) S z := 12 ( a † a − b † b ) , with S + := S x + iS y , S − := S x − iS y and N = a † a + b † b .The operators given in Eq. (3) satisfy the usual spin al-gebra [ S α , S β ] = ǫ αβγ S γ with α, β, γ ∈ x, y, z and ǫ αβγ is Levi-Civita tensor. In the case that g AB = g BA ,the model describes a reciprocal, single-photon exchangetype coupling between two resonant cavities. The lattertype of coupling generally induces genuine mode correla-tions which can be expressed with the covariances of thetwo modes [29, 30] and can further be related to the spinnoise [31].When g AB = g BA the system becomes non-Hermitianeven if the coupling coefficients are real as it can easilybe seen from Eq. (2). The asymmetric coupling betweenthe cavities behaves as a dissipation or an amplificationchannel depending on which direction that the symme-try is broken, as a result the system does not conservethe mean number of photons h N i [6]. However, the non-conservation of the mean number of excitations do re-sult from the applied dynamical approach as well as theinitial preparation of the system [6, 25, 32]. Here, weadopt the approach for which the total number of pho-tons h N i is not conserved. We also verify that if theinitial state of the system is a coherent or a squeezedstate, the non-reciprocal dynamics does not conserve themean spin h S i = h S x + S y + S z i as well.The dynamics of the system may be investigated withthe usual Heisenberg approach [25], however it has beenrecently shown [6] that to capture the effects of non-reciprocal dynamics one should consider a more generalformalism [16, 33–35]. To that end, we first write theHamiltonian (1) as the sum of its Hermitian H + andanti-Hermitian H − parts H = H + + H − , (4) where H ± := 1 / H ± H † ) with H ± = ± H †± . The timeevolution of a state ρ ( t ) of the system can be deter-mined by the modified Liouville-von Neumann masterequation [6] ∂∂t ρ ( t ) = − i [ H + , ρ ( t )] + − i [ H − , ρ ( t )] − , (5)where [ , ] + and [ , ] − represent the commutator and theanti-commutator of the corresponding operators. Due tothe non-unitary character of the Eq. (5), we renormalizethe density operator as ρ ( t ) ′ := ρ ( t ) T r ( ρ ( t )) . (6)It follows that the expectation value of a given observable Q is calculated via the relation h Q i := T r ( ρ ( t ) Q ) T r ( ρ ( t )) . (7)In the following section, we shall first define the measureof quantum entanglement and the parameters that aregoing to be used in our analysis. We, then, present ourresults for initially fock, coherent and squeezed states,separately. III. RESULTS AND DISCUSSIONS
Here, we shall discuss the non-Hermitian quantum dy-namics of mode entanglement between the two cavitymodes. In our numerical analysis, we use the QuTiP:Quantum Toolbox in Python software [36]. We set theHilbert space dimensions of the modes N A = N B = N =25 which we concluded that is sufficient for the analysisof quantum entanglement. We repeated our calculationsup to N = 30 and obtained the same results. In par-ticular, for the dimensions N <
15, we found that theresults are not stable. We note that the latter bounds onthe Hilbert space dimensions are not physical and can bediffer with respect to the preferred numerical algorithmand method.We make our calculations for g AB = g.r , g BA = g with r = 0 . , , r = 1 corresponds to the Hermitianwhereas r = 0 . , E N ( ρ ) to makequantitative discussions on mode entanglement. The log-arithmic negativity is a computable and a non-convexentanglement monotone and it is defined as [26] E N ( ρ ) := log || ρ T A || , (8)where ρ T A stands for the partial transpose with respectto the first subsystem and || ρ T A || is the trace norm of ρ T A .One important property of the logarithmic negativity isthat it does not reduce to the von Neumann entanglemententropy for pure states. It follows that it can detect and ω t ( ∆ S y ) r = 1r = 0.5r = 2 (a) ω t ( ∆ S z ) r = 1r = 0.5r = 2 (b) ω t E N ( ρ ) r = 1r = 0.5r = 2 (c) FIG. 1: Dependence of (a) (∆ S y ) , (b) (∆ S z ) and (c) E N ( ρ ) for r = 1 (black-solid), r = 0 . r = 2(blue-dot-dashed) with respect to the scaled time ω t for an initially Fock state | ψ (0) i = | i| i . ω t h S i r = 1r = 0.5r = 2 (a) ω t ( ∆ S z ) r = 1r = 0.5r = 2 (b) ω t E N ( ρ ) r = 1r = 0.5r = 2 (c) FIG. 2: Dependence of (a) h S i , (b) (∆ S z ) and (c) E N ( ρ ) for r = 1 (black-solid), r = 0 . r = 2 (blue-dot-dashed) with respect to the scaled time ω t for an initially coherent state | ψ (0) i = | α i| i with α = 1. measure mode correlations which are not bipartite. In-deed, the absence of bipartite entanglement between cav-ity modes for Hermitian as well as non-Hermitian caseshas been reported [6].There are subtle relations between quantum coher-ence, correlations, photon localization and delocaliza-tion [31, 37–39]. If the initial state of the system is acoherent one, then such an interference between non-conservation of the mean number of photons and non-Hermitian dynamics has been also verified [6]. Here, weshall discuss whether there is an interplay between non-Hermitian dynamics, spin conservation with the associ-ated spin noise and mode correlations as well. A. Initially Fock state
We first consider an initial state in which the cavity A is in a Fock state with a single photon whereas the cavity B is in its vacuum | ψ (0) i = | i| i . (9)The mean h N i is conserved in both Hermitian and non-Hermitian dynamics [6]. We numerically verified that themean of the total spin operator h S i = h S x + S y + S z i is also conserved. Therefore, we can discriminate theeffects of non-Hermitian dynamics on the mode correla-tions with conserved symmetries.In Figs. 1(a)-1(c) we plot the dynamics of the variances(∆ S y ) , (∆ S z ) and the logarithmic negativity E N ( ρ )with respect to the scaled time ω t , respectively. Wecalculated that (∆ S x ) = 0 .
25 for Hermitian as well as ω t h S i r = 1r = 0.5r = 2 (a) ω t ( ∆ S y ) r = 1r = 0.5r = 2 (b) ω t ( ∆ S z ) r = 1r = 0.5r = 2 (c) FIG. 3: Dependence of (a) h S i , (b) (∆ S y ) and (c) (∆ S z ) for r = 1 (black-solid), r = 0 . r = 2 (blue-dot-dashed) with respect to the scaled time ω t for an initially squeezed state | ψ (0) i = | α, ǫ i| i with α = 1, ǫ = 0 . ω t E N ( ρ ) r = 1r = 0.5r = 2 FIG. 4: Dependence of E N ( ρ ) for r = 1 (black-solid), r = 0 . r = 2 (blue-dot-dashed) with respect tothe scaled time ω t for an initially squeezed state | ψ (0) i = | α, ǫ i| i with α = 1, ǫ = 0 . non-Hermitian cases. Fig. 1(a) depicts the dynamics of(∆ S y ) . When g AB > g BA , the photon excitation rate inthe empty cavity is faster than the case of g AB < g BA ,for which the period of the oscillations is bigger than thatof the Hermitian case g AB = g BA . The variance (∆ S z ) behaves similarly as shown in Fig. 1(b).Fig. 1(c) shows the dynamics of the logarithmic neg-ativity with respect to the scaled time ω t . The modeentanglement oscillates between near death E N ( ρ ) ∼ E N ( ρ ) ∼ h S z i oscillates around zero with equal amplitudes andnever collapses due to the lack of nonlinearity in the sys-tem. As expected, these oscillations have residual effectson the dynamics of spin noise as well as the mode entan-glement [40] as reported in Fig. 1. The large-amplitudeoscillations have the period T which is inversely propor-tional to the non-Hermiticity parameter r , i.e., T ∝ /r which shall later be inherited by initially coherent andsqueezed state cases as well. B. Initially coherent state
Next, we consider an initial state in which the cavity A is in a coherent state with an amplitude of α = 1 andthe cavity B is in its vacuum | ψ (0) i = | α i| i . (10)If g AB = g BA , the mean number of photons h N i is notconserved for an initially coherent state [6].In Figs. 2(a)-2(c), we plot the dynamics of the meanof the total spin h S i , the variance (∆ S z ) and the loga-rithmic negativity E N ( ρ ) with respect to the scaled time ω t , respectively. The variances (∆ S x ) and (∆ S y ) showidentical behaviour to that of (∆ S z ) and thus are notpresented here. Figure 2(a) shows the Hermitian dynam-ics conserves the mean spin h S i . If g AB > g BA , thedeviation from the steady value h S i = 1 is greater inaccordance with the photon number dynamics [6]. Theassociated spin noise behaves similarly and makes negli-gible oscillations around (∆ S z ) ∼ .
25 if g AB = g BA asit is expected from a coherent state.Figure 2(c) shows if g AB = g BA , the mode entangle-ment first increase and then starts to oscillate with lowamplitudes around E N ( ρ ) ∼ .
6. The coherent trappingof mode entanglement in the JO regime has been also re-ported for the two-mode BECs [41]. The non-Hermitianinteractions in the cases g AB > g BA and g AB < g BA am-plify the mode correlations and have constructive effectsin this regard. The number of photons created in theempty cavity differs by the chosen asymmetry in the cou-pling strengths. On the other hand, the amplification ofthe mode correlations is mainly a reaction to the brokensymmetries as the value of E N ( ρ ) is greater than the Her-mitian case and almost equal to each other if g AB < g BA or g AB > g BA . C. Initially squeezed state
We consider an initially squeezed state of the form | ψ (0) i = | α, ǫ i| i , (11)where α = 1 is the coherent state amplitude and ǫ = 0 . h N i as well as the number of photons in eachcavity shows similar behaviours under Hermitian andnon-Hermitian dynamics as reported for initially coher-ent state [6]. The only difference is that for a squeezedstate we have h N i = | α | + sinh ǫ. (12)In Figs. 3(a)-3(c), we plot the dynamics of the mean ofthe total spin h S i and the variances (∆ S y ) , (∆ S z ) with respect to the scaled time ω t , respectively. Thevariance (∆ S x ) shows identical behaviour as in Fig. 2(b)with relatively small amplitudes due to the squeezing andis not presented here. Figure 3 shows that the squeezingleads to the reduction of quantum fluctuations in meanspin h S i as well as in the variances (∆ S y ) and (∆ S z ) .Figures 3(b) and 3(c) depicts that if g AB < g BA ,for which the empty cavity is weakly excited [6], smallplateaus occur where the spin noise is stabilized. If g AB > g BA , the empty cavity is strongly exited. In thatcase, single mode squeezing is not enough to create timeintervals in which the noise is rather steady, though itreduces the amplitudes of the fluctuations.In Fig. 4, we plot the dynamics of logarithmic negativ-ity with respect to the scaled time ω t . If g AB = g BA ,mode entanglement resembles the dynamics as in the ini-tially coherent state, however it oscillates with relatively higher amplitudes in comply with the spin noise dynam-ics. If g AB > g BA , mode entanglement is blighted by thesqueezing. The high amplitude oscillations persist andthe maximum value of the logarithmic negativity E N ( ρ )shrinks in comparison with that of the cases of initiallycoherent and Fock states. Squeezing serves well to thecause in the case of g AB < g BA . The amplitude of theoscillations scale down to a pliable level and the coher-ent entanglement trapping is achieved as in the case of g AB = g BA , though the degree of entanglement reduces. IV. CONCLUSIONS
In summary, we studied the Hermitian and the non-Hermitian dynamics of the mode entanglement in a sys-tem of cavities coupled with a chiral mirror. The modeentanglement, characterized by the logarithmic negativ-ity measure, was investigated for initially Fock, coherentand squeezed states.For an initially Fock state both the total number ofphotons [6] and the mean of the total spin are conservedregardless of the type of the dynamics. The single photonexchange is a delocalizing and mode correlating interac-tion [31]. As a result, the period of oscillations in the timeevolution of the mode entanglement mimics that of thephoton exchange but keeps the degree of entanglementconstant. The former is also inherited by the initiallycoherent and squeezed state cases as well.The interplay between coherence, correlations and thenon-conservation of mean spin as well as mean numberof photons is revealed in the case of an initially coherentstate. The degree of mode entanglement is amplified ifthe coupling between the two cavity is non-reciprocal.The amplification is nearly equal whether g AB > g BA or g AB < g BA whereas the number of photons are quitedifferent depending on the asymmetries.Lastly, we considered an initially squeezed state to di-minish the amplitudes of the oscillations in the dynamicsof the mode entanglement. We found that if the emptycavity is weakly excited squeezing leads to the desiredreduction with the expense in the magnitude of the en-tanglement.Our results demonstrate that the non-reciprocal ex-change interactions may be used to ensure an effectivecontrol over the dynamics as well as the degree of thequantum entanglement which could be desirable from theperspective of quantum information technologies. Acknowledgments
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