Selective protected state preparation of coupled dissipative quantum emitters

Inherent binary or collective interactions in ensembles of quantum emitters induce a spread in the energy and lifetime of their eigenstates. While this typically causes fast decay and dephasing, in many cases certain special entangled collective states with minimal decay can be found, which possess ideal properties for spectroscopy, precision measurements or information storage. We show that for a specific choice of laser frequency, power and geometry or a suitable configuration of control fields one can efficiently prepare these states. We demonstrate this by studying preparation schemes for strongly subradiant entangled states of a chain of dipole-dipole coupled emitters. The prepared state fidelity and its entanglement depth is further improved via spatial excitation phase engineering or tailored magnetic fields.


I. INTRODUCTION
Ensembles of effective two level quantum emitters consisting of single atoms, ions, or defects in solids are ubiquitously employed in quantum optics and quantum information [1].They are the basis for precision spectroscopy or atomic clock setups, as well as in experiments testing fundamental quantum physics concepts or implementations of the strong coupling cavity QED (quantum electrodynamics) regime [2,3].In the absence of direct particle-particle interactions, larger ensembles fundamentally allow for faster, more precise measurements [4].This scales the effective single photon to matter coupling strength g by a factor N 1/2 (with system size N ) and reduces quantum projection noise (by N −1/2 ) [5,6].
For any precise measurement one has to externally prepare, control and measure the particles dynamics.Hence the spins are almost unavoidably also coupled to their environment and a suitable theoretical framework to model such experiments is open system dynamics via coupling to a fluctuating thermal bath.At optical frequencies this can be often approximated by the zero effective temperature electromagnetic vacuum field [7,8].Often extra perturbations by a thermal environment, background gas collisions cannot be avoided.
For any laboratory experiment the particles need to be confined in a finite spatial volume which can be addressed by laser beams.Increasing particle numbers thus will unavoidably lead to higher densities, where direct particle-particle interactions as well as environmentally induced collective decoherence effects can be no longer neglected.For optical transition frequencies a critical density is conventionally assumed at the point where the average particle separation is of the order of an optical wavelength [9].In this limit vacuum fluctuations at the two positions tend to become uncorrelated and decay becomes independent.However, recent calculations have shown that collective states can exhibit superradiance and subradiance even at much larger distance [10] as long as bandwidth of the emission is small enough.
In many typical configurations and in particular in optical lattices, the particle-particle interaction is dominated by binary dipole-dipole coupling, with the real part inducing energy shifts and its imaginary part being responsible for collective decay [11,12].Generally this interaction is associated with dephasing and decay.However, it has been recently found that in special cases also the opposite can be the case and interaction can lead to synchronization [13] and blockade of decay [14].Often it is assumed that while such states exist, they cannot be prepared by lasers as they are strongly decoupled from radiation fields.However, it was recently proposed that individual instead of overall atom addressing can induce the many particle system to evolve towards decay-or dephasing protected subspaces (exhibiting subradiance) [15].When applied to Ramsey metrology such states are shown to exhibit frequency sensitivities superior even to those obtained with non-interacting ensembles [16].However, apart from special cases of optimal lattice size and excitation angle, it is not so obvious how to implement such precise control.
In this work we exhibit the surprising fact that interaction induced level shifts can be used to help preparing such states.In many cases the magnitude of the shifts a state experiences and its lifetime are tightly connected allowing one to identify and address interesting states via energy resolution.As generic ensemble we particularize on a 1D regular chain of quantum emitters coupled by dipole-dipole interactions, with tunable magnitude (by varying the interparticle separation).Collective coupling to the vacuum modes leads to the occurrence of subradiant as well as super-radiant excitonic states [10].In particular the subradiant states should prove extremely useful for quantum information as well as metrology applications as they exhibit robust, multipartite quantum correlations.As mentioned above interaction provides a first handle for target state selection as it leads to energy resolved collective states.Furthermore, using narrow bandwidth laser excitation matched in both energy and symmetry to the target states allows selective population transfer from the ground state via an effective n degeneracy of a given n-excitation manifold is lifted by the dipole-dipole interactions.The target states are then reached by energy resolution (adjusting the laser frequency) and symmetry (choosing the proper m).d) Scaling of the decay rates of energetically ordered collective states starting from the ground state (state index 1) up to the single and double excitation manifolds for 6 particles at distance a = 0.02 × λ0.The circles identify the decay rates for the lowest energy states in the single (A) and double (B) excitation manifolds.e) Numerical results of target state population evolution for N = 6 and a = 0.02 × λ0 during and after the excitation pulse.Near unity population is achieved for both example states A (where we used η = 0.53 × Γ) and B (for η = 2.44 × Γ) followed by subradiant consequent evolution after pulse time T shown in contrast with the independent decay behaviour at rate Γ (dashed line).
Rabi π-pulse.In many cases, however, the required phase structure of the target state is not compatible with the excitation laser phase so that only very weak coupling can be achieved.Increasing laser power on the other reduces spectral selectivity by unwanted coupling to off resonant but strongly coupled states.Hence to address a larger range of states of practical interest, we also propose and analytically study new methods for phase imprinting by a weak spatial magnetic field gradient.The small relative phase shifts then increase the effective coupling to groups emitters via a nonuniform phase distribution.With this method even fully symmetric sub-radiant states with identical phase get a finite laser coupling to the ground state via the magnetic field induced level shifts resulting in efficient population transfer with minimal perturbation on their lifetime.

II. MODEL
The considered set-up is a chain [see Fig. 1(a)] of N identical two-level systems (TLS) with levels |g and |e separated by an energy of ω 0 (transition wavelength λ 0 ) in a geometry defined by the position vectors {r i } for i = 1, ...N .For each i, operations on the corresponding two-dimensional Hilbert space are written in terms of the Pauli matrices σ x,y,z i and raising/lowering operators σ ± i connected via The complete Hamiltonian describing the coherent dynamics is where Ĥ0 (with = 1) is the free Hamiltonian and has degenerate energy levels (degeneracy C N n = N !/(N − n)!n! for level n) ranging from 0 for the ground state to N ω 0 for the highest excited state.The second term Ĥdip describes interactions between pairs of TLS which can be induced either by an engineered bath (such as a common, fast evolving optical cavity field) or by the inherent electromagnetic vacuum.We denote the couplings between emitters i and j by Ω ij and particularize to the case of a free-space one dimensional chain of TLS with small interparticle distances a such that a λ 0 [as depicted in Fig. 1(a)].For sake of simplicity, we use dipole moments perpendicular to the chain for all numerical computations.To a good approximation, in the limit of k 0 a 1, the nearest neighbor (NN) assumption can be used (such that Ω ij = Ωδ ij±1 ) and exact solutions in the single-excitation manifold can be found [17].For an index m running from 1 to N , dipole-induced shifts are given by m = 2Ω cos [πm/(N + 1)] and the corresponding eigenvectors are where and we used |G = |g ⊗N .Spontaneous decay via coupling to the free radiation modes in evolution of the system can be included in a generalized Lindblad form [8].
where the γ ij denote collective damping rates arising from the coupling to a common radiation modes.These rates also strongly depend on the atomic distances a which two prominent limiting cases of γ ij (a → ∞) = Γδ ij (independent emitters limit) and γ ij (a → 0) = Γ (the Dicke limit [18]).In general one can perform a transformation of the Liouvillian into a new basis by diagonalizing the γ ij matrix.This procedure leads to a decomposition into N independent decay channels with both superradiant (> Γ) and subradiant (robust) decay rates (< Γ) [16].Note, however, that these states generally do not coincide with energy eigenstates of the Hamiltonian, so that we cannot reduce the whole decay dynamics to simple rate equations.

III. SELECTIVE STATE PREPARATION
Tailored coherent excitation.As mentioned above our dipole coupled systems possesses states with a large variation of radiative lifetimes and energy shifts.Depending on the desired application particular states can be highly preferable to other.In a first straightforward approach we now illustrate that in principle it is possible to access a desired state simply via selective coherent driving with properly chosen amplitude and phase for each state.This is described by the Hamiltonian with suitably chosen η j for each desired state.For a targeted eigenstate in the single-excitation manifold, some analytical insight how to choose these amplitudes can be gathered from the state's symmetry.For energy eigenstates this can be quite reliably found within the nearest neighbor approximation [19].For an equidistant finite chain this suggests the following choice of driving fields at laser frequency ω l : η m j = η sin [πmj/(N + 1)] chosen to fit the symmetry of a target state m.
The selectivity of the excitation process can be further improved via energetic resolved excitation of a given state m by proper choice of the laser frequency ω l = ω 0 + m and bandwidth.This is possible due to the interaction induced level splitting from Ĥdip [as depicted in Fig. 1(c)].Indeed, in perturbation theory and in a frame rotating at ω l the evolution of the system starting from ground state approximately up to a normalization factor leads to The success of the corresponding process is illustrated in the sequence of plots in Fig. 1, where the m = N state with n = 1 is considered (target state A) and accessed via the combination η N j of pumps lasting for a duration T .
Numerical simulations were performed on a six-atom chain driving strength η = 0.53 × Γ at an atomic separation of a = 0.02 × λ 0 .The time in which the pumps are switched on is T = 1.58 × Γ −1 which is vastly shorter that the time scale governed by the decay rate of the target state, Γ A = 0.0009 × Γ.The resulting dynamics is an effective π-pulse (efficiency of 99.94%) flipping the population into state |m = N followed by slow decay, signalling the robustness of the target state [as seen in the curve A of Fig. 1(e)].
It is of course often of interest to target higher excitation manifolds.In the absence of analytical expressions or good approximations for the target states, we employ phases that yield maximal asymmetry, i.e. ηj = η(−1) j for any j = 1, ..N .Such a driving configuration can be expected to lead address collective states, where the fields emitted by any two neighboring particles interfere destructively [14] (similar to the mechanism employed in [15]).Numerical simulations show that the resulting collective states indeed exhibit the lowest energy shifts of the targeted manifold and can be expected to be long lived.The resonance condition for a specific state |ψ within the manifold n is nω l = nω 0 +δω ψ , where δω ψ = ψ| Ĥdip |ψ .As an illustration, the curve B in Fig. 1(e) shows an almost unit-efficiency (98.36%) two-photon π-pulse allowing the reach of the longest living collective state in the second excitation manifold of N = 6 emitters separated by a = 0.02λ.The chain was driven with a strength of η = 2.44 × Γ for a time T = 3.44 × Γ, which again is significantly shorter than the natural time scale given by the target state decay rate Γ B = 0.0402 × Γ.
Let us add some first comment on the practical implementation of such addressing.In typical current experimental configurations for clocks based on 1D magic wavelength lattices [20,21] the atoms are very close and hardly allow individual direct particle addressing.One is largely limited by a quasi plane wave driving, which typically addresses all particles with equal intensity.If the pump light is applied perpendicular to the trap, the evolution is governed by a symmetric Hamiltonian Ĥsym , obtained from Eq. ( 5) for equal pump amplitude η m j = η for any m and j.Laser excitation from the ground state into state |m is connected to the coupling amplitude We will then refer to states with even m as dark states as they cannot be accessed by the laser excitation and bright otherwise [14].In the limit of large atom numbers N 1, it is of interest to investigate the two cases, where m N and m ∼ N , for states at the top/bottom of the manifold.In the first case, the function for the driving yields χ m ≈ η √ 8N /mπ, whereas in the other case we have χ m ≈ 0.
Note that sometimes geometry can change this behaviour.For a 1D string of equidistant particles illumination at a chosen angle of incidence and polarization leads to a designable phase gradient of the excitation amplitudes.The situation gets even more complex for a 3D cubic lattice, where the phases also differ in the different lattice planes.As a lucky coincidence, a perpendicular plane illumination at the clock frequency in a magic lattice for Strontium targets an almost dark state.This leads to subradiance and in priciple allows a spectral resolution better than the natural linewidth [22].
In not so favorable cases one could also think of specific lattice design to facilitate tailored dark state excitation.
Radiative properties -To be useful resources for quantum information applications, target states should exhibit robustness with respect to the environmental decoherence.To identify states of minimum decay rate, we scan through the eigenstates |ψ k of the Hamiltonian H = H 0 + H dip (for k = 1, ..., 2 N ) and compute their decay rates; to this end we consider the homogeneous part of the differential equation for the state population, derived from the master equation: We find that generally, for a given manifold, the energetic ranking of the states closely indicates their robustness to decay [as illustrated by the color-coding in Fig. 1(c)] ranging from blue for subradiant states to red for superradiant states.This is due to the fact that both radiation and energetic shifts are strongly dependent on the symmetry of the states.In Fig. 1(d), for N = 6, we plot the decay rates of states in the first (n = 1) and second (n = 2) excitation manifold arranged as a function of their increasing energy [corresponding to the level structure of Fig. 1(c)].Superradiant states are found at the upper sides of the manifolds while the ideal robust states lie at the bottom of the manifolds.In Fig. 1(d), the circles single out the optimal decay rates in the single-(Γ A = 0.0009 × Γ) and double-excitation manifolds (Γ B = 0.0402 × Γ) corresponding to target states A and B whose population evolution is analyzed in Fig. 1(e).Within the single-excitation manifold, an analytical expression for the decay rate of a state |m can be found as Γ m = i,j γ ij f m i f m j .For small distances the state m = 1 (upper state) is superradiant, whereas states at the bottom of the manifold m ∼ N show subradiant properties.In the Dicke limit where a = 0 we have γ ij = Γ for any i and j, and we can compute Γ m = 2Γ cot 2 [mπ/(2N + 2)]/(N + 1) for m odd and Γ m = 0 for m even.Note that in this particular limit, these are just the same conditions as for the darkness and brightness of a state.For large numbers of emitters, we recover the expected superradiant scaling with N for the state with m = 1: Γ 1 ≈ 8ΓN/π 2 .On the other hand, large m yield a decay rate of Γ m ≈ 0 (perfect subradiance) in the same limit.From these results there are two important conclusions: i) since in the considered limit the decay rate of the superradiant state |m = 1 scales with Γ 1 ∝ N , whereas its driving is χ 1 ∝ √ N , driving this state becomes more difficult with increasing atom number due to the reduced time-scale and ii) if the number of atoms is not too large, χ m will remain finite, while Γ m already indicates vast subradiance due to its scaling down with N .Hence, there are robust states that remain bright, i.e. they can be driven directly even though the driving is not matched to their symmetry.

IV. ACCESSING DARK STATES VIA MAGNETIC FIELD GRADIENTS
The direct symmetric driving with H sym instead gives access to bright states only.Given that nearby dark states can conceivably be more robust, we are now employing a progressive level shifting mechanism that allows coupling bright and dark states.This is achieved by subjecting the ensemble to a magnetic field with a positive gradient along the chain direction.The increasing energy shifting of the upper atomic levels [as depicted in Fig. 2(a)] plays a role similar to the individual phase imprinting mechanism previously described: for each particle the shift of the excited level induces a time-dependent phase proportional to the value of the magnetic field at its position.We reveal the mechanism for the particular two-atom example where indirect near unity access to the dark subradiant asymmetric collective state is proven and extend it to the single-excitation manifold of N atoms.2b)), where where ∆ B is tunable and quantifies the per-emitter shift for a given magnetic field amplitude.We first analyze the dynamics in the absence of decay by solving the time-dependent Schrödinger equation under the Hamiltonian H = H 0 + H dip + H sym + H B , where We reduce the dynamics to 3 states, and assume a quasi-resonant Raman-like scheme where the population of |E is at all times negligible.In the collective basis, where , and i ċG = ηc S , where Ω = Ω 12 is the coherent interaction between the FIG. 2. Coupling to dark states via magnetic field gradient.a) Linearly increasing level shifts along the chain occur in the presence of the magnetic field gradient.b) Illustration of level structure and indirect dark-state access for two coupled emitters.While symmetry selects the |S state, off-resonant addressing combined with bright-dark state coupling of strength ∆B allows for near-unity population transfer into state |A .c) Dynamics in the single-excitation manifold of N coupled emitters when symmetric driving reaches the bright states with amplitudes χm while the magnetic field couples neighbouring dark and bright states.d) Plot of asymmetric state population for the two-atom case as a function of increasing magnetic field compared to the steady-state approximation (dashed) at numerically optimized time T = 16.19/Γ, with parameters η = Γ and a = 0.05 × λ0.e) For a chain of N = 4 emitters, an almost perfect π-pulse to the most robust state can be achieved as evidenced in the population evolution plot.The separation is chosen as a = 0.025 × λ0 while η = 8Γ and numerical optimization is used to find ∆B = 4.923 × Γ. atoms and ∆ is the detuning between the atomic resonance frequency and the driving laser.For efficient driving of |A the population of state |S needs to be negligible which allows us to set steady-state conditions ċS = 0.An effective two-level system arises (between the ground state and the asymmetric state) and the resonance condition can be identified as ∆ (2) 1, we need to restrict the driving to a parameter regime where η, ∆ B Ω. A scan over the magnetic field is performed and the exact numerical results for the asymmetric state population are plotted in Fig. 2(d) against the adiabatic solution showing near unity population transfer for optimized ∆ B .Further restrictions are added when decay is considered.These stem from the consideration that the coherent process described by ν R should be faster than the incoherent one characterized by γ A .For close particles, the tunability of the distance ensures that the scaling down of γ A is very fast and the above conditions are readily fulfilled.For the particular example illustrated in Fig. 2(d) we took a = 0.05 × λ 0 , resulting in Ω = 23.08 × Γ, γ A = 0.019 × Γ.The 0.994 population is reached in T = 16.19/Γ, which is very close to the theoretical estimate of T = π/2ν (2) R = 16.179/Γobtained from the adiabatic solution under the conditions of a π-pulse transfer of population to the target state.
Many-atom case -For a chain of N atoms, we consider the progressive shifting of excited levels along the chain depicted in Fig. 2(a); this is realized by the application of a magnetic field with a constant gradient and is described by the following Hamiltonian and was obtained in the limit where the coupling of the dark state to the other adjacent bright state |d + 1 was neglected owing to the relation χ d−1 χ d+1 .The effective transition strength between the ground state and state |d is The addition of decay imposes a new constraint on the timescale of the process ν (N ) R γ d necessary to ensure near unity population in the dark state.The fulfillment of this condition depends on the individual system considered.As an illustration of the procedure, Fig. 2(e) shows the targeting of a robust dark state in the single excitation manifold of 4 particles.Note that the numerical results are performed in an exact regime beyond the NN approximation and show excellent agreement with conclusions obtained from the NN treatment.2e) and as an indicator for the efficiency of the driving scheme with a magnetic field gradient the corresponding populations were plotted as data point (blue circle), clearly showing more than two-atom entanglement.

V. DISCUSSIONS
Entanglement properties -To justify the usefulness of collective states for quantum information purposes, we make use of the von Neumann entropy to analyze their entanglement properties.More specifically, we compute the von Neumann entropy of the reduced density matrix ρ at of a single two-level atom (showing the degree of its bipartite entanglement with the rest of the system) defined by S(ρ at ) = − i λ i log 2 λ i , where λ i is the i-th eigenvalue of ρ at and 0 log 2 0 ≡ 0. We furthermore minimize the set of values for all atoms to obtain a lower bound on the entanglement contained in the system.We the compare numerical results with single-atom entropy of the symmetric Dicke state |−N/2, N/2 + n [18].For these particular states the entropy can be obtained in analytical form S(ρ It follows that it is highly desirable to drive the system into robust states as close as possible to n = N/2 excitations (where N/2 is the largest integer smaller or equal to N/2), since this manifold holds the highest entangled state.A comparison of the exact numerical data and the analytical expression for the entropy is shown in Fig. 3(a).
Another way to characterize the entanglement of the prepared state is to investigate their depth of entanglement [23] [24], which does not quantify the entanglement itself but rather shows how many atoms of an ensemble are involved in the present entanglement.This measure has been used in recent experiments [24,25] since it is a readily measurable quantity.This depth of entanglement is computed as follows: given a target state that shows N -atom entanglement, we compute the limit of how much population one can drive into this state such that the resulting density matrix ρ (in our case a superposition of the ground and target state) remains separable into a subset of density matrices that exhibit no more than k-atom entanglement, i.e. ρ = i ρ ki i with k i ≤ k and at least one k i = k.In order to compute the boundaries of separability (k = 1) and two-atom entanglement (k = 2), we generalized the maximum-likelihood algorithm from [24] (where it was used in the case of preparation of a w-state) to an arbitrary state in the single-excitation manifold of four atoms.
Obviously, for the pure target states considered in the above computation all atoms contribute to the entanglement, since otherwise the minimal von Neumann entropy as shown in Fig. 3(a) would be zero.For a more interesting result, we can compute the depth of entanglement in order to demonstrate the efficiency of the driving procedure using a magnetic field gradient as shown in Fig. 2(e).As Fig. 3(b) shows, the resulting state (blue dot) lies vastly above the boundaries for k = 1, 2 indicating that at least three atoms are entangled.

VI. CONCLUSIONS
While decay is typically regarded as detrimental to most applications in many-particle quantum systems, we focus here on examples where the collective nature of the decoherence process combined with the inherent coherent binary or multiple dipole-dipole interactions allows for the controlled selective preparation of target states exhibiting both entanglement as well as robustness.We particularized on a one-dimensional system of tightly-packed equidistant quantum emitters in a naturally occurring coupling environment; a state preparation technique has been exemplified via the continuous application of a spatially varying magnetic field.The general phase imprinting technique can be potentially applicable to artificial environments such as atoms/ions pinned within one or more evolving/decaying common optical cavity modes [26,27], NV-centers or superconducting qubits coupled to CPW transmission lines or resonantors [28,29].

FIG. 1 .
FIG. 1. Selective state preparation procedure.a) A chain of N closely spaced quantum emitters (separation a with ka 1, k being the laser wave-vector) are individually driven with a set of pumps {η m j }. b) The lasers are turned on for time T , optimized such that an effective π-pulse into the desired subradiant target state is achieved.c) Level structure for the Nsystems where the C Nn degeneracy of a given n-excitation manifold is lifted by the dipole-dipole interactions.The target states are then reached by energy resolution (adjusting the laser frequency) and symmetry (choosing the proper m).d) Scaling of the decay rates of energetically ordered collective states starting from the ground state (state index 1) up to the single and double excitation manifolds for 6 particles at distance a = 0.02 × λ0.The circles identify the decay rates for the lowest energy states in the single (A) and double (B) excitation manifolds.e) Numerical results of target state population evolution for N = 6 and a = 0.02 × λ0 during and after the excitation pulse.Near unity population is achieved for both example states A (where we used η = 0.53 × Γ) and B (for η = 2.44 × Γ) followed by subradiant consequent evolution after pulse time T shown in contrast with the independent decay behaviour at rate Γ (dashed line).

2 .
Two-atom case -The eigenstates of the Hamiltonian H 0 + H dip are |E = |ee , |G = |gg , and in the single-excitation subspace |S = (|eg + |ge )/ √ 2 and |A = (|eg − |ge )/ √ The symmetric state |S is superradiant (γ S = Γ 1 = Γ + γ 12 ) and bright, directly accessible via symmetric driving with strength χ 1 = √ 2η.The asymmetric state |A , on the other hand, is subradiant (γ A = Γ − γ 12 ) and dark.Indirect access can be gained by shifting the second atom's excited state by 2∆ B (see schematics in Fig. us consider a dark state |d (d even) and the immediate upper bright state |b = d − 1 ; their coupling via H B is quantified by ∆ db = 2∆ B i (i − 1)f d i f b i , as shown in Fig. 2(c).We develop a protocol where direct off-resonant driving into the bright state (amplitude χ b ) combined with coupling between the bright and dark states via the magnetic field leads to close to unity population transfer into the dark state.Given a sufficient energy separation, the problem can be reduced to solving the time-dependent Schrödinger equation for the three coupled state amplitudes c b , c d and c G .Following the same adiabatic procedure as in the two-atom case we reduce the general dynamics i ċb = [∆ + b + ∆ B (N − 1)] c b + ∆ db c d + χ b c G , i ċd = [∆ + d + ∆ B (N − 1)] c d + ∆ db c b and i ċG = χ b c b to an effective two-level system between the states meant to be connected with an effective π-pulse: |d and |G .The generalized resonance condition (with db = d − b ) reads ∆

FIG. 3 .
FIG. 3. Entanglement properties.a) Comparison of the numerically computed entropy (black squares) of the reduced density matrix of the chain minimized over the atom index and the analytical expression for the entropy of the Dicke state (green circles), both for excitations n = 1 and n = N/2 as a function of the atom number N .b) Boundaries where k = 1, 2 atoms are entangled, numerically computed by maximizing the target state population Pt as a function of the ground state population Pg.The computation was done in the single-excitation manifold of a N = 4 atom chain separated by a = 0.025 × λ0 (see Fig.2e) and as an indicator for the efficiency of the driving scheme with a magnetic field gradient the corresponding populations were plotted as data point (blue circle), clearly showing more than two-atom entanglement.