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Phase-imprinted multiphoton subradiant states

We propose to generate the multiphoton subradiant states and investigate their fluorescences in an array of two-level atoms. These multiphoton states are created initially from the timed-Dicke states. Then we can use either a Zeeman or Stark field gradient pulse to imprint linearly increasing phases on the atoms, and this phase-imprinting process unitarily evolves the system to the multiphoton subradiant states. The fluorescence engages a long-range dipole-dipole interaction which originates from a system-reservoir coupling in the dissipation. We locate some of the subradiant multiphoton states from the eigenmodes, and show that an optically thick atomic array is best for the preparation of the state with the most reduced decay rate. This phase-imprinting process enables quantum state engineering of the multiphoton subradiant states, and realizes a potential quantum storage of the photonic qubits in the two-level atoms.

We propose to generate the multiphoton subradiant states and investigate their fluorescences in an array of two-level atoms. These multiphoton states are created initially from the timed-Dicke states. Then we can use either a Zeeman or Stark field gradient pulse to imprint linearly increasing phases on the atoms, and this phaseimprinting process unitarily evolves the system to the multiphoton subradiant states. The fluorescence engages a long-range dipole-dipole interaction which originates from a system-reservoir coupling in the dissipation. We locate some of the subradiant multiphoton states from the eigenmodes, and show that an optically thick atomic array is best for the preparation of the state with the most reduced decay rate. This phase-imprinting process enables quantum state engineering of the multiphoton subradiant states, and realizes a potential quantum storage of the photonic qubits in the two-level atoms.
Introduction.-Recent progress on the collective light scattering [1,2] and subradiant fluorescence measurements [3], and proposals of preparing such subradiant states [4][5][6][7][8][9] opens up a renewed study of strong and collective light-matter interactions [10][11][12][13][14] in neutral atoms. This cooperative phenomenon of the reduced or prolonged spontaneous emission time results from a long-range dipole-dipole interaction [11,12] in a confined atomic system. When the atoms are close enough to each other and become aware of each other's presence through exchanging photons in the dissipation process, the decay rate can be collectively enhanced [15][16][17] along with the associated frequency shift in the emission [13,[18][19][20][21][22][23][24]. This collective light-matter interaction can also initiates a subradiant lifetime in nanocavities [25] and molecules [26]. Furthermore, the atom-atom correlations induced from the dipole-dipole interaction play an important role in determining the emission properties of the cold atoms where the mean-field approach fails [27,28]. This pairwise correlation also has an effect on the superradiant laser [29,30] in a confined and driven atom-cavity system.
In general a driven light-matter interacting system accesses a total number of L N states for N atoms with L atomic levels. This exponential dependence of number of atoms imposes a limitation on classical computer simulations, where several thousands of atoms are able to be simulated only when, for example, a single excitation is assumed. Even such simplification in a reduced Hilbert space of N excited states, a singlephoton emission can be superradiant [31][32][33][34] or subradiant [4,6], which results from the dynamical couplings between these excited states. Single-photon superradiance and subradiance, therefore, provide the first step to investigate even more intricate couplings in the multiply-excited Hilbert space. For a multiply-excited and subradiant state, it is less explored systematically. It is also experimentally difficult to prepare and manipulate these states because of the weak dipole-coupling and intractably large subspace. That is, a total of possible states would become O(N M ) for M excitations in large N two-level atoms if M ≪ N .
In this Letter we propose to prepare the phase-imprinted multiphoton subradiant states by applying a Zeeman or Stark field gradient pulse to imprint linearly increasing phases in an array of two-level atoms. This multiphoton state can have an extremely low decay rate, which essentially can be stored with a long lifetime [35], and is potentially useful in quantum information processing [36] with maximal multiphoton entanglement [37].
Phase-imprinted multiphoton states.-We consider a setting where a resonant multiphoton is absorbed by two-level atoms with |g and |e for the ground and excited states respectively. On absorption, the atoms form a symmetrical and multiplyexcited state or timed Dicke state [31], where µ 0 = 0. We define k · M m=1 r µm ≡ k · R M which is the traveling phase in the absorption of the incoming multiphoton with the transition wave vector k and atomic positions r µm . The M atomic excitations are distributed symmetrically among N atoms with C N M possible configurations where C denotes the binomial coefficients. C N M represents the normalization constant for the above collectively excited state and also the number of Hilbert space for the subspace of M excitations in N atoms. We note that the initially-prepared state |φ is not the eigenstate of the system since the dissipation in general involves a resonant long-range dipole-dipole interaction [12] which we will show later.
For a concise notation, we define |ψ (M) N ( µ) ≡ |e µ1 |e µ2 · · · |e µM |g ⊗(N −M) which denote bare state bases of multiple excitations with a configuration µ ≡ (µ 1 , µ 2 , · · · , µ M ). If there is single excitation (M = 1), we can construct a complete basis |φ (1) n by imprinting linearly increasing phases on a one-dimensional (1D) [6] or three-dimensional (3D) atomic array [7] of arbitrary N atoms, can be imprinted by Zeeman or Stark field gradient pulse, as in Fig. 1. The orthogonality of the state bases can be satisfied by using De Moivre's formula φ For M ≥ 2, the linear phaseimprinted multiphoton states in general cannot construct an orthogonal basis except for N = 3 and M = N . We take two excitations, M = 2, in three equidistant atoms of an array, for example, and The indices in f ( µ) simply represent a discrete increase of the phases from linear phase-imprinting, that is f (µ 1 , µ 2 ) = µ 1 + µ 2 . This special case of multiphoton states construct a complete Hilbert space of two excitations in a N = 3 atomic array, which are orthogonal to each other, guaranteed by De Moivre's formula. For M = 3 and N = 3, the fully-excited state |φ n=1 becomes symmetrical, as is true for all fully-excited multiphoton states.
On the other hand, for M ≥ 2 and N > 3, we obtain the phase-imprinted multiphoton states as µ i represents the incremental increases of the imprinted phases in a 3D atomic array as we propose in Fig. 1. These µ's need to be labeled also linearly in a specific order along three axes. The order of the axes matters only on the relative gradient strength, and without loss of generality, we choose the labeling order first alonĝ x, thenŷ andẑ as shown in Fig. 1. The above states |φ (M) n in general cannot construct the orthogonal states in a multiphoton Hilbert space of M excitations since De Moivre's formula does not apply here. For example N = 4 and M = 2, the bare states |e 1 g 2 g 3 e 4 and |g 1 e 2 e 3 g 4 in Eq. (4) acquire the same phase of e i4nπ/3 , which violates the N -periodic requirement of De Moivre's formula. Nevertheless, our scheme to imprint the linear phases on the atoms can still access the super-and subradiant modes of the system.
Pairwise and long-range dipole-dipole interactions.-The theoretical analysis for the fluorescence of these subradiant states is based on the Lindblad forms of the dissipation. Here we consider a general spontaneous emission process that involves long-range dipole-dipole interaction [6,12]. This interaction originates from the absorption and reabsorption events in the commonly mediating quantized bosonic field. For an arbitrary quantum operatorQ, the Heisenberg equation reads where the Lindblad form for the spontaneous emission is The annihilation (creation) operator isσ − µ (σ + µ ) whereσ − µ ≡ |g µ e| andσ − µ ≡ (σ + µ ) † . The pairwise frequency shift G µν and decay rate F µν are [12] F µν (ξ) ≡ 3Γ 2 1 − (d ·r µν ) 2 sin ξ ξ where Γ is the single-particle decay rate of the excited state, ξ = |k|r µν , and r µν = |r µ − r ν |.
We then turn to the Schrödinger equations which can be projected from the above Lindblad form, to study the time evolution of the phase-imprinted multiphoton states. First we define the state of the system with M excitations in Schrödinger picture as where µ n ≡ (µ 1 , µ 2 , ..., µ M ) n with n here denoting a specific order. Each n requires µ j ∈ [µ j−1 + 1, N − M + j] with µ j < µ j+1 . It increases first with µ M = µ M−1 + 1 up to N while fixes the other µ j 's. Next it keeps increasing with an increment in µ M−1 = µ M−2 + 2 along with µ M = µ M−1 + 1 up to N , until µ 1 reaches (N − M + 1). Taking an example of N = 4 and M = 3, the bare states |ψ (M) N ( µ n ) with n = 1-4 are ordered as |e 1 e 2 e 3 g 4 , |e 1 e 2 g 3 e 4 , |e 1 g 2 e 3 e 4 , and |g 1 e 2 e 3 e 4 . Next we obtain the coupled equations of the probability amplitudes according to Eq. (5), where A nm couples all C N M states in the subspace. The matrix elements are A nn = − MΓ 2 , and A n,m =n = (−F s1s2 /2 + iG s1s2 )e −ik·(rs 1 −rs 2 ) where (s 1 , s 2 ) can be obtained from a sorting function S(n, m). We use S to sort out two numbers, s 1 and s 2 , after comparing the nth and mth bare states of |ψ (M) N ( µ) , which correspond to exactly one different excited atomic index respectively in these bare states. Again for the example of N = 4 and M = 2, to determine A 1,2 , S(1, 2) = S(|e 1 e 2 g 3 g 4 , |e 1 g 2 e 3 g 4 ) gives (2, 3), showing that a dipoledipole interaction lowers the third atomic excited state while raising the second atom to the excited one. This reflects the nature of pairwise couplings in the multiphoton state bases, and thus a matrixÂ represents the dynamically couplings between every two relevant states in |ψ (M) N ( µ) . If (s 1 , s 2 ) gives (0, 0), it leaves a null A n,m =n when more than one distinct atomic indices appear in the nth or mth bare states. Using the same example, A 1,6 (A 2,5 ) = 0, since there is no dipoledipole coupling between the states |e 1 e 2 g 3 g 4 and |g 1 g 2 e 3 e 4 (|e 1 g 2 e 3 g 4 and |g 1 e 2 g 3 e 4 ).
We can apply the similarity transformation to diagonalizê A in terms of the eigenvalues λ l and eigenvectorsÛ , such that the bare state probability amplitude can be solved as where the initial condition, c m (0) = 1/ C N M , is for the symmetrical Dicke state on absorption of M photons. When we evolve the atoms to one of the phase-imprinted multiphoton states |φ where Here v l (n) is the inner product of nth phase-imprinted multiphoton state and lth eigenvector inÛ , and therefore, |v n (m)| 2 indicates the fidelity to the eigenstate. We also define a normalized weighting of wt(l) ≡ |v l (n)w l (n)| 2 as a measure of the contribution for a specific λ l which determines the time evolution of the phase-imprinted multiphoton states.  Fig. 2(a) also appears in the single-photon subradiant state [6], and its beating frequency can be determined by the difference of the frequency shifts in the eigenstates. Five most significant weightings [wt(l)] on the eigenstates are plotted in the inset of Fig.  2(a), where l = 1 and 2 at d s = 0.1λ correspond to 2λ n ′ /Γ = 0.004 − i18.19 and 0.0068 − i17.53 respectively. The difference of 0.33Γ in the frequency shifts thus gives the oscillation period T = 2π/0.33Γ, or ΓT ∼ 20. Taking rubidium atoms as an example, the envelope of this oscillation can be fitted  . In contrast to this oscillatory feature, the phase-imprinted state at d s = 0.25λ simply decays with a subradiant rate. This is due to the extended distribution of the weightings, which smooths out various beating frequencies in these eigenvalues. As expected, a shorter lattice spacing allows to have more enhanced superradiant modes and frequency shifts, which can be seen in Fig. 2(b) at n ′ 90. For even more photons in different geometries, we investigate the phase-imprinted three-photon subradiant states in Fig.  3, generated from an array, rectangular cuboid, to a square. This shows an effective transition from optically thick to optically thin atoms. The lifetimes of these subradiant states can be estimated from the time evolutions in Fig. 3(a). The atomic array shows either 120 or 200 times the lifetime [1/(3Γ)] of the intrinsic spontaneous decay e −3Γt , with a fitting to the intercept at the origin or not. This fitting deviation is due to an initial rapid drop and a following flattened decay, indicating an non-exponential behavior in the subradiance. For the optically thin shapes, fitting an exponential decay is sufficiently well, and their lifetimes are only about four to five times longer, where the square lattice shows less. The weightings on the eigenvalues in Fig. 3(b) rather spread out. Therefore, this limits the sole access to the lowest possible decay constant in Fig. 3(c), where the lowest decay rates can be as reduced as a hundred or ten times smaller respectively for the array or optically thin geometry.
To access even lower subradiant decay rates, we can either increase the total number of atoms or consider an optically thick system (longer atomic chain). For N = 27 and M = 3, we obtain the lowest decay rates of 4.5 × 10 −4 and 0.04Γ respectively in the array and the cube. For a lattice spacing d s 0.5λ, the effect of resonant dipole-dipole interaction in the dissipation is not significant, and therefore the fluorescence of the multiphoton states does not distinguish much from the noninteracting ones. Due to the limitation of available Zeeman field gradient in several mG/µm, practically it is not possible to efficiently prepare all the phase-imprinted subradiant states proposed in Eq. (4). For a longer atomic array, a more demanding field gradient is required to imprint the phase difference between two adjacent sites. For example, preparing the nth multiphoton subradiant state requires a phase difference of 2nπ/C N M . However only a fraction of 2π phase difference is needed to show significant subradiance, as shown in Figs. 2 and 3 where n ≪ C N M . Other issue for phase fluctuations can be removed if the stability of atomic chains and field gradients can be sustained. In experiment, a few percent level of phase error can be controlled, and therefore the phase-imprinted scheme should be robust enough to prepare the multiphoton subradiant state with a high fidelity [6]. On the other hand, the low efficiency due to the short lifetime in the initialized timed-Dicke state can be restored by heralded multiphoton source and post selection. The phase-imprinted scheme promises a dynamical control over the subradiant states, which can be implemented as quantum memory of a multiphoton source. The radiation properties can be also engineered via manipulating atomic geometries and lattice spacings. We note that a complexity of larger N and M arises quite fast already for three atomic excitations in twenty-seven atoms. In such case, a dimension of 2925 states appears in the eigenvalue algorithm, comparing several hundreds of states in Figs. 2 and 3. We estimate the time consumed in simulating the case for N = 20 and M = 10 to longer than seven days, using a typical Intel(R) Xeon CPU E5. Therefore even for such a few-body system with an order of ten atomic excitations, the whole Hilbert space has challenged the capability of present computations.
In conclusion, we propose a scheme to prepare phaseimprinted multiphoton subradiant states in two-level atomic arrays. The fluorescences of these states decay with significantly reduced rates and can be oscillatory with a beating frequency determined by the difference of the frequency shifts. The optically thick atomic system with the appropriate imprinted phases is best for the preparation of the subradiant state with the longest lifetime, making a promising twolevel atomic quantum memory of multiphoton. Future study can also apply this phase-imprinted scheme to a ring-shaped atomic array or the atoms distributed on the side of a cylinder. Using light excitations with m orbital angular momentum [38][39][40][41][42] automatically imprints a phase of e imφ on the atoms, and essentially the phase-imprinted multiphoton states proposed here can be the candidates for quantum storage of orbital angular momentum photonic qubits [43][44][45].
Acknowledgements.-This work is supported by the Ministry of Science and Technology (MOST), Taiwan, under the Grant No. MOST-104-2112-M-001-006-MY3. We are grate-ful for stimulating discussions with Pye-Ton How on a sorting algorithm in assigning multiphoton coupling matrix elements, and with M.-S. Chang and Y.-C Chen on a potential experimental implementation in the ring-shaped atomic arrays.