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Nonlinear pi phase shift for single fiber-guided photons interacting with a single atom

Realizing a strong interaction between individual optical photons is an important objective of research in quantum science and technology. Since photons do not interact directly, this goal requires, e.g., an optical medium in which the light experiences a phase shift that depends nonlinearly on the photon number. Once the additional phase shift for two photons reaches pi, such an ultra-strong nonlinearity could even enable the direct implementation of high-fidelity quantum logic operations. However, the nonlinear response of standard optical media is many orders of magnitude too weak for this task. Here, we demonstrate the realization of an optical fiber-based nonlinearity that leads to an additional two-photon phase shift close to the ideal value of pi. Our scheme employs a whispering-gallery-mode resonator, interfaced by an optical nanofiber, where the presence of a single rubidium atom in the resonator results in a strongly nonlinear response. We experimentally show that this results in entanglement of initially independent incident photons. The demonstration of this ultra-strong nonlinearity in a fiber-integrated system is a decisive step towards scalable quantum logics with optical photons.

Optical photons are a key ingredient for investigations and applications in modern quantum information science. They are well decoupled from their environment and can, nevertheless, be conveniently manipulated with high precision. In conjunction with the ease of transmitting them over long distances using optical fibers, this makes photons prime candidates for the distribution (4) and processing of quantum information in scalable quantum networks (5) as well as for metrology beyond the standard quantum limit (6). These and many other applications require the photons to interact with each other in order to prepare and probe entanglement or to perform quantum logic operations. However, a direct photon-photon interaction does not exist in free space. It has been shown that an effective photon-photon interaction can be implemented probabilistically using linear optical elements in combination with projective measurements. (7) Yet, for now, this technique cannot considered to be scalable in a practical sense because it requires high fidelity single photon sources and detectors which still go beyond present capabilities. (8) Alternatively, a deterministic interaction can be realized by means of an optical medium that exhibits a nonlinearity down to the level of individual photons. Such strong optical nonlinearities have been demonstrated, e.g., in atomic ensembles where the nonlinearity either stems from direct interaction between the excited atoms (9,10) or from the generation of an intrinsic Kerr nonlinearity via electromagnetically induced transparency. (11,12) An alternative approach is based on enhancing the nonlinearity of individual quantum emitters by coupling them to optical resonators. With these systems, nonlinear phase shifts up to a few ten degrees, single-photon controlled on-and off-switching of light, as well as photon number dependent redirection of light have been demonstrated. (13)(14)(15)(16)(17) In this work, we demonstrate the realization of an optical fiber-based nonlinearity that leads to an unprecedented two-photon phase shift close to the ideal value of π, following a proposal by Hofmann et al. (3). The nonlinearity is induced by a single 85 Rb atom that is strongly coupled to a whispering-gallery-mode (WGM) resonator while the latter is efficiently interfaced by an optical nanofiber. Our experimental setup is sketched in Fig. 1. The central element is a so-called bottle microresonator, (18)(19)(20) a novel type of WGM microresonator which is conceptually similar to other WGM microresonators (21,22) but has the additional advantage of being fully tunable (20). We interface the resonator with a tapered fiber coupler that comprises a nanofiber waist (23). The distance between the nanofiber and the resonator is chosen such that the system operates in the overcoupled regime, i.e., that the coupling rate κ f between the nanofiber and the resonator is larger than the intrinsic resonator loss rate κ i . In this regime, resonant, horizontally (H-) polarized light that is guided in the nanofiber will couple into the empty resonator and then couple back into the fiber, thereby acquiring a phase shift of π due to the interaction with the resonator mode. The situation is different if the photon is vertically (V -) polarized or if an atom is strongly coupled to the resonator. In the former case, the light cannot enter due to the large birefringence of the resonator while, in the latter case, the presence of the atom changes the resonance condition. Thus, with an atom present, H-polarized photons that arrive one by one cannot enter the resonator mode. In both cases, the photons therefore do no acquire a phase shift. However, due to the nonlinearity of the interaction, the system behaves differently when two H photons are incident on the resonator at the same time: The presence of one photon saturates the atom-resonator system and thus modifies the resonance condition for the other photon. Since this process is coherent and the two photons are indistinguishable, this results in an additional phase shift of π for the two-photon state compared to the case where only a single photon interacts with the atom-resonator system. This can be measured experimentally by comparing it with the uncoupled V -polarized light field which acts as phase reference. Neglecting optical losses as well as the spontaneous emission of the atom into the resonator mode (see supplementary information) and choosing an input polarization along the +45 • (i.e., H+V ) direction, the quantum state of the two photons before and after the interaction with the atom-resonator system is given by respectively. Here we introduced the creation operators for horizontal, vertical, plus (P ), and respectively. Due to the nonlinear interaction, an additional phase occurs for the |HH term which, in the case of a π phase shift, results in the minus in Eq. (2). As a result, the final state is no longer separable. In the experiment, optical losses, fluctuating detunings, birefringence, and spontaneous emission into the resonator mode (3) will inevitably change the probabilities for the outcome of the individual states and lead to decoherence of their relative phase. One thus has to verify the fidelity with which the nonlinear interaction leads to the ideal output |ψ final and if the entanglement is indeed prepared under realistic conditions.
In order to optimize our atom-resonator system, we first analyze its performance at the  Fig. 2). In order to model this behavior, we assume a normal distribution of the atom-resonator coupling strength g, which originates from the motion of the atom through the resonator mode. From the transmission properties of our atom-resonator system, we calculate the corresponding polarization change (see supplementary material). We then fit this distribution to the measured data, with the mean coupling strength,ḡ, and the standard deviation, σ g , as the only free parameters. From this fit we obtainḡ = 2π × (13.5 ± 1.5) MHz and σ g = 2π × 4 MHz.
We now choose the working point κ f /κ i = 2.8 (see dashed line in Fig. 2) at which we expect photon number-independent loss and, as a consequence, an effectively dispersive nonlinearity (see supplementary material). At this point, the atom-resonator system yields a polarization dependent transmission between 20% for Hand close to 100% for V -polarization.
In order to experimentally determine the nonlinear phase shift, we analyze the transmitted light in three complementary polarization bases by recording coincidence counts between the different detectors (see Fig. 1). Figure  We find a maximum overlap of 0.57 ± 0.01 and a nonlinear phase shift of (1.05 ± 0.01)π, close to the value of π expected under ideal conditions.
As is apparent from Eq. (2), the π nonlinear phase shift ideally generates a non-separable state of light from the initially independent incident photons. In order to quantify the nonclassical character of the experimentally prepared two-photon state, we calculate its concurrence (26,27) (see supplementary material). We find a maximum concurrence of 0.27 ± 0.01 at zero mean detection time difference, thereby demonstrating that ρ meas corresponds to a highly entangled state. The concurrence and, thus, the entanglement vanishes when the time delay between the photons increases beyond the resonator lifetime (see Fig. 4). This is expected because, in this case, the photons interact one-by-one with the atom-resonator system.
In summary, we demonstrated an ultra-strong optical nonlinearity that leads to a nonlinear π phase shift for coincident photons. The employed experimental platform is fully fiber-integrated and, thus, compatible with optical fiber communication networks. The scheme relies on a single atom which can, in contrast to atomic ensembles, be straightforwardly prepared a quantum mechanical superposition of two states which are coupled and not coupled to the resonator mode, respectively. This is the basis for realizing a deterministic controlled π-phase gate (28)      E Real (red) and imaginary part (blue) of the reconstructed density matrix for zero time delay and F of the density matrix ρ final of the ideal state.

Supplementary Material
Atom detection and experimental sequence.
In order to couple atoms to the resonator, we use an atomic fountain that delivers a cloud of Here κ f is the fibre-resonator coupling strength, κ L the resonator loss rate, and ∆ rl the detuning between the resonator and the light field. This expression allows us to treat both cases -the empty resonator and the coupled atom-resonator system -by introducing two loss rates: For the empty resonator, the loss rate is given by the intrinsic loss rate, κ L = κ i = 2π × 8.4 MHz, while for the coupled atom-resonator system in the single photon limit, the losses are governed by an effective loss rate that also depends on the atom: Here, γ is the atomic decay rate, g the atom-resonator coupling strength, and ∆ al is the detuning between the atom and the resonator-mode. Using these expressions, we model the phase shift and transmission properties of the resonator and calculate the expected polarisation change of the transmitted light field.

Atom-resonator interaction beyond the single-photon limit
In order to simulate the behavior of the atom-resonator system for the case with more than one photon present, we perform a full numerical calculation of our system by solving the master equations of the pumped atom-resonator system using the approximation of a two-level atom interacting with a single resonator mode (24). In this case, the master equation is given by where is the atom-resonator Hamiltonian in the rotating wave approximation and b (b † ) is the annihilation (creation) operator of a resonator photon, σ + (σ − ) is the atomic excitation (deexcitation) operator, ρ is the atom-resonator density matrix and a H,in is the amplitude of the incident H-polarized fibre guided light. The final state of the light field after the interaction with the resonator is then given by the interference of the incident field with the field that is coupled out of the resonator: The input light field corresponds to a coherent state which allows us to replace the input operator a H,in and a V,in by their expectation value (30) and to calculate the expectation values for the output fields. Using the measured distribution of coupling strengths, this allows us to predict the optical losses in the system and to estimate the optimal working point, where the two-photon transmission is independent of the photon time delay.

State reconstruction
For the reconstruction of the density matrix of the two-photon state after the interaction with the resonator, we use the 19 non-trivial coincidence detections obtained from the polarisation setup. Note that we did not record coincidences for the settings RR and V V (see supplementary information). We define a coincidence window and sum up all two-photon events within this window. The incident photons originate from a coherent laser beam with a coherence time on the order of one µs and are thus, to a very good approximation, indistinguishable within the interaction time which is given by the resonator lifetime of κ −1 ≈ 7 ns. As a consequence, the two-photon state is limited to the symmetric subspace of a two-qubit Hilbert space. The corresponding basis states are |HH , |S = (|HV + |V H )/ √ 2, and |V V and the density can be written as The matrix element ρ A,A and all other matrix elements which contain contributions from the asymmetric basis state |A = (|HV −|V H )/ √ 2 vanish (25) and the 3×3 sub-matrix contains the full information on the experimentally prepared state. The 3 × 3 sub-matrix is reconstructed by performing a quantum state tomography using a maximum likelihood estimation. (27) Concurrence and nonlinear phase shift In order to quantify the degree of entanglement, we transform the full 4 × 4 density matrix into an unentangeld basis defined by the basis states |HH , |HV , |V H and |V V and subsequently calculate the concurrence (26). The nonlinear phase shift is determined directly from the mea- where these two density matrix elements were chosen due to their higher signal to noise ratio.

Error estimation
The statistical errors of the measured quantities were determined by adding Poissonian noise to the measured coincidences, followed by a density matrix reconstruction and the subsequent evaluation of the concurrence, nonlinear phase shift, and overlap with ρ ideal . For each data point in Fig. 4D in the main text, we generate a set of 100 random density matrices and use the resulting standard deviations as our error estimate.  From this measurement, we obtain the data shown in Fig. 2 of the main text.

Set of measured coincidences
Using the setup shown in Fig. 1 of the main manuscript, we measure 19 non-trivial coincidences. The coincidence settings RR and V V are not accessible due to the limited number of single photon counting modules. The full set of measured coincidences is shown in Fig. 2.

Influence of spontaneous emission
For clarity, we used a strongly simplified description of the nonlinear interaction of the photons with the atom-resonator system in the main text. In particular, we neglected the spontaneous emission of the atom into the resonator mode. However, as described in a semiclassical model

Implementation of a photon-photon sign-flip gate
While the nonlinearity demonstrated in our work cannot directly be exploited for the realization of a high-fidelity photon-photon gate, (1,3) a deterministic controlled sign-flip gate for photons can be realized by storing the first photon in the atom-resonator system prior to the interaction with the second photon: We assume that the mode of incident photons is defined by their temporal degree of freedom, i.e., the single photon pulses have a relative delay which is much larger than their coherence time. The atom is initially prepared in a state |g coupl that couples to the resonator (e.g., the F = 3 hyperfine ground state of 85 Rb). We consider the case where the two incident photons are P -polarized. Thus, the total state before the interaction with the resonator is given by When the first photon arrives at the resonator we apply an additional control laser which performs a Raman transition of the atom from |g coupl to |g uncoupl if and only if the incident photon 1 is H-polarized and enters the resonator. Here, |g uncoupl | is a second ground state of the atom that does not couple to the resonator (e.g., the F = 2 hyperfine ground state of 85 Rb). As a consequence, the H-photon is stored in the atom and the state is now given by Thus, if the first photon is H polarized, the atom is switched to a state that is off-resonant with respect to the resonator mode and the H component of the second photon will acquire a phase shift of π. Otherwise, the atom is resonant with the resonator mode and no phase shift occurs.
The total state after the interaction of the second photon with the resonator thus reads |Ψ = 1 2 (−|H 2 |0 |g uncoupl + |V 2 |0 |g uncoupl + |H 2 |V 1 |g coupl + |V 2 |V 1 |g coupl ). (13) In the next step, the stored photon can be released using a second Raman process and ones end up in the final state which is the desired state. We note that other implementations of a controlled phase-flip gate which do not require the implementation of Raman transitions have been proposed. (4)