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Conditional creation of superpositions of coherent states with an ineffective photon

К.Р.Саттаркулов, M.M. Komilov, З.Н. Oбидoва, N.T. Imankulov, A.K. Fayzullayev

Published Jan 1, 2024• 0 citations
Publisher

Abstract

The article shows that a single photon source and straightforward, purely optical processes can be used to produce a linear superposition of two macroscopically identifiable optical coherent states. A superposition of coherent states in a freely propagating optical field with high coherence amplitude can be generated via weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detection with imperfect threshold detectors. (α>2) and F=0.99, which is extremely precise. Our technique does not require a precise photon count, resolving measurements, or nonlinear Kerr-type interactions, in contrast to all existing schemes for obtaining such a state. It also exhibits some tolerance to inefficiencies in photon generation and is robust against detection of inefficiencies.

DOI
10.1051/e3sconf/202459201015

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The article shows that a single photon source and straightforward, purely optical processes can be used to produce a linear superposition of two macroscopically identifiable optical coherent states. A superposition of coherent states in a freely propagating optical field with high coherence amplitude can be generated via weak squeezing on a single photon, beam mixing with an auxiliary coherent state, and photon detection with imperfect threshold detectors. (α>2) and F=0.99, which is extremely precise. Our technique does not require a precise photon count, resolving measurements, or nonlinear Kerr-type interactions, in contrast to all existing schemes for obtaining such a state. It also exhibits some tolerance to inefficiencies in photon generation and is robust against detection of inefficiencies.
Conditional Measurement scheme which employs linear optical elements and photon detection is the fertile ground for nonclassical state generation. We consider a simple setup that requires a coherent state and a number state as inputs of the beam splitter, and a photon detector. We show that by tuning the parameters involved in the setup, we can achieve optimal squeezing from the setup. This is facilitated by writing the output state of the conditional measurement as displaced qudits. Setting aside displacement which plays no role in squeezing, the finite-dimensional representation makes it possible to calculate the maximal amount of squeezing. By fixing the detection at one photon level irrespective of any number state input and carefully chosen coherent parameter and beam splitter reflectivity values, one can reach the maximal squeezing at least for lower number state inputs. This is in contrast to the earlier attempts in atom field interaction models etc., where the squeezing obtained was far from saturation. To accommodate the experimental imperfections, we consider the impure nature of the photon source and detector inefficiency.
The generation of non-Gaussian quantum states is a key requirement for universal continuous-variable quantum information processing. We report the experimental generation of large-amplitude squeezed coherent-state superpositions (squeezed cat states) on free-space optical pulses, reaching an amplitude of α=2.47α= 2.47α=2.47, which, to our knowledge, exceeds all previously reported values. Our protocol relies on the controlled mixing of the Fock states ∣1⟩|1\rangle∣1⟩ and ∣2⟩|2\rangle∣2⟩ through a tunable beam splitter, followed by heralding via homodyne detection. The resulting state displays three well-resolved negative regions in its Wigner function and achieves a fidelity of 0.530.530.53 with the target state ∝S^(z)(∣α⟩−∣−α⟩)\propto \hat{S}(z)(|α\rangle - |-α\rangle)∝S^(z)(∣α⟩−∣−α⟩), with α=2.47α= 2.47α=2.47 and squeezing parameter z=0.56z = 0.56z=0.56. These results constitute a significant milestone for temporal breeding protocols and for the iterative generation of optical GKP states, opening new perspectives for scalable and fault-tolerant photonic quantum architectures.
Optical Schrödinger cat states are non-Gaussian states with applications in quantum technologies, such as for building error-correcting states in quantum computing. Yet the efficient generation of high-fidelity optical Schrödinger cat states is an outstanding problem in quantum optics. Here, we propose using squeezed superpositions of zero and two photons, ∣θ⟩=cos⁡(θ/2)∣0⟩+sin⁡(θ/2)∣2⟩|θ\rangle = \cos{(θ/2)}|0\rangle + \sin{(θ/2)}|2\rangle∣θ⟩=cos(θ/2)∣0⟩+sin(θ/2)∣2⟩, as ingredients for protocols to efficiently generate high-fidelity cat states. We present a protocol using linear optics with success probability P≳50%P\gtrsim 50\%P≳50% that can generate cat states of size ∣α∣2=5|α|^2=5∣α∣2=5 with fidelity F>0.99F>0.99F>0.99. The protocol relies only on detecting single photons and is remarkably tolerant of loss, with 2%2\%2% detection loss still achieving F>0.98F>0.98F>0.98 for cats with ∣α∣2=5|α|^2=5∣α∣2=5. We also show that squeezed θθθ states are ideal candidates for nonlinear photon subtraction using a two-level system with near deterministic success probability and fidelity F>0.98F>0.98F>0.98 for cat states of size ∣α∣2=5|α|^2=5∣α∣2=5. Schemes for generating θθθ states using quantum emitters are also presented. Our protocols can be implemented with current state-of-the-art quantum optics experiments.
Photon subtraction is useful to produce nonclassical states of light addressed to applications in photonic quantum technologies. After a very accelerated development, this technique makes possible obtaining either single photons or optical cats on demand. However, it lacks theoretical formulation enabling precise predictions for the produced fields. Based on the representation generated by the two-mode SU(2) coherent states, we introduce a model of entangled light beams leading to the subtraction of photons in one of the modes, conditioned to the detection of any photon in the other mode. We show that photon subtraction does not produce nonclassical fields from classical fields. It is also derived a compact expression for the output field from which the calculation of conditional probabilities is straightforward for any input state. Examples include the analysis of squeezed-vacuum and odd-squeezed states. We also show that injecting optical cats into a beam splitter gives rise to entangled states in the Bell representation.
Gottesman-Kitaev-Preskill (GKP) qubits, known for their exceptional error-correction capabilities, are highly coveted in quantum computing. However, generating optical GKP qubits has been a significant challenge. Measurement-based methods, where a portion of entangled squeezed vacuum modes are measured with photon number resolving detectors heralding a desired state in the undetected modes, have emerged as leading candidates for optical GKP qubit generation due their minimal resource requirements. While the current measurement-based methods can produce high-quality GKP qubits, they suffer from low success probabilities limiting experimental realization. The heart of the problem lies in the duality of photon number resolving measurements, being both the source of nonlinearity needed to generate quality GKP qubits and the component driving down their probability of successful production. Our method, breeding approximate squeezed coherent state superpositions created by generalized photon subtraction, overcomes this problem by supplementing two photon number resolving measurements with a single high success probability homodyne measurement. This scheme achieves success probabilities $\ensuremath{\ge}{10}^{\ensuremath{-}5}$, two orders of magnitude higher than strictly photon number resolving measurement based methods, while still producing states with high fidelity, possessing error-correction capabilities equivalent to up to a 10 dB squeezed GKP qubit. This breakthrough significantly advances the practical use of the optical GKP qubit encoding.
State preparation via postselected weak measurements in a three-wave mixing process is studied. We assume the signal input mode is prepared in a vacuum state, coherent state, or squeezed vacuum state, while the idler input is prepared in a weak coherent state and passes the medium characterized by the second-order nonlinear susceptibility. It is shown that when the single photon is detected at one of the output channels of the idler beam's path, the signal output channel is prepared in a single-photon Fock state, single-photon-added coherent state, or single-photon-added squeezed vacuum state with very high fidelity, depending upon the input signal states and related controllable parameters. The properties of squeezing, signal amplification, second-order correlation, and the Wigner functions of the weak-measurement-based output states are also investigated. Our scheme promises to provide an alternate effective method for producing useful nonclassical states in quantum information processing.