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arXiv:2408.07403 · Physical Review A

Generating Fock-state superpositions from coherent states by selective measurement

Chen-yi Zhang, Jun Jing

Published Oct 18, 2024• 7 citations
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Abstract

Fock states and their superpositions are exotic test beds for nonclassical physics and valuable resources for quantum technologies. We provide a simple protocol for the quantum measurement to generate an arbitrary Fock state and certain superposed Fock states from a coherent state of a target resonator, without any carefully tailored driving. This conditional protocol can be efficiently constructed by a sequence of joint free evolution of the resonator and an ancillary qubit, which are coupled via a Jaynes-Cummings interaction, and projective measurements on the qubit. By properly choosing the duration of each evolution-measurement cycle and the initial state of the resonator, we can generate a desired Fock state $|n\ensuremath{\rangle}$ and a superposed Fock state $(|0\ensuremath{\rangle}+|n\ensuremath{\rangle})/\sqrt{2}, n\ensuremath{\sim}10$, with a fidelity over $99%$ in less than 30 measurements. Moreover, our protocol can be extended straightforwardly to the generation of a Bell-like state $(|00\ensuremath{\rangle}+|nn\ensuremath{\rangle})/\sqrt{2}$ with multiple excitations in a double-resonator system. We also calculate the outcome fidelity and the success probability of our protocol in the presence of decoherence.

DOI
10.1103/physreva.110.042421

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Fock states and their superpositions are exotic test beds for nonclassical physics and valuable resources for quantum technologies. We provide a simple protocol for the quantum measurement to generate an arbitrary Fock state and certain superposed Fock states from a coherent state of a target resonator, without any carefully tailored driving. This conditional protocol can be efficiently constructed by a sequence of joint free evolution of the resonator and an ancillary qubit, which are coupled via a Jaynes-Cummings interaction, and projective measurements on the qubit. By properly choosing the duration of each evolution-measurement cycle and the initial state of the resonator, we can generate a desired Fock state ∣n\ensuremath⟩|n\ensuremath{\rangle}∣n\ensuremath⟩ and a superposed Fock state (∣0\ensuremath⟩+∣n\ensuremath⟩)/2,n\ensuremath∼10(|0\ensuremath{\rangle}+|n\ensuremath{\rangle})/\sqrt{2}, n\ensuremath{\sim}10(∣0\ensuremath⟩+∣n\ensuremath⟩)/2​,n\ensuremath∼10, with a fidelity over 9999%99 in less than 30 measurements. Moreover, our protocol can be extended straightforwardly to the generation of a Bell-like state (∣00\ensuremath⟩+∣nn\ensuremath⟩)/2(|00\ensuremath{\rangle}+|nn\ensuremath{\rangle})/\sqrt{2}(∣00\ensuremath⟩+∣nn\ensuremath⟩)/2​ with multiple excitations in a double-resonator system. We also calculate the outcome fidelity and the success probability of our protocol in the presence of decoherence.
Nonclassical states of bosonic modes, especially the large number states, are valuable resources for quantum information processing and quantum metrology. It is however intricate to generate a desired Fock state of bosonic systems by unitary protocols due to their uniform energy spectrum. We here propose a nonunitary protocol that is based on the resonant Jaynes-Cummings interaction of the bosonic mode with an ancillary two-level atom and sequential projective measurements on the atom. Using the generalized parity-measurement operator constructed by several rounds of free evolution with stepwise halved intervals and measurement, we can efficiently filter out the unwanted population and push the target resonator conditionally toward the desired Fock state. In the ideal situation, a Fock state ∣nt≈2000⟩|n_t\approx2000\rangle∣nt​≈2000⟩ can be prepared with a fidelity over 98%98\%98% using only eight rounds of measurements. Under qubit dissipation and dephasing and cavity decay in the current circuit-QED platforms, a Fock state ∣nt≈100⟩|n_t\approx100\rangle∣nt​≈100⟩ can be prepared with a fidelity of about 80%80\%80% by six measurements. It is found that the number of measurement rounds for preparing a large Fock state ∣nt⟩|n_t\rangle∣nt​⟩ scales roughly as log⁡2nt\log_2\sqrt{n_t}log2​nt​​, which is similar to the number of ancillary qubits required in the state preparation via the quantum phase estimation algorithm and yet costs much less in gate operations. Our protocol can also be used to prepare a large Dicke state ∣J≃1000,0⟩|J\simeq1000,0\rangle∣J≃1000,0⟩ of a spin ensemble with a sufficiently high fidelity by less than six measurements. It is qualified by the quantum Fisher information approaching the Heisenberg scaling in sensing the rotation phase along the xxx axis.
Fock states are fundamental quantum states with a precisely defined integer number of excitations, serving as the core basis for describing bosonic modes. Large Fock states provide irreplaceable non-classical resources for quantum information processing and quantum metrology. The deterministic generation of macroscopic photon-number Fock states has long been a difficult problem in the field of quantum optics. We propose an adaptive generalized parity measurement (GPM) protocol for generating Fock states with more than 100001000010000 excitations, avoiding low success probability subject to postselection and high cost under complex coherent controls. For general discrete-spectrum systems, e.g., a bosonic mode coupled to an ancillary qubit, we derive a construction rule in which the intervals between repeated measurements on qubit are updated adaptively based on the last outcome. It means that our protocol does not discard any measurement trajectory, dramatically different from the probabilistic protocols that retain only one prescribed trajectory of postselection. In the resonant Jaynes-Cummings model, a large coherent state can be almost deterministically transformed to a large Fock state of nt=O(104)n_t=\mathcal{O}(10^4)nt​=O(104) excitations by 101010 rounds of measurements, the average fidelity of which is about 87%87\%87%. The success probability for obtaining ∣20000−20000≤nt≤20000+20000⟩|20000-\sqrt{20000}\leq n_t\leq20000+\sqrt{20000}\rangle∣20000−20000​≤nt​≤20000+20000​⟩ with a fidelity above 99%99\%99% is about 35%35\%35% with respect to the ensemble sampling. Our protocol is fault-tolerant in the presence of moderate measurement error and parametric imperfection. Also it remains effective when the system is prepared as displaced thermal states, showing reliable performance regardless of initial state.
Entangled states are self-evidently important to a wide range of applications in quantum communication and quantum information processing. We propose an efficient and convenient two-step protocol for generating Bell states and NOON states of two microwave resonators from merely coherent states. In particular, we derive an effective Hamiltonian for resonators coupled to a superconducting $\mathrm{\ensuremath{\Lambda}}$-type qutrit in the dispersive regime. By the excitation-number-dependent Stark shifts of the qutrit transition frequencies, we are able to individually control the amplitudes of specified Fock states of the resonators associated with relevant qutrit transition, using carefully tailored microwave drive signals. Thereby an arbitrary bipartite entangled state in Fock space can be generated by a typical evolution-and-measurement procedure. We analyze the undesired state transitions and the robustness of our protocol against the systematic errors from the microwave driving intensity and frequency, the quantum decoherence of all components, and the crosstalk of two resonators. In addition, we demonstrate that our protocol can be extended to a similar scenario with a $\mathrm{\ensuremath{\Xi}}$-type qutrit.
Large Fock states are important resources for bosonic quantum information and quantum-enhanced metrology, but preparing them with high probability at large excitation numbers remains challenging, as deterministic methods become increasingly control-intensive, while measurement-based approaches typically suffer from low heralding probabilities. Here we propose a protocol that combines quantum nondemolition photon-number encoding with quantum amplitude amplification to enable high-probability heralded generation of large Fock states. Starting from a cavity mode prepared in a coherent state, Quantum Phase Estimation encodes photon-number information into a multi-qubit register, while Quantum Amplitude Amplification boosts the probability of a desired target outcome before measurement. The scheme has an immediate implementation in dispersive circuit-QED, but can be analogously adapted to other bosonic platforms with QND photon-number readout, such as cavity-QED. With a register of up to eight qubits, near-deterministic preparation of Fock states with hundreds of excitations is possible. We also show that the protocol can serve as the first stage of an extension toward generating a two-mode NOON state via a conditional beam-splitter operation.
We introduce a protocol capable of generating a general measurement operator for a mechanical resonator. The technique requires a qubit-resonator interaction and uses a coherent pulse to drive qubit transitions. This is followed by projective measurement of the qubit's energy, constraining the resonator in a state that depends on the pulse shape. The freedom to choose a pulse shape for the coherent drive enables an arbitrary position-basis measurement operator. Using this measurement operator, we outline a two pulse protocol that probabilistically generates a pure mechanical state with a desired wavefunction, with near unit fidelity for realizable parameters.
The scalable preparation of large photon-number (Fock) states is a long-standing frontier in quantum science, with direct implications for quantum metrology and bosonic quantum information processing. Despite substantial progress at small photon numbers, extending state generation to large photon numbers while maintaining high fidelity and operating deterministically remains a significant challenge. Here we demonstrate a scalable and experimentally accessible control protocol for generating large photon-number states using only native spin--oscillator operations. The protocol alternates Jaynes--Cummings interactions with phase-space displacements to imprint photon-number--dependent phases and convert them into selective interference in photon-number space. It already achieves high preparation fidelity unconditionally, while an optional final qubit projection removes residual qubit--field correlations and further enhances the fidelity. Conditioned on this final projection, photon-number state preparation with fidelities exceeding 0.950.950.95 is achieved for photon numbers in the few-hundred regime, with a success probability exceeding 0.900.900.90, placing the protocol in a near-deterministic operating regime. The resulting control sequences remain shallow and are robust against detuning, control noise, and experimentally relevant dissipation. Our results establish a practical route to scalable, high-fidelity photon-number state preparation at large photon numbers and provide a versatile interference-engineering toolbox for nonclassical bosonic state synthesis.