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arXiv:0808.3279 · Physical Review Letters

Measurement of the Decay of Fock States in a Superconducting Quantum Circuit

H. Wang, M. Hofheinz, M. Ansmann, Radoslaw C. Bialczak, Erik A. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, Andrew N. Cleland, John M. Martinis

Published Dec 8, 2008• 161 citations
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Abstract

We demonstrate the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit. The subsequent decay of the Fock states, due to dissipation, is then monitored by varying the time delay between preparing the state and performing a number-state analysis. We find that the decay dynamics can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory. We have also generated a coherent state in the microwave resonator, and monitored its decay process. We demonstrate that the coherent state maintains a Poisson distribution as it decays, with an average photon number that decreases with the same characteristic decay time as the one-photon Fock state.

DOI
10.1103/physrevlett.101.240401

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We demonstrate the controlled generation of Fock states with up to 15 photons in a microwave coplanar waveguide resonator coupled to a superconducting phase qubit. The subsequent decay of the Fock states, due to dissipation, is then monitored by varying the time delay between preparing the state and performing a number-state analysis. We find that the decay dynamics can be described by a master equation where the lifetime of the n-photon Fock state scales as 1/n, in agreement with theory. We have also generated a coherent state in the microwave resonator, and monitored its decay process. We demonstrate that the coherent state maintains a Poisson distribution as it decays, with an average photon number that decreases with the same characteristic decay time as the one-photon Fock state.
Two-level systems, or qubits, afford complete quantum control using only classical electrical signals. Achieving the same degree of control over harmonic resonators and photons has remained elusive, due to their infinite number of equally spaced energy levels. We exploit our complete qubit control by using a superconducting phase qubit to pump microwave photons into a high-Q coplanar wave guide resonator and, subsequently, use the qubit to read out the resonator state. This scheme has previously allowed us to prepare and detect photon number states (Fock states [1]) in the resonator and to measure their decay. Using a generalization of a scheme developed by Law and Eberly [2], we can now create arbitrary quantum states of the photon field with up to approximately 10 photons [3]. We analyze the prepared states by directly mapping out the corresponding Wigner function, which is the phase-space equivalent of the density matrix, and provides a complete description of the quantum state. Movies of the decay of a variety of non-classical states show a loss of quantum coherence over the energy decay time of the resonator.
We study the implementation of quantum phase measurement in a superconducting circuit, where two Josephson phase qubits are coupled to the photon field inside a resonator. We show that the relative phase of the superposition of two Fock states can be imprinted in one of the qubits. The qubit can thus be used to probe and store the quantum coherence of two distinguishable Fock states of the single-mode photon field inside the resonator. The effects of dissipation of the photon field on the phase detection are investigated. We find that the visibilities can be greatly enhanced if the Kerr nonlinearity is exploited. We also show that the phase measurement method can be used to perform the Gauss sum factorization of numbers ($\ensuremath{\geqslant}{10}^{4}$) into a product of prime integers, as well as to precisely measure both the resonator's frequency and the nonlinear interaction strength. The largest factorizable number is mainly limited by the coherence time. If the relaxation time of the resonator were to be  10~10 10 $\ensuremath{\mu}$s ( 1~1 1 ms), then the largest factorizable number can be $\ensuremath{\geqslant}{10}^{4}N$ ($\ensuremath{\geqslant}{10}^{7}N$), where NNN is the number of photons in the resonator.
Quantum states inevitably decay with time into a probabilistic mixture of classical states due to their interaction with the environment and measurement instrumentation. We present the first measurement of the decoherence dynamics of complex photon states in a condensed-matter system. By controllably preparing a number of distinct quantum-superposed photon states in a superconducting microwave resonator, we show that the subsequent decay dynamics can be quantitatively described by taking into account only two distinct decay channels: energy relaxation and pure dephasing. Our ability to prepare specific initial quantum states allows us to measure the evolution of specific elements in the quantum density matrix in a very detailed manner that can be compared with theory.
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We demonstrate a superconducting resonator with variable coupling to a measurement transmission line. The resonator coupling can be adjusted through zero to a photon emission rate 1000 times the intrinsic resonator decay rate. We demonstrate the catch and release of photons in the resonator, as well as control of nonclassical Fock states. We also demonstrate the dynamical control of the release waveform of photons from the resonator, a key functionality that will enable high-fidelity quantum state transfer between distant resonators or qubits.
We propose and experimentally demonstrate a new method to generate arbitrary Fock state superpositions in a superconducting quantum circuit, where a qubit is dispersively coupled to a microwave cavity mode. Here, the qubit is used to conditionally modulate the probability amplitudes of the Fock state components of a coherent state to those of the desired superposition state, instead of pumping photons one by one into the cavity as in previous schemes. Our method does not require the adjustment of the qubit frequency during the cavity state preparation and is more robust to noise and accumulation of experimental errors compared to previous ones. Using the method, we experimentally generate phase eigenstates under various Hilbert-space dimensions and squeezed states, which are useful for the quantum walk and high-precision measurement.