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Quantum limits on linear amplifiers What’s the problem?

Quantum limits on linear amplifiers What’s the problem? Quantum limits on noise in phase-preserving linear amplifiers. The whole story Completely positive maps and physical ancilla states IV. Nondeterministic linear amplifiers Carlton M. Caves

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Quantum limits on linear amplifiers What’s the problem?

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  1. Quantum limits on linear amplifiers • What’s the problem? • Quantum limits on noise in phase-preserving linear amplifiers. The whole story • Completely positive maps and physical ancilla states • IV. Nondeterministic linear amplifiers • Carlton M. Caves • Center for Quantum Information and Control, University of New Mexico • Centre for Engineered Quantum Systems, University of Queensland • http://info.phys.unm.edu/~caves • Co-workers: Z. Jiang, S. Pandey, J. Combes Center for Quantum Information and Control

  2. I. What’s the problem? View from Cape Hauy Tasman Peninsula Tasmania

  3. Phase-preserving linear amplifiers added-noise operator output noise input noise added noise gain Refer noise to input Added noise number C. M. Caves, PRD 26, 1817 (1982). C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, arXiv:1208.5174 [quant-ph] Noise temperature

  4. Ideal phase-preserving linear amplifier Parametric amplifier

  5. Ideal phase-preserving linear amplifier Parametric amplifier The noise is Gaussian. Circles are drawn at half the standard deviation of the Gaussian. A perfect linear amplifier only has the (blue) amplified input noise.

  6. Ideal phase-preserving linear amplifier Models • ● Parametric amplifier with ancillary mode in vacuum • ● Simultaneous measurement of x and p followed by creation • of amplified state • ● Negative-mass (inverted-oscillator) ancillary mode in vacuum • ● Master equation E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). R. J. Glauber, in New Techniques and Ideas in Quantum Measurement Theory, edited by D. M. Greenberger (NY Acad Sci, 1986), p. 336. C. W. Gardiner and P. Zoller, Quantum Noise, 3rd Ed. (Springer, 2004). ● Op-amp: another kind of linear amplifier A. A. Clerk et al., Rev. Mod. Phys. 82, 1155 (2010).

  7. Phase-preserving linear amplifiers Microwave-frequency amplifiers using superconducting technology are approaching quantum limits and are being used as linear detectors in photon-coherence experiments. This requires more than second moments of amplifier noise. What about nonGaussian added noise? What about higher moments of added noise? THE PROBLEM What are the quantum limits on the entire distribution of added noise?

  8. Initial coherent state

  9. Ideal amplification of initial coherent state

  10. NonGaussian amplification of initial coherent state Are these legitimate linear amplifiers?

  11. II. Quantum limits on noise in phase-preserving linear amplifiers. The whole story Per-gyr falcon Jornada del Muerto New Mexico Harris hawk Near Bosque del Apache New Mexico

  12. What is a phase-preserving linear amplifier? Amplification of input coherent state Smearing probability distribution. Smears out the amplified coherent state and includes amplified input noise and added noise. For coherent-state input, it is the P function of the output. THE PROBLEM Given that the amplifier map must be physical (completely positive), what are the quantum restrictions on the smearing probability distribution?

  13. Addressing the problem Tack 1 This is hopeless.

  14. Addressing the problem Tack 2 But we have no way to get from this to the standard input-output relation and thus no way to untangle the primary mode from the ancilla and to derive constraints on the smearing distribution. Our definition of a linear amplifier appears to be strictly weaker than the standard input-output relation.

  15. Addressing the problem Tack 3

  16. Addressing the problem Tack 3 THE PROBLEM TRANSFORMED Given that the amplifier map must be physical (completely positive), what are the quantum restrictions on the ancillary mode’s initial “state” σ?

  17. Addressing the problem Tack 3 THE ANSWER Any phase-preserving linear amplifier is equivalent to a two-mode squeezing paramp with the smearing function being a rescaled Q function of a physical initial state σ of the ancillary mode.

  18. NonGaussian amplification of initial coherent state To IV

  19. Quantum limits on phase-preserving linear amplifiers The problem of characterizing an amplifier’s performance, in absolute terms and relative to quantum limits, becomes a species of “indirect quantum-state tomography” on the effective, but imaginary ancillary-mode state σ.

  20. Completely positive maps and • physical ancilla states Western diamondback rattlesnake My front yard, Sandia Heights

  21. When does the ancilla state have to be physical? Z. Jiang, M. Piani, and C. M. Caves, arXiv:1203.4585 [quant-ph]. (orthogonal) Schmidt operators

  22. When does the ancilla state have to be physical?

  23. Why does the ancilla state for a linear amplifier have to be physical? To End

  24. IV. Nondeterministic linear amplifiers On top of Sheepshead Peak, Truchas Peak in background Sangre de Cristo Range Northern New Mexico

  25. Nondeterministic linear amplifier Original idea (Ralph and Lund): When presented with an input coherent state, a nondeterministic linear amplifier amplifies immaculately with probability p and punts with probability 1 – p. T. C. Ralph and A. P. Lund, in QCMC, edited by A. Lvovsky (AIP, 2009), p. 155. . This is an immaculate linear amplifier, more perfect than perfect; it doesn’t even have the amplified input noise. If the probability of working is independent of input and the amplifier is described by a phase-preserving linear-amplifier map when it does work, then the success probability is zero, unless when it works, it is a standard linear amplifier, with the standard amount of noise.

  26. Nondeterministic linear amplifier phase-preserving symmetry Projector onto subspace of first N + 1 number states

  27. Nondeterministic linear amplifier independent of k

  28. Nondeterministic linear amplifier

  29. Nondeterministic linear amplifiers The bottom line

  30. Nondeterministic linear amplifiers The bottom line Lessons from the bottom line ● These should not be thought of as amplifiers. They are useful for other jobs, where something more immaculate than perfection, even though rarely attained, is valued more highly than partial achievement, but they are not useful for amplification. ● Results on nondeterministic devices should always be assessed in the light of success probabilities.

  31. That’s it, folks! Thanks for your attention. Echidna Gorge Bungle Bungle Range Western Australia

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