Time-delayed quantum coherent Pyragas feedback control of photon squeezing in a degenerate parametric oscillator

Manuel Kraft, Sven M. Hein, Judith Lehnert, Eckehard Schöll, Stephen Hughes, and Andreas Knorr

Phys. Rev. A 94, 023806 – Published 1 August 2016

Abstract

Quantum coherent feedback control is a measurement-free control method fully preserving quantum coherence. In this paper we show how time-delayed quantum coherent feedback can be used to control the degree of squeezing in the output field of a cavity containing a degenerate parametric oscillator. We focus on the specific situation of Pyragas-type feedback control where time-delayed signals are fed back directly into the quantum system. Our results show how time-delayed feedback can enhance or decrease the degree of squeezing as a function of time delay and feedback strength.

Received 24 March 2016

DOI:https://doi.org/10.1103/PhysRevA.94.023806

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Coherent control Nonlinear optics Optical parametric oscillators & amplifiers Quantum control Quantum fluctuations & noise

Atomic, Molecular & OpticalQuantum Information, Science & TechnologyNonlinear Dynamics

Authors & Affiliations

Manuel Kraft^1,*, Sven M. Hein¹, Judith Lehnert², Eckehard Schöll², Stephen Hughes³, and Andreas Knorr¹

¹Technische Universität Berlin, Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Hardenbergstraße 36, 10623 Berlin, Germany
²Technische Universität Berlin, Institut für Theoretische Physik, Nichtlineare Dynamik und Kontrolle, Hardenbergstraße 36, 10623 Berlin, Germany
³Department of Physics, Engineering Physics and Astronomy, Queen's University, Kingston, Ontario K7L 3N6, Canada

^*kraft@itp.tu-berlin.de

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Vol. 94, Iss. 2 — August 2016

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Figure 1
Schematic of our physical setup and model. A degenerate parametric oscillator (DPO) is embedded in a cavity and driven by a laser with pumping strength $| ε |$ . The cavity mirror $M_{2}$ with loss rate $γ_{2}$ is coupled to an external reservoir denoted by the input $b_{in}^{2}$ and output $b_{out}^{2}$ . (a) The mirror $M_{1}$ is coupled to a photonic waveguide terminated by a mirror at a distance $L / 2$ from the coupling point. The waveguide mirror will introduce a phase shift $ϕ = π$ for the reflected field and time-delayed feedback into the system. For the case of imperfect coupling to the waveguide, we additionally implement another loss reservoir, graphically denoted by the blue cloud and coupling strength $γ_{3}$ . (b) Instead of a waveguide, we let the outgoing radiation from $M_{1}$ directly emit onto an external mirror.
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Figure 2
Stability analysis in the case of constructive (a) and destructive (b) interference as a function of $γ_{1} τ$ and the effective pump strength $\tilde{α} = [| ε | - 0.5 (γ_{2} + γ_{3})] / γ_{1}$ , Eq. (30). The stable regions are the blue (shaded) ones, where $S_{W}^{1} < 0$ . (a) Constructive interference $[ω_{0} τ = (2 n - 1) π]$ . The boundary line between the stable and unstable region is first constant with $\tilde{α} = 2$ for $γ_{1} τ < 1$ . At $γ_{1} τ = 1$ there is a stability change. From that point, the stability boundary decreases with growing $γ_{1} τ > 1$ toward $\tilde{α} \to 0$ . This means that for longer delays the stability becomes independent of the coupling $γ_{1}$ and of the delay time $τ$ . We call the region where $γ_{1} τ < 1$ the short-delay regime and the region $γ_{1} τ > 1$ the long-delay regime. The dashed line marks the boundary between the short and long feedback regions. The point $P$ represents a fixed value of $| ε | - (γ_{2} + γ_{3}) / 2$ , which is investigated in Fig. 4. (b) Destructive interference $[ω_{0} τ = (2 n - 1) π]$ . The dotted line marks the stability boundary given by Eq. (33), below which the system is guaranteed to be stable independent of delayed feedback effects. We observe that as long $\tilde{α} < 0$ , i.e., $| ε | < (γ_{2} + γ_{3}) / 2$ , the system is stable.
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Figure 3
Long-delay regime. Maximal squeezing spectra (in a frame rotating with $ω_{0}$ ) at the $M 2$ output with and without feedback at threshold (in the feedback case) $| ε | = γ_{2} / 2$ in the cases $ω_{0} τ = 2 n π$ and $ω_{0} τ = (2 n - 1) π$ for $τ \neq 0$ . The parameters are $S = 0.5, γ_{1} = γ_{2} = 2 {ns}^{- 1}, | ε | = 1 {ns}^{- 1}$ , and $γ_{3} = 0$ (ideal feedback coupling). The spectrum is highly structured as a function of $ω$ and $τ$ . For the case of destructive interference ( $ω_{0} τ = 2 n π$ ) maximal squeezing is achievable at threshold at the central frequency $ω_{0}$ ( $ω = 0$ in the plots). On the other hand, constructive interference cannot enhance the squeezing at $ω_{0}$ . The side peaks show that the feedback scheme shifts the frequencies where squeezing occurs; however, it is never maximal for other frequencies than $ω_{0}$ . The case without feedback corresponds to a double-sided lossy cavity as discussed in Ref. [29] with respective loss rates $γ_{1} = γ_{2} = 2 {ns}^{- 1}$ pumped below threshold at $| ε | = 1 {ns}^{- 1}$ .
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Figure 4
Short-delay regime. Comparison of the squeezing spectra $P_{1}$ and $P_{2}$ (in a frame rotating with $ω_{0}$ ) in the case of constructive interference, i.e., $ω_{0} τ = (2 n - 1) π$ for $τ \neq 0$ . The parameters are $S = 0.1, γ_{1} = 2.75 {ns}^{- 1}, γ_{2} \in {0.5, 3, 9} {ns}^{- 1}$ , and a fixed loss-pump strength $| ε | - γ_{2} / 2 = 5 {ns}^{- 1}$ . The situation corresponds to the stable point denoted by $P_{1}$ in Fig. 2. We see that while $γ_{2}$ increases, the squeezing in the output of the waveguide is diminished, whereas the squeezing at the mirror $M 2$ output is enhanced. Note that for the present parameters the uncontrolled case ( $γ_{1} = 0$ ) is not stable and therefore cannot be shown (stability would require $| ε | < γ_{2} / 2$ ).
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Figure 5
Long-delay regime. Maximal squeezing spectra (in a frame rotating with $ω_{0}$ ) at the waveguide output with and without feedback at threshold $| ε | = γ_{2} / 2$ in the cases $ω_{0} τ = 2 n π$ and $ω_{0} τ = (2 n - 1) π$ for $τ \neq 0$ . The parameters are $S = 0.5, γ_{1} = γ_{2} = 2 {ns}^{- 1}, | ε | = 1 {ns}^{- 1}$ , and $γ_{3} = 0$ (ideal feedback coupling). The spectrum is highly structured as a function of $ω$ and $τ$ . For the case of destructive interference ( $ω_{0} τ = 2 n π$ ) only noise remains as an output at threshold at the central frequency $ω_{0}$ ( $ω = 0$ in the plots), and thus $P_{1} (ω_{0}) = 1$ corresponding to the quantum noise limit and no squeezing is achieved. On the other hand, for constructive interference we observe squeezing of the output field at $ω_{0}$ . However, for long delays $τ$ in both cases $γ_{2} / 2 > | ε |$ is required for stability, which decreases squeezing below $50 %$ . For comparison, the green dashed line marks the squeezing spectrum for the case in which the feedback channel is replaced by a Markovian reservoir with loss rates $γ_{1} = γ_{2} = 2 {ns}^{- 1}$ pumped below threshold at $| ε | = 1 {ns}^{- 1}$ .
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Figure 6
Our feedback scheme. Top: A system is coherently driven in a cascaded fashion by a past version of itself via the feedback in-loop field $b_{in, 2}^{1} (t) = e^{i ϕ} b_{out, 1}^{1} (t - τ)$ . The additional input $b_{in}^{2}$ takes account of additional drives or losses of the system independent of $b_{in}^{1}$ . Left bottom: Same physical setup represented by a loop. Right bottom: The in-loop field can be eliminated; its only purpose is to modify the internal dynamic. Note that $b_{out}^{1}$ never influences the system again.
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