Switching on and off of ultrastrong light-matter interaction: Photon statistics of quantum vacuum radiation

L. Garziano, A. Ridolfo, R. Stassi, O. Di Stefano, and S. Savasta

Phys. Rev. A 88, 063829 – Published 17 December 2013

Abstract

We study the extracavity quantum vacuum radiation generated by a sudden switch on and off of the vacuum Rabi frequency of a single two-level system coupled to a single cavity mode in the strong and ultrastrong coupling regime. Dissipation is described by including the qubit-resonator coupling in the derivation of the master equation. The output stream of photons and their second-order correlation functions are calculated by exploiting a recently developed input-output formulation of resonators valid for arbitrary light-matter couplings. The obtained output photon correlations provide direct information on the quantum correlations among the virtual excitations of the dressed vacuum.

Received 15 April 2013

DOI:https://doi.org/10.1103/PhysRevA.88.063829

Authors & Affiliations

L. Garziano¹, A. Ridolfo², R. Stassi¹, O. Di Stefano¹, and S. Savasta¹

¹Dipartimento di Fisica e Scienze della Terra, Università di Messina Viale F. Stagno d'Alcontres 31, I-98166 Messina, Italy
²Physik Department, Technische Universität München, James-Franck-Strasse, 85748 Garching, Germany

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Vol. 88, Iss. 6 — December 2013

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Images

Figure 1
Energy spectrum of $H_{R}$ as function of the coupling strength in the nondispersive regime $(Δ = ω_{c} - ω_{e g} = 0)$ for the Jaynes-Cummings model (dashed red line) and the Rabi Hamiltonian (solid blue line) containing the nonrotating terms.
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Figure 2
Logarithmic plot of time evolution of the intracavity mean photon number $〈 X^{-} X^{+} 〉$ calculated for different coupling strengths $Ω_{R} / ω_{c} = 0.1$ (solid red line), 0.2 (dashed blue line), 0.5 (dotted green line), 0.9 (dot-dashed black line). Larger signals correspond to larger couplings. The typical exponential decay is due to the cavity losses described by the standard master equation.
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Figure 3
Probability distribution $P_{n}$ for the ground state $| \tilde{G} 〉$ to contain an $n$ -photon state calculated for different coupling strengths $Ω_{R} / ω_{c}$ . It can be observed that the main contribution to the dressed ground state generally comes from the $| g 0 〉$ state (with values of $P_{0}$ close to unity) for a wide range of values of $Ω_{R} / ω_{c}$ . For very high values of the coupling strength, however, the system presents indeed a different behavior and the main contribution to the ground state $| \tilde{G} 〉$ comes from the two-photon state $| g 2 〉$ . Once the interaction is switched off, the probability distribution $P_{n}$ of virtual photons in the ground state $| \tilde{G} 〉$ converts to physical photons that can be detected.
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Figure 4
Temporal evolution of the mean cavity number of physical photons $〈 X^{-} X^{+} 〉$ for different values of the coupling strength $Ω_{R} / ω_{c} = 0.1$ (a), 0.2 (b), 0.3 (c), 0.5 (d). Parameters are: $ω_{c} = ω_{e g}; γ_{a} = γ_{x} = 0.01 ω_{c}$ . For $t < t_{0}$ no emission is observed, the system being in its ground state $| g 0 〉$ . Once the interaction has been switched on, we observe a finite emission of physical photons corresponding to an actual output photon rate $γ_{a} 〈 X^{-} X^{+} 〉$ , which can be experimentally detected. In the Bloch-Siegert regime (a) and (b), dynamics of released cavity photons will originate from the superposition of states $| 3 〉$ and $| 4 〉$ , which will give rise to typical Rabi oscillations. As shown in Figs. 4 and 4, for higher values of the coupling strength ( $Ω_{R} \geq 0.3 ω_{c}$ ) a more complex dynamics can be observed. As contributions from higher states transitions become relevant, ${〈 X^{-} X^{+} 〉}_{t}$ does not exhibit an exact periodicity at the transition frequency $Δ ω_{43} \equiv ω_{4} - ω_{3}$ but is now a sum of oscillatory terms with different frequencies. These contributions become more relevant as the coupling strength increases.
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Figure 5
Temporal evolution of the correlation function $G (t, t)$ (solid blue line) and the cross-correlation function $G_{cross} (t, t)$ (dashed red line) for different values of the coupling strength $Ω_{R} / ω_{c} = 0.01$ (a), 0.1 (b), 0.2 (c), 0.3 (d). For weak values of the coupling strength (a) it can be observed that for $t \to t_{0}^{+}$ the cross-correlation function $G_{cross} (t, t) = 1$ while the corresponding value of $G (t, t)$ is zero. Due to the perturbation $V ≅ ℏ g (a σ_{-} + a^{†} σ_{+})$ the system, initially prepared in the bare ground state, undergoes the transition $| g 0 〉 \to V | g 0 〉 = | e 1 〉$ . In this case, then, no pairs of photon can be released $(G (t, t) = 0)$ and the value $G_{cross} (t, t) = 1$ is the signature of the presence of $| e 1 〉$ . As the system undergoes Rabi oscillations between the two states $| e 1 〉$ and $| g 2 〉$ we observe that when the system is exactly in the $| g 2 〉$ state $G (t, t) = 1$ while the cross correlation goes to zero. Figures 5–5 show that in the ultrastrong-coupling regime both correlation functions present a different behavior. As the contributions from the upper dressed states become more relevant, we observe that $G (t, t) \neq 0$ even for $t \to t_{0}^{+}$ and reaches values larger than one. For the same reason the minima of the cross-correlation function never goes to zero due to the non-negligible contribution of the states $| 7 〉, | 8 〉$ .
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Figure 6
Temporal evolution of the equal-time normalized second-order correlation function $g^{(2)} (t, t)$ for different values of the coupling strength $Ω_{R} / ω_{c} = 0.01$ (a), 0.1 (b), 0.2 (c), 0.3 (d). When the ratio $Ω_{R} / ω_{c}$ is small (a), $g^{(2)} (t, t)$ is exactly zero immediately after the switch on confirming the fact that the system is essentially in a superposition of the states $| g 0 〉, | e 1 〉$ so that no pairs of photons can be released at $t = 0$ . A maximum is observed when the system is in the $| g 2 〉$ state where the second-order correlation function certifies a highly super-Poissonian statistics. Figures 6–6 show that in the ultrastrong-coupling regime, where the contribution from the upper dressed states becomes more relevant, there is a finite possibility for photons to be emitted in pairs even immediately after the switch on of the interaction. The more complex dynamics observed for increasing value of the coupling strength originates from contributions from higher states transitions.
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