From blockade to transparency: Controllable photon transmission through a circuit-QED system

Yu-xi Liu, Xun-Wei Xu, Adam Miranowicz, and Franco Nori

Phys. Rev. A 89, 043818 – Published 14 April 2014

Abstract

A strong photon-photon nonlinear interaction is a necessary condition for a photon blockade. Moreover, this nonlinearity can also result in a bistable behavior in the cavity field. We analyze the relation between the detecting field and the photon blockade in a superconducting circuit-QED system, and show that a photon blockade cannot occur when the detecting field is in the bistable regime. This photon blockade is the microwave-photonics analog of the Coulomb blockade. We further demonstrate that the photon transmission through such a system can be controlled (from photon blockade to transparency) by the detecting field. Numerical calculations show that our proposal is experimentally realizable with current technology.

Received 31 January 2014

DOI:https://doi.org/10.1103/PhysRevA.89.043818

Authors & Affiliations

Yu-xi Liu^1,2,3,*, Xun-Wei Xu¹, Adam Miranowicz^3,4, and Franco Nori^3,5

¹Institute of Microelectronics, Tsinghua University, Beijing 100084, China
²Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing 100084, China
³CEMS, RIKEN, Saitama 351-0198, Japan
⁴Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
⁵Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*yuxiliu@mail.tsinghua.edu.cn

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Vol. 89, Iss. 4 — April 2014

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Figure 1
Schematic diagram for the circuit-QED system of a superconducting charge qubit (denoted by the small purple square) coupled to a transmission line resonator, indicated by the black dashed box. The input (output) is denoted by the arrows going to (the arrow leaving from) the black dashed box. We assume that the input field, including the classical driving field $E (t)$ and the vacuum noise $a_{in} (t)$ , is applied to the cavity at the left port. The output field is measured at the right port.
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Figure 2
(a) Eigenvalues $E_{k} (n_{c})$ (green curves) of the free Hamiltonian $ℏ χ (n - n_{c})$ in Eq. (7) vs the rescaled detuning $n_{c} = (χ - Δ_{d}) / (2 χ)$ for the photon numbers $n = 0$ and $n = 1$ inside the cavity, in energy units of $ℏ χ$ . Blue curves denote the eigenvalues $E_{0} (n_{c})$ and $E_{1} (n_{c})$ of the ground and first excited states for the Hamiltonian in Eq. (7) vs the rescaled detuning $n_{c}$ near the point $n_{c} = 0.5$ , with an effective Hamiltonian $ℏ χ (n_{c} - 0.5) (| 1 〉 〈 1 | - | 0 〉 〈 0 |) + ℏ (Ω | 1 〉 〈 0 | + Ω^{*} | 0 〉 〈 1 |)$ , when the external field is included. The degeneracy of the two eigenvalues in the free Hamiltonian is lifted at the point $n_{c} = 0.5$ , with distance $ℏ | Ω |$ , by the external field. All dotted-dashed curves mean $n_{c} > 0.5$ , however, solid curves mean $n_{c} < 0.5$ . (b) Several eigenvalues $E_{k} (n_{c})$ (up to four) of the Hamiltonian in Eq. (7) vs the detuning $n_{c}$ for the ratio $χ / | Ω | = 10$ . (c) The mean photon number in the ground state $| ψ_{0} 〉$ of the Hamiltonian in Eq. (7) vs the detuning $n_{c}$ for $χ / | Ω | = 1$ (green dashed-dotted curve), 10 (red dashed curve), and 100 (blue solid curve). This is the photon analog of the Coulomb staircase.
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Figure 3
Logarithm $\log (| A_{s} |)$ of the steady-state solution $| A_{s} |$ vs the strength of the driving field $| Ω |$ , for different detunings $Δ_{d} / 2 π$ equal to (a) $- 1$ MHz (solid black), 0 MHz (dashed red), and 1 MHz (dotted-dashed green), and (b) 36 MHz (solid black), 37 MHz (dashed red), and 38 MHz (dotted-dashed green curves). Here, $N_{up}$ (double-dotted-dashed blue lines) is the upper-bound photon number for the photon blockade, $Δ / 2 π = 1$ GHz, $κ / 2 π = 0.1$ MHz, and $g / 2 π = 200$ MHz. For the above given parameters, $| A_{s} |$ has two stable states when $9.1 kHz < Δ_{d} < 36.95 MHz$ .
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Figure 4
(a) Coherence $g^{(2)} (0)$ vs the driving field strength $| Ω |$ . (b) $g^{(2)} (τ)$ vs the delay time $τ$ , where the solid black and dashed red curves correspond to the driving strengths $| Ω | / 2 π = 0.01$ and 1 MHz, respectively. Also, $Δ_{d} = 38.5$ MHz, $Δ / 2 π = 1$ GHz, $g / 2 π = 200$ MHz, and $κ / 2 π = 0.1$ MHz. Also, $g^{(2)} (0)$ vs the detuning $Δ_{d}$ in (c) for $g / 2 π = 100$ MHz and (d) for $g / 2 π = 200$ MHz, with $Δ / 2 π = 1$ GHz, $| Ω | / 2 π = 0.01$ MHz, and $κ / 2 π = 0.1$ MHz. For the parameters given in (c) and (d), the condition of the stable states for the driving field is $18 kHz < Δ_{d} < 9.627 MHz$ and $9.1 kHz < Δ_{d} < 36.95 MHz$ , respectively.
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Figure 5
The real $A_{R} (ω^{'})$ and imaginary $A_{I} (ω^{'})$ parts $A_{in} (ω^{'}) / ξ$ as a function of $ω^{'} = ω_{d} - ω$ are plotted in (a) and (b), respectively, for the detuning $Δ_{d} / 2 π$ equal to 9.74 MHz (solid black curves) and 9.96 MHz (dashed red curves) assuming $Δ / 2 π = 1$ GHz, $g / 2 π = 100$ MHz, and $κ / 2 π = | Ω | / 2 π = 0.1$ MHz.
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