State-dependent photon blockade via quantum-reservoir engineering

An arbitrary initial state of an optical or microwave field in a lossy driven nonlinear cavity can be changed into a partially incoherent superposition of only the vacuum and the single-photon states. This effect is known as single-photon blockade, which is usually analyzed for a Kerr-type nonlinear cavity parametrically driven by a single-photon process assuming single-photon loss mechanisms. We study photon blockade engineering via a nonlinear reservoir, i.e., a quantum reservoir, where only two-photon absorption is allowed. Namely, we analyze a lossy nonlinear cavity parametrically driven by a two-photon process and allowing two-photon loss mechanisms, as described by the master equation derived for a two-photon absorbing reservoir. The nonlinear cavity engineering can be realized by a linear cavity with a tunable two-level system via the Jaynes-Cummings interaction in the dispersive limit. We show that by tuning properly the frequencies of the driving field and the two-level system, the steady state of the cavity field can be the single-photon Fock state or a partially incoherent superposition of several Fock states with photon numbers, e.g., (0,2), (1,3), (0,1,2), or (0,2,4). At the right (now fixed) frequencies, we observe that an arbitrary initial coherent or incoherent superposition of Fock states with an even (odd) number of photons is changed into a partially incoherent superposition of a few Fock states of the same photon-number parity. We find analytically approximate formulas for these two kinds of solutions for several differently tuned systems. A general solution for an arbitrary initial state is a weighted mixture of the above two solutions with even and odd photon numbers, where the weights are given by the probabilities of measuring the even and odd numbers of photons of the initial cavity field, respectively. This can be interpreted as two separate evolution-dissipation channels for even and odd-number states. Thus, in contrast to the standard predictions of photon blockade, we prove that the steady state of the cavity field, in the engineered photon blockade, can depend on its initial state. To make our results more explicit, we analyze photon blockades for some initial infinite-dimensional quantum and classical states via the Wigner and photon-number distributions.

Figure 1

Model 1: Dissipation-free evolution of the photon-number probabilities $p_n\left(t\right)=|\langle n|{\psi }_{02}^{\left(m\right)}\left(t\right)\rangle {|}^2$ and the photon-blockade fidelities $F\left(t\right)={\sum }_n{p}_n\left(t\right)$ for the Hamiltonian ${\widehat{H}}_{02}$, given by Eq. (14), for several initial Fock states $|m\rangle$ (as indicated in the panel titles). We set $\epsilon =\chi /6=5$. Panels (a) and (c) show Rabi-type oscillations between the levels $|0\rangle$ and $|2\rangle$, if at least one of them is initially populated. Rabi-type oscillations are not observed if the other levels are the only initially populated states, such as $|1\rangle$ and $|3\rangle$, as shown in panels (b) and (d), respectively. These results have a simple physical explanation in terms of the resonances shown in Fig. 2.

Figure 2

Model 1: Explanation of the occurrence of photon blockade via the energy levels of the Hamiltonian $\widehat{H}$ corresponding to ${\widehat{H}}_{\mathrm{s}}({\Omega }_{02}=\Omega ,{\Sigma }_{02}=0)$, given by Eq. (12), in the limit of a very small driving strength, $\epsilon \ll \chi$. The Kerr-nonlinear term, proportional to $\chi$, changes the harmonic spectrum (shown in the left spectrum) into an anharmonic non-equidistant one (right side) with $E_{n+1}-E_n\ne$const., where $E_n=n\hslash \Omega +n(n-2)\hslash \chi$ (with $n=0,1,...)$ are the eigenvalues of the Hamiltonian $\widehat{H}$. It is seen that the two-photon transitions between the levels $|0\rangle$ and $|2\rangle$ (shown by a solid double arrow in the right spectrum) can be induced by a driving field with frequency ${\omega }_{\mathrm{d}}=(E_2-E_0)/\hslash =2\Omega$, which is the same as for the harmonic system. The other transitions between the levels, e.g., $|1\rangle$ and $|3\rangle$, as well as $|2\rangle$ and $|4\rangle$ (as shown by the dashed double arrows) are off-resonance with $\Omega$ or its multiples.

Figure 3

Model 2: Same as in Fig. 1, but for the probabilities $p_n\left(t\right)=|\langle n|{\psi }_{13}^{\left(m\right)}\left(t\right)\rangle {|}^2$ obtained for the Hamiltonian ${\widehat{H}}_{13}$, given by Eq. (17) with $k=1,l=3$. Panel (a) [(b) and (d)] shows Rabi-type oscillations between the levels $|0\rangle$ and $|4\rangle$ ($|1\rangle$ and $|3\rangle$), if at least one of these levels is initially populated. Rabi-type oscillations are not observed if the other levels are the only ones, which are initially populated, such as $|2\rangle$, shown in panel (c). The physical meaning of these results, analogously to those in Fig. 1, can be simply understood in terms of the resonances shown in Fig. 4.

Figure 4

Model 2: Same as in Fig. 2, but for the Hamiltonian $\widehat{H}$ corresponding to ${\widehat{H}}_{\mathrm{s}}({\Omega }_{13}=\Omega ,{\Sigma }_{13}=0)$, given by Eq. (15) with $k=1,l=3$ assuming $\epsilon \ll \chi$. Here the two-photon (four-photon) transitions between the levels $|1\rangle$ and $|3\rangle$ ($|0\rangle$ and $|4\rangle$), shown by solid double arrows in the right spectrum, can be induced by a driving field with frequency ${\omega }_{\mathrm{d}}=(E_3-E_1)/\hslash =2\Omega$ [${\omega }_{\mathrm{d}}=(E_4-E_0)/\hslash =4\Omega$], which are the multiples of the same frequency $\Omega$ of the harmonic system.

Figure 5

An intuitive explanation of the engineered photon blockades in models 1 and 2 with two-photon dissipation, and the standard photon blockade in model 3 with single-photon dissipation. The diagrams schematically show the energy levels of three Kerr-type nonlinear systems driven by a classical field with frequency ${\omega }_{\mathrm{d}}$ in resonance with the desired transitions, as shown in Fig. 2 for model 1 and Fig. 4 for model 2. The red ellipses with arrows describe these [(a),(b)] two-photon and (c) single-photon drivings, together with the Rabi-type oscillations between the corresponding levels. The systems are described by the Hamiltonians (a) ${\widehat{H}}_{02}$, given by Eq. (14), (b) ${\widehat{H}}_{13}$, given by Eq. (17) for $k=1,l=3$, and ${\widehat{H}}_{\mathrm{usual}}$, given by Eq. (20). The system dissipation is governed by the master equations describing either [(a), (b)] two-photon or (c) single-photon absorption for $\gamma \ll \epsilon \ll \chi$. The numerous green single arrows pointing down describe these dissipations (absorptions). These figures intuitively explain the occurrence of several kinds of the engineered photon blockades, as well as two independent evolutions of the initial Fock states with even and odd numbers of photons for models 1 and 2. This implies that the engineered PB effects in panels (a) and (b) can depend on the initial state of a cavity, although in a limited way, as they depend solely on the ratio of the probabilities of measuring the photon numbers of different parity. In contrast to this, the steady state generated in standard PB, as shown in (c), is independent of the cavity initial state.

Figure 6

Model 1 with two-photon dissipation: (a), (c) The Wigner functions $W\left(\beta \right)$ and (b), (d) the photon-number probabilities $p_n$ for the steady-state solutions ${\widehat{\rho }}_{\mathrm{ss}}^{02}$ of the master equation (26) with the Hamiltonian ${\widehat{H}}_{02}$, given by Eq. (14), assuming the cavity field to be initially in an arbitrary [(a), (b)] even- or [(c), (d)] odd-number state. We set the ratios of the driving field strength $\epsilon$ and the Kerr nonlinear coupling $\chi$, and of the damping constant $\gamma$ and $\epsilon$ to be small and equal to $\delta =\epsilon /\chi =1/6$ and ${\delta }^{\prime }=\gamma /\epsilon =1/25$. The color codes in panel (c) (and all other figures of the Wigner functions) are the same as in panel (a). Note that the negative regions of the Wigner functions are marked in blue.

Figure 7

Model 2 with two-photon dissipation: Same as in Fig. 6, but for the steady-state solutions ${\widehat{\rho }}_{\mathrm{ss}}^{13}$ of the master equation (26) with the Hamiltonian ${\widehat{H}}_{13}$, given by Eq. (17) for $k=1,l=3$.

Figure 8

Model 1 with two-photon dissipation: The photon-number probabilities $p_n=\langle n|{\widehat{\rho }}_{\mathrm{ss}}^{02}|n\rangle$ and the fidelity $F=p_0+p_2$ of the photon blockade versus the driving field strength $\epsilon$, in units of the damping constant $\gamma$, assuming the initial state to have an even number of photons and $\gamma /\chi =1/150$. The analogous figure for an initial odd-number state is omitted since $p_1\approx 1$ [see Fig. 6], at least, for $\epsilon /\gamma \in [0,10]$. For brevity, analogous plots for model 2 are not presented here either.

Figure 9

Model 1 in panel (a) and model 2 in (b), (c) with two-photon dissipation: The photon-number probabilities $p_n$ and the fidelity $F={\sum }_n{p}_n$ of the photon blockade versus the tuning frequency ${\Omega }_{kl}$ for the steady-state solutions ${\widehat{\rho }}_{\mathrm{ss}}$ of the master equation (26), for the Hamiltonian ${\widehat{H}}_{\mathrm{s}}({\Omega }_{kl},{\Sigma }_{kl})$ with fixed ${\Sigma }_{kl}=0$, assuming the initial state to have [(a), (b)] an even or (c) odd number of photons. Here we set $\delta =\epsilon /\chi =1/6$ and ${\delta }^{\prime }=\gamma /\epsilon =1/25$. The corresponding curves $p_n$ versus ${\Omega }_{02}$ for ${\widehat{H}}_{\mathrm{s}}({\Omega }_{02},{\Sigma }_{02}=0)$ and the initial state with odd number of photons are not presented here because $p_1\approx 1$ and $p_3\approx 0$ in the whole studied interval, and this is fully apparent from Fig. 6 as well. It is seen that even if ${\Omega }_{kl}\ne 0$, PB can still occur. Nevertheless, for an initial even (odd) number state, the output steady state approaches the vacuum (single-photon) state even for a relatively small ${\Omega }_{kl}$. Note that plots in panels (a) and (c) look very similar but they correspond to different probabilities.

Figure 10

Model 1 with two-photon dissipation: Same as in Fig. 6, but for different initial states: (a), (b) coherent state $|\alpha =3/4\rangle$, with $p_0\approx p_1\approx p_2$, (c), (d) coherent state $|\alpha =2\rangle$, with $p_1>p_0\approx p_2$, and (e), (f) the cat state $|\alpha =2,\varphi =\pi /4\rangle$, with $p_1<p_0\approx p_2$.

Figure 11

Model 2 with two-photon dissipation: Same as in Fig. 10, but for the steady-state solutions ${\widehat{\rho }}_{\mathrm{ss}}^{13}$ of the master equation (26) with the Hamiltonian ${\widehat{H}}_{13}$, given by Eq. (17).

Figure 12

Model 1 with two-photon dissipation: Photon-number probabilities $p_n=\langle n|{\widehat{\rho }}_{\mathrm{ss}}^{02}|n\rangle$ and ratio $r$ versus mean photon number $\langle n\rangle$ for different initial cavity fields: (a) chaotic state ${\widehat{\rho }}_{\mathrm{ch}}$ and (b) single-photon-added chaotic state ${\widehat{\rho }}_{{}_{\mathrm{AT}}}$.

Figure 13

Model 1 with two-photon dissipation: The figure shows how the steady state ${\widehat{\rho }}_{\mathrm{ss}}^{02}$ in the engineered photon blockade depends on initial cavity field $|{\psi }_0\rangle$. The photon-number probabilities $p_n=\langle n|{\widehat{\rho }}_{\mathrm{ss}}^{02}|n\rangle$ (for $n=0,1,2$) and the ratio $r=p_{\mathrm{odd}}\left(\right|{\psi }_0\rangle )/p_{\mathrm{even}}({\psi }_0\rangle )$ versus real amplitude $\alpha$ for various initial states $|{\psi }_0\rangle$: (a) coherent state $|\alpha \rangle$ or, equivalently, the Yurke-Stoler cat state $|{\alpha }_{{}_{\mathrm{YS}}}\rangle =|\alpha ,\varphi =\pi /2\rangle$, (b) the cat state $|\alpha ,\varphi =\pi /4\rangle$, and (c) the ideal squeezed state $|\alpha ,\xi =1/2\rangle$, and [(d)–(f)] the displaced number states $|\alpha ,n_0\rangle$ with $n_0=1,2,3$. The steady-state solutions ${\widehat{\rho }}_{\mathrm{ss}}^{02}$ of the master equation (26) for the Hamiltonian ${\widehat{H}}_{02}$ are obtained assuming $\delta =\epsilon /\chi =1/6$ and ${\delta }^{\prime }=\gamma /\epsilon =1/25$.

Figure 14

Models 3 in panels (a), (b) and 5 in (c), (d) with single-photon dissipation for any initial states: Wigner functions $W\left(\beta \right)$ and photon-number probabilities $p_n$ for the steady-state solutions of the master equation (27) with ${\gamma }_2={\gamma }_{\perp }=0$ for [(a), (b)] the single-photon driven Hamiltonian ${\widehat{H}}_{\mathrm{usual}}$, given by Eq. (20), and [(c), (d)] the two-photon driven Hamiltonian ${\widehat{H}}_{01}$, given by Eq. (17) with $k=0,l=1$. We set $\delta =\epsilon /\chi =1/6$ and ${\gamma }_1/\epsilon =1/25$. Model 3 corresponds to the standard description of photon blockade. It is worth noting that the same solutions, as shown in panels (a), (b), can be obtained for the two-photon absorption master equation, given by Eq. (26) with ${\delta }^{\prime }=\gamma /\epsilon =1/25$. This case is referred to as model $3^{\prime }$ in Table 1.

Figure 15

Models 1 (a) and 2 [(b)–(d)]: Dissipative evolutions of the photon-number probabilities $p_n\left(t\right)$ and the photon-blockade fidelities $F\left(t\right)={\sum }_n{p}_n\left(t\right)$ for several initial Fock states $|m\rangle$ (as indicated in the panel titles) assuming $\delta =\epsilon /\chi =1/6$, ${\delta }^{\prime }=\gamma /\epsilon =1/25$, and $\epsilon =5$. The decay of Rabi-type oscillations is clearly seen. Panels (a) and (b), (c), (d) should be compared with Figs. 1 and 2, 2, 2 demonstrating the corresponding dissipation-free evolutions in models 1 and 2, respectively. It is seen that the decaying states rapidly approach the steady states shown in Figs. 6 and 7.