Conditional quantum-state engineering using ancillary squeezed-vacuum states

We investigate an optical scheme to conditionally engineer quantum states using a beam splitter, homodyne detection, and a squeezed vacuum as an ancillar state. This scheme is efficient in producing non-Gaussian quantum states such as squeezed single photons and superpositions of coherent states (SCSs). We show that a SCS with well defined parity and high fidelity can be generated from a Fock state of $n⩽4$, and conjecture that this can be generalized for an arbitrary $n$ Fock state. We describe our experimental demonstration of this scheme using coherent input states and measuring experimental fidelities that are only achievable using quantum resources.

Figure 1

(Color online) (a) Schematic of the post-selection protocol. ${\widehat{X}}_{\mathrm{in}}^{\pm }$: amplitude (+) and phase (−) quadratures of the input state; anc: ancilla squeezed vacuum state; $r$: reflected; $t$: transmitted; out: post-selected output state. $R$: beam-splitter reflectivity; GD: gate detector; PS: post-selection protocol. (b) Standard deviation contours of the Wigner functions of an input coherent state (blue) and post-selected output states (green) for $R=0.75$ and varying ancilla state squeezing of (i) $s=0$, (ii) $s=0.35$, (iii) $s=0.69$, (iv) $s=1.03$, and (v) ideal squeezing.

Figure 2

(a) The average fidelity ${\mathcal{F}}_{\mathrm{ave}}^{\left(1\right)}$ between the post-selected output state of an input single photon state ∣1⟩ and the ideal squeezed single photon target state $\widehat{S}\left(s^{\prime }\right)\mid 1⟩$. (b) The success probability ${{\mathcal{P}}_s}^{\left(1\right)}$ against the post-selection threshold $x_0$. Solid line: beam-splitter reflectivity $R=0.98$, ancillary state squeezing $s=0.7$, and target state squeezing of $s^{\prime }=0.67$. Dashed line: beam-splitter reflectivity of $R=0.5$, ancillary state squeezing $s=-0.7$, and target state squeezing of $s=-0.464$.

Figure 3

(Color online) The average Wigner function $W_{\mathrm{ave}}^{\left(1\right)}\left(\alpha \right)$ of the output state corresponding to ${\mathcal{F}}_{\mathrm{ave}}^{\left(1\right)}=0.99$ (a) for $x_0=0.025$, $R=0.98$, $s=0.7$, and ${\mathcal{P}}_s^{\left(1\right)}=0.003$ and for (b) $x_0=0.04$, $R=1∕2$, $s=-0.7$, and ${\mathcal{P}}_s^{\left(1\right)}=0.016$.

Figure 4

(Color online) (a) The average Wigner function $W_{\mathrm{ave}}^{\left(1\right)}\left(\alpha \right)$ of the output state obtained from the single-photon input with $s=0.7$ and $R=0.98$ using the feedforward scheme with the unity gain. The non-Gaussian quantum characteristics are washed away. (b) If the result obtained by the feedforward method is postselected for $x=0.3$, the non-Gaussian characteristic begins to emerge. (c) The average fidelity ${\mathcal{F}}_{\mathrm{ave}}^{\left(1\right)}=0.87$ and the minimum negative value $W_{\mathrm{ave}}^{\left(1\right)}\left(0\right)\approx -0.48$ are obtained for threshold $x_0=0.1$.

Figure 5

(Color online) The Wigner function $W_{\mathrm{out}}^{\left(1\right)}\left(\alpha \right)$ of the output state obtained from the single-photon input with $s=0.7$ and $R=0.98$. (a) If the measurement result is $X_r^{+}=0$, an ideal squeezed single photon with $s^{\prime }=0.67$ is obtained, i.e., the fidelity is ${\mathcal{F}}_{\left(1\right)}(X_r^{+}=0)=1$. However, the non-Gaussian shape is distorted as the measurement results deviate from zero as (b) ${\mathcal{F}}_{\left(1\right)}(X_r^{+}=-0.1)=0.67$ and (c) ${\mathcal{F}}_{\left(1\right)}(X_r^{+}=-0.5)=0.03$. The non-Gaussian characteristics cannot be recovered by the classical feedforward methods after such distortions. It is clear from this figure that the postselection is necessary to preserve the non-Gaussian feature for the output state.

Figure 6

The fidelity ${\mathcal{F}}_{\left(n\right)}$ between the ideal SCSs $W_{\mathrm{SCS}}^{\pm }\left(\alpha \right)$ with amplitude $\mid \gamma \mid$ and the output states $W_{\mathrm{out}}^{\left(n\right)}(\alpha ;X_r^{+}=0)$ from the $n$-photon input Fock states for $n=2$ (solid line), $n=3$ (dashed line), and $n=4$ (dotted line). Note that the target state is the ideal even (odd) SCS when $n$ is an even (odd) number.

Figure 7

(a) The average fidelity ${\mathcal{F}}_{\mathrm{ave}}^{\left(n\right)}$ between the ideal SCS and the output state of the input Fock state $\mid n⟩$ and (b) the success probability ${\mathcal{P}}_s^{\left(n\right)}$ for varying threshold $x_0$. The beam splitter reflectivity is $R=1∕2$. Solid line: $n=2$, the ancillar squeezing is $s=-0.37$, and the amplitude of the target SCS is $\mid \gamma \mid =1.1$. Dashed line: $n=3$, the ancillar squeezing is $s=-0.34$, and the amplitude of the target SCS is $\mid \gamma \mid =1.29$. Dotted line: $n=4$, the ancillar squeezing is $s=-0.37$, and the amplitude of the target SCS is $\mid \gamma \mid =1.49$.

Figure 8

(Color online) The average Wigner functions $W_{\mathrm{ave}}^{\left(n\right)}$ of the output state corresponding to ${\mathcal{F}}_{\mathrm{ave}}^{\left(n\right)}=0.99$ for (a) $n=2$, $x_0=0.084$, and ${\mathcal{P}}_s^{\left(2\right)}=0.055$, for (b) $n=3$, $x_0=0.0583$, and ${\mathcal{P}}_s^{\left(3\right)}=0.0309$, and for (c) $n=4$, $x_0=0.0555$, and ${\mathcal{P}}_s^{\left(4\right)}=0.0254$. It is clearly observed that the average output Wigner functions (left-hand side) are virtually identical to those of ideal even $(n=2,4)$ and odd $(n=3)$ SCSs (right-hand side) with the corresponding amplitudes.

Figure 9

The average fidelity ${\mathcal{F}}_{\mathrm{ave}}$ between the output state from a coherent state input $\mid \gamma ⟩$ and an ideally squeezed coherent state by the feedforward scheme (solid line) and the post-selection scheme (dashed lime) against the ancillar squeezing $s$ of the resource squeezed vacuum. $R=1∕2$, $\mid X\mid <0.025$, and $s^{\prime }=0.346574$. The coherent amplitudes are (a) $\mid \gamma \mid =0.5$, (b) $\mid \gamma \mid =1$, and (c) $\mid \gamma \mid =2$. The threshold for the postselection scheme is $x_0=0.025$.

Figure 10

(a) Schematic of the postselection protocol. ${\widehat{X}}_{\mathrm{in}}^{\pm }$: amplitude (+) and phase (−) quadratures of the input state; anc: ancilla state; $r$: reflected; $t$: transmitted; and out: postselected output state. $R$: beam-splitter reflectivity; GD: gate detector; RM: removable mirror; OPA: optical parametric amplifier; LO: local oscillator; AM/PM: amplitude/phase modulator; ${\theta }_1∕{\theta }_2$: optical phase delay; HP/LP: high/low pass filters; $G$: radio frequency amplifier; $\omega$: frequency of displaced coherent state at $6.81\mathrm{MHz}$; $\Omega$: electronic local oscillator at $6.875\mathrm{MHz}$; DAQ: data acquisition system.

Figure 11

(Color online) Experimental fidelity of the postselected output state for varying amplitude quadrature expectation values of the input coherent state $\mid {\gamma }^{+}\mid \equiv ⟨{\widehat{X}}_{\mathrm{in}}^{+}⟩$, for $R=0.75$ and $x_0=0.009$. Dark grey region: classical fidelity limit for an ancilla vacuum state, Light grey region: classical fidelity limit. Dot-dashed line: calculated theoretical curve including experimental losses and the finite postselection threshold.

Figure 12

(Color online) Experimental fidelity of the postselected output state for varying amplitudes $⟨{\widehat{X}}_{\mathrm{in}}^{+}⟩$ of the input coherent state, for $R=0.75$ and $x_0=0.009$.

Figure 13

(Color online) (a) Experimental fidelity of the postselected output state for varying success probability corresponding to varying postselection threshold. $R=0.5$ and $⟨{\widehat{X}}_{\mathrm{in}}^{+}⟩=0.71$. Light grey region: classical fidelity region. (b) Experimental optical quadrature gains $g^{\pm }=⟨{\widehat{X}}_{\mathrm{out}}^{\pm }⟩∕⟨{\widehat{X}}_{\mathrm{in}}^{\pm }⟩$ for varying success probability. (c) Experimental quadrature variances of the postselected output state $V_{\mathrm{out}}^{\pm }$ for varying success probability.

Figure 14

(Color online) (a) Experimental normalized purity for varying input states $⟨{\widehat{X}}_{\mathrm{in}}^{+}⟩$, for $R=0.75$ and $x_0=0.009$. Dash arrows show purity prior to (diamonds) and after (circles) postselection. Dot-dashed line: calculated theoretical prediction of the experiment. (b) Experimental purity for varying input state $⟨{\widehat{X}}_{\mathrm{in}}^{+}⟩$, for $R=0.5$ and $x_0=0.005$.