Quantum-state engineering with continuous-variable postselection

We present a scheme to conditionally engineer an optical quantum system via continuous-variable measurements. This scheme yields high-fidelity squeezed single photons and a superposition of coherent states, from input single- and two-photon Fock states, respectively. The input Fock state is interacted with an ancilla squeezed vacuum state using a beam splitter. We transform the quantum system by postselecting on the continuous-observable measurement outcome of the ancilla state. We experimentally demonstrate the principles of this scheme using coherent states and experimentally measure fidelities that are only achievable using quantum resources.

Figure 1

(Color online) (a) Schematic of the postselection protocol. ${\widehat{X}}_{\mathrm{in}}^{\pm }$: amplitude (+) and phase (−) quadratures of the input state; (anc) ancilla state; $\left(r\right)$ reflected; $\left(t\right)$ transmitted; and (out) postselected output state. $R$: beam-splitter reflectivity; GD: gate detector; PS: postselection protocol. (b) Standard deviation contours of the Wigner functions of an input coherent state (blue) and postselected 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

(Color online) (a) The average fidelity ${\mathcal{F}}_{\mathrm{ave}}$ between the postselected output state of an input single-photon state ∣1⟩, and the squeezed single-photon state $\widehat{S}\left(s^{\prime }\right)\mid 1⟩$, for varying threshold $x_0$. The beam-splitter reflectivity is $R=0.98$, ancilla state squeezing is $s=0.7$, and target state squeezing is $s^{\prime }=0.67$. (b) The average Wigner function $W_{\mathrm{ave}}$ of the output state for $F_{\mathrm{ave}}=0.99$, for $x_0=0.025$, and for $P_s=0.003$.

Figure 3

(Color online) (a) The average fidelity ${\mathcal{F}}_{\mathrm{ave}}$ between the output state of the input two-photon state, ∣2⟩, and the ideal SCS, for the varying threshold $x_0$. The beam-splitter reflectivity is $R=1∕2$, the ancilla state squeezing is $s=-0.37$, and the amplitude of the SCS is $\gamma =1.1i$. (b) The average Wigner function $W_{\mathrm{ave}}$ of the output state for $F_{\mathrm{ave}}=0.99$, for $x_0=0.084$, and for $P_s=0.052$.

Figure 4

(Color online) Experimental fidelity for varying amplitudes $\mid {\gamma }^{+}\mid$ of the input coherent state, for $R=0.75$ and $x_0=0.01$. Dark gray region: classical fidelity limit for an ancilla vacuum state; Light gray region: classical fidelity limit. Dot-dashed line: calculated theoretical prediction of experiment, with experimental losses and inefficiencies.

Figure 5

(Color online) (a) Experimental fidelity $\left({\mathcal{F}}_{\mathrm{ave}}\right)$ for varying postselection success probability, for an amplitude $\mid {\gamma }^{+}\mid =1.07$ of the input state and $R=0.75$. (b) Experimental purity $\left({\mathcal{P}}_{\mathrm{norm}}\right)$ for the varying input state $\mid {\gamma }^{+}\mid$, for $x_0=0.01$. Dashed arrows show purity prior to (diamonds) and after (circles) postselection. Dot-dashed line: calculated theoretical prediction of the experiment.