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arXiv:0804.4499 · Physical Review A

Coherent-states engineering with linear optics: Possible and impossible tasks

Bing He, János A. Bergou

Published May 19, 2008• 6 citations

Abstract

The general transformation of the product of coherent states ${\ensuremath{\prod}}_{i=1}^{N}|{\ensuremath{\alpha}}_{i}⟩$ to the output state ${\ensuremath{\prod}}_{i=1}^{M}|{\ensuremath{\beta}}_{i}⟩$ ($N=M$ or $N\ensuremath{\ne}M$), which is realizable with linear optical circuit, is characterized with a linear map from the vector $({\ensuremath{\alpha}}_{1}^{\ensuremath{\ast}},\dots{},{\ensuremath{\alpha}}_{N}^{\ensuremath{\ast}})$ to $({\ensuremath{\beta}}_{1}^{\ensuremath{\ast}},\dots{},{\ensuremath{\beta}}_{M}^{\ensuremath{\ast}})$. A correspondence between the transformations of a product of coherent states and those of a single photon state is established with such linear maps. It is convenient to apply this linear transformation method to design any linear optical scheme working with coherent states. The examples include message encoding and quantum database searching. The limitation of manipulating entangled coherent states with linear optics is also discussed.

DOI: 10.1103/physreva.77.053818

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The general transformation of the product of coherent states

{\ensuremath{\prod}}_{i=1}^{N}|{\ensuremath{\alpha}}_{i}⟩

to the output state

{\ensuremath{\prod}}_{i=1}^{M}|{\ensuremath{\beta}}_{i}⟩

(

N=M

or

N\ensuremath{\ne}M

), which is realizable with linear optical circuit, is characterized with a linear map from the vector

({\ensuremath{\alpha}}_{1}^{\ensuremath{\ast}},\dots{},{\ensuremath{\alpha}}_{N}^{\ensuremath{\ast}})

to

({\ensuremath{\beta}}_{1}^{\ensuremath{\ast}},\dots{},{\ensuremath{\beta}}_{M}^{\ensuremath{\ast}})

. A correspondence between the transformations of a product of coherent states and those of a single photon state is established with such linear maps. It is convenient to apply this linear transformation method to design any linear optical scheme working with coherent states. The examples include message encoding and quantum database searching. The limitation of manipulating entangled coherent states with linear optics is also discussed.

Summary form only given. We discuss an efficient scheme which is elegant in its simplicity. By encoding the quantum information in multi-photon coherent states, rather than single photon states, simple optical manipulations acquire unexpected power. The required resource, which may be produced non-deterministically, is a superposition of the vacuum and a coherent state. Given this, the scheme is deterministic and requires only simple linear optics and photon counting. Qubit readout uses homodyne detection.

Photon-number Fock states play important roles in quantum optics. However, the generation of the states are very difficult because of the lack of large effective nonlinearities. In this Rapid Communication, I show a scheme to produce the Fock state from an input coherent state by using only linear optics method and projection measurements. I analyze the case of five-photon number state generation from coherent states for a specific case of the scheme. It is possible to remove the vacuum state from the output state when one introduces quantum nondemolition measurement that can also be implemented with all linear optics setup.

The evolution of quantum light through linear optical devices can be described by the scattering matrix

S

of the system. For linear optical systems with

m

possible modes, the evolution of

n

input photons is given by a unitary matrix $U={\ensuremath{\varphi}}_{m,M}(S)$, derived from a known homomorphism, ${\ensuremath{\varphi}}_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states,

M

. We present a method to decide whether a given unitary evolution

U

for

n

photons in

m

modes can be achieved with linear optics or not and the inverse transformation ${\ensuremath{\varphi}}_{m,M}^{\ensuremath{-}1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map between the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

Programmable photonic integrated circuits are emerging as increasingly large and versatile interferometric processors operating at room temperature. However, their native operations are linear, while many computational tasks require nonlinear input-output transformations that typically rely on nonlinear optical materials or resonant devices. Here, we establish a direct mapping between quantum signal processing (QSP) and the native two-mode operations of programmable interferometric photonic integrated circuits, showing that the

SU(2)

structure required by QSP can be realized through programmable phase control and mode mixing. We experimentally demonstrate this correspondence using dual-rail single-photon encoding on a 24-mode programmable photonic integrated circuit, realizing QSP sequences up to depth

L=11

and synthesizing STEP, ReLU and SELU functions through programmable phase control. The optical transformation remains linear, while repeated encoding of the input variable and coherent interference produce a nonlinear dependence of the output probabilities on the encoded variable. Across all accessible circuit depths, the measured responses follow the programmed QSP transformations, with hardware-induced mean squared errors between

10^{-3}

and

U

, all remaining below the intrinsic finite-depth approximation error. These results establish an algorithm-to-hardware mapping of QSP on programmable photonic processors. They also demonstrate an algorithmic route to programmable nonlinear functions of encoded variables on linear photonic hardware.