Tuning photon statistics with coherent fields

Casalengua, Eduardo Zubizarreta; Carreño, Juan Camilo López; Laussy, Fabrice P.; Valle, Elena del

doi:10.1103/physreva.101.063824

Cited by 24 publications

(30 citation statements)

References 113 publications

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“…The mixing of two quantum fields is most simply achieved by passing them through the two ports of a balanced beam‐splitter. In the homodyning case that involves a coherent state

| α ⟩

, with complex amplitude

α = false| α false| e^{i ϕ}

, mixed with a field of general nature, with annihilation operator d , the normally ordered correlators of the resulting field

s = α + d

is expressed in terms of the inputs as [ 109 ]

false⟨ s^{† n} s^{m} false⟩ = \sum_{p = 0}^{n} \sum_{q = 0}^{m} (\frac{0pt}{n}) (\frac{0pt}{m}) α^{* p} α^{q} false⟨ d^{† n - p} d^{m - q} false⟩

up to some unimportant normalization and phase‐shift factors. In practice, the polarization degree of freedom is involved with one of the two optical polarizations used as the local oscillator, with a polarizer to mix them afterwards, [ 58 ] but this needs not enter our simple theoretical picture.…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…

g_{s}^{(N)} = \frac{\sum_{k = 0}^{2 N} c_{k} {| α |}^{k}}{{false⟨ n_{s} false⟩}^{N}}

where

c_{k}

are coefficients that depend on the phase of the coherent field ϕ and mean values of the type

⟨ d^{† μ} d^{ν} ⟩

with

μ + ν \leq N^{2}

. In particular, the 2nd‐order correlation function, Equation (), can be rearranged as

g_{s}^{(2)} = 1 + {scriptI}_{0} + {scriptI}_{1} + {scriptI}_{2}

with

{scriptI}_{m} \sim {| α |}^{m}

, [ 35,110–112 ] where 1 represents the coherent contribution of the total signal, and the incoherent contributions read [ 109 ]

{scriptI}_{0} = \frac{false⟨ d^{† 2} d^{2} false⟩ - {⟨ d^{†} d ⟩}^{2}}{{false⟨ n_{s} false⟩}^{2}}

{scriptI}_{1} = 4 \frac{ℜ [α^{*} false(⟨ d^{†} d^{2} ⟩ - ⟨ d^{†} d ⟩ ⟨ d ⟩ false)]}{{false⟨ n_{s} false⟩}^{2}}

...…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…Furthermore, if

⟨ d ⟩ = 0

, the numerator of

I_{2}

simplifies to

{4 | α |}^{2} false(false⟨ : X_{d}^{2} : false⟩ - {| α |}^{2} false)

. An analogous procedure likewise decomposes the higher‐order coherence functions, that is,

g_{s}^{(n)} = 1 + \sum_{m = 0}^{2 n - 2} {scriptJ}_{m}^{(n)}

, with closed‐form expressions for

J_{m}^{false(3 false)}

given in ref. [109].…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…The interference at the beam splitter mixes these two states. Writing the operators in the output basis gives [ 109 ]

{scriptD}_{a} false(α false) = {scriptD}_{o} false(α_{o} false) {scriptD}_{s} false(α_{s} false) = {scriptD}_{s} false(α_{s} false) {scriptD}_{o} false(α_{o} false)

where

α_{o} = i R α

α_{s} = T α

and

{scriptD}_{j} false(α_{j} false) = \exp false(α_{j} j - α^{*} j^{†} false)

for

j = o, s

. Similarly, the squeezing operator in the output basis reads:

\begin{matrix} true \\ right S_{d} ((), ξ) & = & left \exp ([] \frac{1}{2} (ξ_{o}^{*} o^{2} - ξ_{o} o^{†}) + \frac{1}{2} (ξ_{s}^{*} s^{2} - ξ_{s} s^{†}) + (ξ_{o s}^{*} o s - ξ_{o s} o^{†} s^{†})) \\ = & left \exp false(S_{o} + S_{s} + S_{o s} false) \end{matrix}

where

ξ_{o} = {normalT}^{2} ξ

ξ_{s} = - {normalR}^{2} ξ

and

ξ_{o s} = i R T ξ

.…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…The thermal population in terms of the squeezed population of the input signal

false⟨ n_{d} false⟩ = \sinh^{2} r

becomes

{normalp}_{normalth} = \frac{1}{2} false(\sqrt{1 + false⟨ n_{d} false⟩} - 1 false)

From

ρ_{s}

, one can compute the observables for the mixed signal, for example,

g_{s}^{(2)} = Tr false[ρ_{s} s^{† 2} s^{2} false] / Tr {[ρ_{s} s^{†} s]}^{2}

( g (3) is given in ref. [109]):

false⟨ n_{s} false⟩ = \frac{{false| α false|}^{2}}{2} + \frac{⟨ n_{d} ⟩}{2}, false| ⟨ s^{2} ⟩ false| = (p_{th} + \frac{1}{2}) \sinh false(r false)

\begin{matrix} true \\ right g_{s}^{false(2 false)} = 1 + {false⟨ n_{s} false⟩}^{- 2} \sinh^{2} r ([] \cosh 2 r + {2 | α |}^{2} (1 - \cos false(θ - 2 ϕ false) \coth r)) \end{matrix}

The second‐order correlation for the total () can be decomposed as in Equation () into

{scriptI}_{0} = \frac{\sinh [4] (r)}{false...}

…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

See 4 more Smart Citations

Conventional and Unconventional Photon Statistics

Casalengua

Carreño

Laussy

et al. 2020

Laser & Photonics Reviews

Self Cite

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The photon statistics emitted by a large variety of light‐matter systems under weak coherent driving can be understood, to lowest order in the driving, in the framework of an admixture of (or interference between) a squeezed state and a coherent state, with the resulting state accounting for all bunching and antibunching features. One can further identify two mechanisms that produce resonances for the photon correlations: i) conventional photon blockade describes cases that involve a particular quantum level or set of levels in the excitation/emission processes with interferences occurring to all orders in the photon numbers, while ii) unconventional photon blockade describes cases where the driving laser is far from resonance with any level and the interference occurs for a particular number of photons only, yielding stronger correlations but only for a definite number of photons. Such an understanding and classification allows for a comprehensive and transparent description of the photon statistics from a wide range of disparate systems, where optimum conditions for various types of photon correlations can be found and realized.

show abstract

“…The mixing of two quantum fields is most simply achieved by passing them through the two ports of a balanced beam‐splitter. In the homodyning case that involves a coherent state

| α ⟩

, with complex amplitude

α = false| α false| e^{i ϕ}

, mixed with a field of general nature, with annihilation operator d , the normally ordered correlators of the resulting field

s = α + d

is expressed in terms of the inputs as [ 109 ]

false⟨ s^{† n} s^{m} false⟩ = \sum_{p = 0}^{n} \sum_{q = 0}^{m} (\frac{0pt}{n}) (\frac{0pt}{m}) α^{* p} α^{q} false⟨ d^{† n - p} d^{m - q} false⟩

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…

g_{s}^{(N)} = \frac{\sum_{k = 0}^{2 N} c_{k} {| α |}^{k}}{{false⟨ n_{s} false⟩}^{N}}

where

c_{k}

are coefficients that depend on the phase of the coherent field ϕ and mean values of the type

⟨ d^{† μ} d^{ν} ⟩

with

μ + ν \leq N^{2}

. In particular, the 2nd‐order correlation function, Equation (), can be rearranged as

g_{s}^{(2)} = 1 + {scriptI}_{0} + {scriptI}_{1} + {scriptI}_{2}

with

{scriptI}_{m} \sim {| α |}^{m}

, [ 35,110–112 ] where 1 represents the coherent contribution of the total signal, and the incoherent contributions read [ 109 ]

{scriptI}_{0} = \frac{false⟨ d^{† 2} d^{2} false⟩ - {⟨ d^{†} d ⟩}^{2}}{{false⟨ n_{s} false⟩}^{2}}

{scriptI}_{1} = 4 \frac{ℜ [α^{*} false(⟨ d^{†} d^{2} ⟩ - ⟨ d^{†} d ⟩ ⟨ d ⟩ false)]}{{false⟨ n_{s} false⟩}^{2}}

...…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…Furthermore, if

⟨ d ⟩ = 0

, the numerator of

I_{2}

simplifies to

{4 | α |}^{2} false(false⟨ : X_{d}^{2} : false⟩ - {| α |}^{2} false)

. An analogous procedure likewise decomposes the higher‐order coherence functions, that is,

g_{s}^{(n)} = 1 + \sum_{m = 0}^{2 n - 2} {scriptJ}_{m}^{(n)}

, with closed‐form expressions for

J_{m}^{false(3 false)}

given in ref. [109].…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…The interference at the beam splitter mixes these two states. Writing the operators in the output basis gives [ 109 ]

{scriptD}_{a} false(α false) = {scriptD}_{o} false(α_{o} false) {scriptD}_{s} false(α_{s} false) = {scriptD}_{s} false(α_{s} false) {scriptD}_{o} false(α_{o} false)

where

α_{o} = i R α

α_{s} = T α

and

{scriptD}_{j} false(α_{j} false) = \exp false(α_{j} j - α^{*} j^{†} false)

for

j = o, s

. Similarly, the squeezing operator in the output basis reads:

\begin{matrix} true \\ right S_{d} ((), ξ) & = & left \exp ([] \frac{1}{2} (ξ_{o}^{*} o^{2} - ξ_{o} o^{†}) + \frac{1}{2} (ξ_{s}^{*} s^{2} - ξ_{s} s^{†}) + (ξ_{o s}^{*} o s - ξ_{o s} o^{†} s^{†})) \\ = & left \exp false(S_{o} + S_{s} + S_{o s} false) \end{matrix}

where

ξ_{o} = {normalT}^{2} ξ

ξ_{s} = - {normalR}^{2} ξ

and

ξ_{o s} = i R T ξ

.…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

“…The thermal population in terms of the squeezed population of the input signal

false⟨ n_{d} false⟩ = \sinh^{2} r

becomes

{normalp}_{normalth} = \frac{1}{2} false(\sqrt{1 + false⟨ n_{d} false⟩} - 1 false)

From

ρ_{s}

, one can compute the observables for the mixed signal, for example,

g_{s}^{(2)} = Tr false[ρ_{s} s^{† 2} s^{2} false] / Tr {[ρ_{s} s^{†} s]}^{2}

( g (3) is given in ref. [109]):

false⟨ n_{s} false⟩ = \frac{{false| α false|}^{2}}{2} + \frac{⟨ n_{d} ⟩}{2}, false| ⟨ s^{2} ⟩ false| = (p_{th} + \frac{1}{2}) \sinh false(r false)

\begin{matrix} true \\ right g_{s}^{false(2 false)} = 1 + {false⟨ n_{s} false⟩}^{- 2} \sinh^{2} r ([] \cosh 2 r + {2 | α |}^{2} (1 - \cos false(θ - 2 ϕ false) \coth r)) \end{matrix}

The second‐order correlation for the total () can be decomposed as in Equation () into

{scriptI}_{0} = \frac{\sinh [4] (r)}{false...}

…”

Section: Homodyne and Self‐homodyne Interferencesmentioning

confidence: 99%

See 3 more Smart Citations

Conventional and Unconventional Photon Statistics

Casalengua

Carreño

Laussy

et al. 2020

Laser & Photonics Reviews

Self Cite

View full text Add to dashboard Cite

show abstract

Unconventional Photon Blockade in Quantum‐Well Microcavities with Squeezed Light

Jabri

Eleuch

2021

Physica Status Solidi (b)

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The single-photon blockade effect arising in a cavity containing a quantum well and interacting with squeezed light, resulting from a down-conversion process through a nonlinear medium, is investigated. Optical parametric oscillator (OPO) materials with positive and negative second-order susceptibilities are considered. By solving the master equation analytically in the weak-driving limit and calculating the second-order equal-time correlation function, it is found that strong photon antibunching can be achieved in this system with weak excitonic nonlinearity or squeezed light via a destructive quantum interference mechanism. The optimal conditions for the photon blockade are derived analytically and discussed in details. Unlike the excitonic nonlinearity, a strong photon blockade occurs with the squeezed light at total or quasitotal resonance, even in the weak coupling regime, using an OPO material with negative second-order nonlinear susceptibility. By acting on the driving coherent pump frequency, the control of the cavity output is achieved and the scheme can serve as a tunable singlephoton emission source.

show abstract

Two-Photon Blockade with Second-Order Nonlinearity in Cavity Systems

Zhang

Wang

et al. 2022

Int J Theor Phys

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Tuning photon statistics with coherent fields

Cited by 24 publications

References 113 publications

Conventional and Unconventional Photon Statistics

Conventional and Unconventional Photon Statistics

Unconventional Photon Blockade in Quantum‐Well Microcavities with Squeezed Light

Two-Photon Blockade with Second-Order Nonlinearity in Cavity Systems

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