Werner Vogel

Universitat RostockGermany

Contents

Introduction

Nonclassical phase-space functions

Nonclassical characteristic functions

General nonclassicality condition

Nonclassical moments of two quadratures

Measuring moments of two quadratures

Nonclassical moments of number and quadrature

Comments on entangled states

Summary

Introduction

Characterization of quantum states

Balanced homodyne detection:

Introduction

Measured quantities:

• Difference statistics⇔ quadrature operator:

xϕ = aeiϕ + a†e−iϕ

• Perfect detection, strong LO:

P∆m =1|α|

p(x =∆m|α|

, ϕ)

Introduction

Experimental realization :

Introduction

Experimental realization :

→ squeezed vacuum state

[Smithey, Beck, Raymer, Faridani, Phys. Rev. Lett. 70, 1244 (1993)]

Introduction

Tomographic quantum-state reconstruction:

• measuring p(x, ϕ) for ϕ . . . ϕ + π

→ Wigner function: W(α)

→ Density matrix

[K. Vogel and H. Risken, Phys. Rev. A40, 2847 (1989)]

Nonclassical phase-space functions

P-representation of the density operator:

ρ =

∫d2αP(α) |α〉〈α|

• expectation values:

〈: F(a†, a) :〉 =∫

d2αP(α)F(α∗, α)

Nonclassical phase-space functions

P-representation of the density operator:

ρ =

∫d2αP(α) |α〉〈α|

• expectation values:

〈: F(a†, a) :〉 =∫

d2αP(α)F(α∗, α)

Correspond to classical mean values:

(1) ”subtracting” ground-state noise via F→ : F :

(2) P corresponds to classical probability: P(α) ≡ Pcl(α)[U.M. Titulaer and R.J. Glauber, Phys. Rev. 140, B676 (1965)]

Nonclassical phase-space functions

A state is nonclassical, if:

(a) ground-state noise is substantial;cf. also nonclassicality in weak measurements[L.M. Johansen, Phys. Lett. A 329, 184 (2004)]

Nonclassical phase-space functions

A state is nonclassical, if:

(a) ground-state noise is substantial;cf. also nonclassicality in weak measurements[L.M. Johansen, Phys. Lett. A 329, 184 (2004)]

(b) P fails to be a classical probability: P(α) , Pcl(α);

– The only classical pure states are coherent ones![M. Hillery, Phys. Lett. A111, 409 (1985)]

– Squeezing: 〈: (∆xϕ)2 :〉 < 0– sub-Poissonian photon statistics: 〈: (∆n)2 :〉 < 0

Nonclassical phase-space functions

A state is nonclassical, if:

(a) ground-state noise is substantial;cf. also nonclassicality in weak measurements[L.M. Johansen, Phys. Lett. A 329, 184 (2004)]

(b) P fails to be a classical probability: P(α) , Pcl(α);

– The only classical pure states are coherent ones![M. Hillery, Phys. Lett. A111, 409 (1985)]

– Squeezing: 〈: (∆xϕ)2 :〉 < 0– sub-Poissonian photon statistics: 〈: (∆n)2 :〉 < 0

• Sought: observable conditions for P(α) , Pcl(α)

• Problem: P(α) may be strongly singular!

Nonclassical characteristic functions

Characteristic function of P(α):

Φ(β) =∫

d2αP(α) exp[(αβ∗ − α∗β)]

• Bochner Theorem (1933):

Φ(β) is a classical characteristic function, if and only ifn∑

i, j=1

Φ(βi − β j) ξi ξ∗

j ≥ 0,

for any integer n and all complex βi, ξk (i, k = 1 . . . n).

Nonclassical characteristic functions

• Define matrix: Φi j = Φ(βi − β j)

• Theorem:

Φ(β) is a classical characteristic function, if and only if

Dk ≡ Dk(β1, . . . βk) =

∣∣∣∣∣∣∣∣∣∣∣1 Φ12 · · · Φ1kΦ∗12 1 · · · Φ2k. . . . . . . . . . . . . . . .Φ∗1k Φ

∗

2k · · · 1

∣∣∣∣∣∣∣∣∣∣∣ ≥ 0

for any order k = 1, . . . ,+∞.

Nonclassical characteristic functions

• Define matrix: Φi j = Φ(βi − β j)

• Theorem:

Φ(β) is a classical characteristic function, if and only if

Dk ≡ Dk(β1, . . . βk) =

∣∣∣∣∣∣∣∣∣∣∣1 Φ12 · · · Φ1kΦ∗12 1 · · · Φ2k. . . . . . . . . . . . . . . .Φ∗1k Φ

∗

2k · · · 1

∣∣∣∣∣∣∣∣∣∣∣ ≥ 0

for any order k = 1, . . . ,+∞.

⇒ P(α) is not a probability if and only if there exist va-lues of k and βk (k = 2 . . .∞) with

Dk(β1, . . . βk) < 0

[Th. Richter and W. Vogel, Phys. Rev. Lett. 89, 283601 (2002)]

Nonclassical characteristic functions

Observable characteristic functions of quadratures

G(k, ϕ) = Ggr(k)Φ(ike−iϕ),with Φ = 1 in the ground state

• First-order nonclassicality:[W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)]

|G(k, ϕ)| > Ggr(k)

Nonclassical characteristic functions

Observable characteristic functions of quadratures

G(k, ϕ) = Ggr(k)Φ(ike−iϕ),with Φ = 1 in the ground state

• First-order nonclassicality:[W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)]

|G(k, ϕ)| > Ggr(k)

• applies to many nonclassical states:Fock, squeezed, even/odd coherent states, . . .

• experimental demonstration:mixture of a single photon with the vacuum state

ρ = η|1〉〈1| + (1 − η)|0〉〈0|

[A.I. Lvovsky and J.H. Shapiro, Phys. Rev. A 65, 033830 (2002)]

Nonclassical characteristic functions

Direct observation via fluorescence

resonance fluorescence

Nonclassical characteristic functions

Direct observation via fluorescence

resonance fluorescence

• Hamiltonian: Hint =12~(ΩA12 +Ω

∗A21

)x(ϕ)

[S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995)]

⇒ experimental realization

[P.C. Haljan, K.-A. Brickman, L. Deslauriers, P.L. Lee, and C. Monroe (2004)]

General nonclassicality condition

Reformulation

• Hermitian Operator: f † f

• Normally ordered expectation value:

〈: f † f :〉 =∫

d2α | f (α)|2P(α),

⇒ nonnegative for P(α) = Pcl(α), for any operator f

General nonclassicality condition

Reformulation

• Hermitian Operator: f † f

• Normally ordered expectation value:

〈: f † f :〉 =∫

d2α | f (α)|2P(α),

⇒ nonnegative for P(α) = Pcl(α), for any operator f

• Quantum state nonclassical, iff there exists f with

〈: f † f :〉 < 0

⇒ various choices of representations of f !

General nonclassicality condition

Sufficient Conditions for nonclassicality:

• Sub-Possonian number statistics:

f ≡ ∆n = n − 〈n〉, n = a†a

⇒ condition:〈: f † f :〉 → 〈: (∆n)2 :〉 < 0

• Quadrature Squeezing:

f ≡ ∆xϕ = xϕ − 〈xϕ〉, xϕ = aeiϕ + a†e−iϕ

⇒ condition:〈: (∆xϕ)2 :〉 < 0

General nonclassicality condition

Fourier representation

f =∫

d2α f (α) :D(−α) :

• condition〈: f † f :〉 < 0

• now reads as:∫d2α

∫d2β f (α) f ∗(β)Φ(α − β) < 0

→ continuous version of the Bochner condition!

→ criteria for characteristic functions: special represen-tation!

General nonclassicality condition

Taylor expansion

f ≡ f (A, B) =∑n,m

fnm : AnBm :

Choice of A, B for complete description:

• Hermitian operators:

(a) A = xϕ, B = pϕ, pϕ ≡ xϕ+π/2(b) A = xϕ, B = n

• non-Hermitian operators:

(c) A = a†, B = a⇒ different types of complete sets of criteria!

Nonclassical moments of two quadratures

Taylor expansion in quadratures

f = f (xϕ, pϕ) =∑n,m

fnm : xnϕpm

ϕ :

• nonclassicality condition

〈: f † f :〉 ⇒∑

n,m,k,l

fnm f ∗klMnm,kl(ϕ) < 0

whereMnm,kl(ϕ) = 〈: xn+k

ϕ pm+lϕ :〉

[E. Shchukin, Th. Richter, and W. Vogel, to be published]

Nonclassical moments of two quadratures

In terms of determinants:

• determinants under study:

d(N)ϕ =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 〈: xϕ :〉 〈: pϕ :〉 〈: x2ϕ :〉 〈: xϕpϕ :〉 〈: p2

ϕ :〉 . . .〈: xϕ :〉 〈: x2

ϕ :〉 〈: xϕpϕ :〉 〈: x3ϕ :〉 〈: x2

ϕpϕ :〉 〈: xϕp2ϕ :〉 . . .

〈: pϕ :〉 〈: xϕpϕ :〉 〈: p2ϕ :〉 〈: x2

ϕpϕ :〉 〈: xϕp2ϕ :〉 〈: p3

ϕ :〉 . . .〈: x2

ϕ :〉 〈: x3ϕ :〉 〈: x2

ϕpϕ :〉 〈: x4ϕ :〉 〈: x3

ϕpϕ :〉 〈: x2ϕp2

ϕ :〉 . . .〈: xϕpϕ :〉 〈: x2

ϕpϕ :〉 〈: xϕp2ϕ :〉 〈: x3

ϕpϕ :〉 〈: x2ϕp2

ϕ :〉 〈: xϕp3ϕ :〉 . . .

〈: p2ϕ :〉 〈: xϕp2

ϕ :〉 〈: p3ϕ :〉 〈: x2

ϕp2ϕ :〉 〈: xϕp3

ϕ :〉 〈: p4ϕ :〉 . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣• Necessary and sufficient nonclassicality conditions:

there exist values of N (N ≥ 2) and ϕ with

d(N)ϕ < 0

Nonclassical moments of two quadratures

Sufficient conditions:

(1) Restriction to second-order determinant:

d(2)ϕ = 〈: (∆xϕ)2 :〉 < 0

→ quadrature squeezing!

Nonclassical moments of two quadratures

Sufficient conditions:

(1) Restriction to second-order determinant:

d(2)ϕ = 〈: (∆xϕ)2 :〉 < 0

→ quadrature squeezing!

(2) Third-order determinant:

d(3)ϕ = 〈: (∆xϕ)2 :〉〈: (∆pϕ)2 :〉 − 〈: ∆xϕ∆pϕ :〉2 < 0

→ moments of two quadratures, but:

d(3)ϕ = 〈: (∆xϕ)2 :〉min〈: (∆pϕ)2 :〉max

→ no new effect!

Nonclassical moments of two quadratures

(3) Elimination of one quadrature:

q(n)ϕ =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1⟨: xϕ :

⟩. . .⟨: xn−1

ϕ :⟩⟨

: xϕ :⟩ ⟨

: x2ϕ :⟩. . .

⟨: xn

ϕ :⟩

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .⟨: xn−1

ϕ :⟩ ⟨

: xnϕ :⟩. . .⟨: x2n−2

ϕ :⟩

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣→ nonclassicality conditions due to Agarwal:

q(n)ϕ < 0

[G.S. Agarwal, Opt. Commun. 95, 109 (1993)]

Nonclassical moments of two quadratures

(4) Sub-determinants, for example:

s(2)ϕ =

∣∣∣∣∣∣∣⟨: x2

ϕ :⟩ ⟨

: x2ϕpϕ :

⟩⟨: x2

ϕpϕ :⟩ ⟨

: x2ϕp2

ϕ :⟩∣∣∣∣∣∣∣ < 0

→ Illustration for the quantum state:

|ψ〉 =|0〉 + c |3〉√

1 + |c|2

→ nonclassical for a larger parameter range!

Nonclassical moments of two quadratures

→ no squeezing (q(2) > 0), but q(3), s(2) < 0 !

Measuring moments of two quadratures

Basic measurement scheme:

see also [J.W. Noh, A. Fougeres, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991)]

Measuring moments of two quadratures

• effective photon-number operators:

n1,2 =14

(n ± |α| pϕ + |α|2

)n3,4 =

14

(n ± |α| xϕ + |α|2

)• detecting correlations, such as:

〈: nin j :〉, 〈: nin jnk :〉, . . .

• advantage: insensitive to efficiencies of detectors!

• extension to high orders possible!

[M. Beck, C. Dorrer, I. A. Walmsley, Phys. Rev. Lett. 87, 253601 (2001)]

Nonclassical number-quadrature moments

Taylor expansion in number and quadrature

• Reformulate the condition 〈: f † f :〉 < 0

• with the representation

f = f (xϕ, n) =∑

k,l

fkl : xkϕnl :

• Conditions in terms of number-quadrature moments:

Mk,l = 〈: xkϕnl :〉

⇒ Homodyne correlation measurements[W. Vogel, Phys. Rev. Lett. 67, 2450 (1991); Phys. Rev. A51, 4160 (1995);

H.J. Carmichael, H.M. Castro-Beltran, G.T. Foster, L.A. Orozco, Phys. Rev.

Lett. 85, 1855 (2000)]

Nonclassical number-quadrature moments

⇒ Observables of dissimilar types:xϕ continuous and n discrete and non-negative!

⇒ Two different types of nonclassicality conditions:∣∣∣∣∣∣∣∣∣∣∣1 M0,1 M1,0 · · ·

M0,1 M0,2 M1,1 · · ·

M1,0 M1,1 M2,1 · · ·

. . . . . . . . . . . . . . . . . . .

∣∣∣∣∣∣∣∣∣∣∣ < 0

∣∣∣∣∣∣∣∣∣∣∣2M0,1 −M2,0 2M0,2 −M2,1 2M1,1 −M3,0 · · ·

2M0,2 −M2,1 2M0,3 −M2,2 2M1,2 −M3,1 · · ·

2M1,1 −M3,0 2M1,2 −M3,1 2M2,1 −M4,0 · · ·

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∣∣∣∣∣∣∣∣∣∣∣ < 0

Comment on entangled states

Criteria for continuous variable entanglement

• Conditions based on second-order moments[L.M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722

(2000); R. Simon, Phys. Rev. Lett. 84, 2726 (2000)]

• Negative partial transposition of density matrix

Comment on entangled states

Criteria for continuous variable entanglement

• Conditions based on second-order moments[L.M. Duan, G. Giedke, J.I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722

(2000); R. Simon, Phys. Rev. Lett. 84, 2726 (2000)]

• Negative partial transposition of density matrix

⇒ General test via NPT condition:

〈 f † f 〉PT < 0

⇒ Fourier representation (two modes):

f =∫

d2α1 f (α1, α2) :D(−α1)D(−α2) :

Comment on entangled states

Complete condition for negative PT

• Discrete version of 〈 f † f 〉PT < 0:n∑

i, j=1

eα∗

iα j+β∗

iβ jΦ(αi − α j, β∗

j − β∗

i ) ξi ξ∗

j < 0

Comment on entangled states

Complete condition for negative PT

• Discrete version of 〈 f † f 〉PT < 0:n∑

i, j=1

eα∗

iα j+β∗

iβ jΦ(αi − α j, β∗

j − β∗

i ) ξi ξ∗

j < 0

⇒ Conditions for determinants of characteristic functions

⇒ Observable conditions

⇒ Systematic check of NPT for non-Gaussian continuousquantum states!

⇒ Only sufficient criterion for entanglement!

Summary

• Nonclassical P-functions

• Nonclassical characteristic functions

• Nonclassical conditions for quadrature moments

• Measurement of quadrature moments

• Nonclassical number-quadrature moments

• Criteria for NPT of entangled states