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Werner Vogel
Universitat RostockGermany
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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
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Introduction
Characterization of quantum states
Balanced homodyne detection:
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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)
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Introduction
Experimental realization :
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Introduction
Experimental realization :
→ squeezed vacuum state
[Smithey, Beck, Raymer, Faridani, Phys. Rev. Lett. 70, 1244 (1993)]
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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)]
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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)
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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)]
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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)]
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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
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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!
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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).
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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, . . . ,+∞.
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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)]
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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)
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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)]
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Nonclassical characteristic functions
Direct observation via fluorescence
resonance fluorescence
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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)]
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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
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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 !
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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
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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!
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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!
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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]
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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
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Nonclassical moments of two quadratures
Sufficient conditions:
(1) Restriction to second-order determinant:
d(2)ϕ = 〈: (∆xϕ)2 :〉 < 0
→ quadrature squeezing!
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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!
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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)]
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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!
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Nonclassical moments of two quadratures
→ no squeezing (q(2) > 0), but q(3), s(2) < 0 !
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Measuring moments of two quadratures
Basic measurement scheme:
see also [J.W. Noh, A. Fougeres, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991)]
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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)]
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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)]
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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
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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
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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) :
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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
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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!
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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