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Nonclassical Harmonic Oscillator

Werner Vogel

Universitat Rostock, Germany

Contents

• Introduction

• Nonclassical phase-space functions

• Nonclassical characteristic functions

• Nonclassical moments

• Recent experiments

– Reconstruction of a nonclassical P function

– Characteristic functions and moments

• Nonclassical correlation properties

• Quantifying nonclassicality

Introduction

Squeezing: 〈(∆xϕ)2〉 < 〈(∆xϕ)2〉gr, xϕ = aeiϕ + a†e−iϕ

nonclassical?

”Subtracting” ground-state (vacuum) noise:

〈: (∆xϕ)2 :〉 < 0 ⇔ 〈(∆xϕ)2〉cl 6< 0

⇒ observable (sufficient) condition for nonclassicality!

Nonclassical phase-space functions

P -representation of the density operator: 1

ρ =

∫d2αP (α) |α〉〈α|

→ resembles classical mixture!

Expectation values:

〈: F (a†, a) :〉 =

∫d2αP (α)F (α∗, α)

Correspondence to classical mean values:

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

(2) P corresponds to classical probability:2

P (α) ≡ Pcl(α)

1E. C. G. Sudarshan, Phys. Rev. Lett. 10, 227 (1963); R. J. Glauber, Phys. Rev. 131, 2766 (1963)2U.M. Titulaer and R.J. Glauber, Phys. Rev. 140, B676 (1965)

Nonclassical phase-space functions

A state is nonclassical, if: 3

(a) ground-state noise is substantial,cf. nonclassicality in weak measurements4

→ alternatively:5 small photon numbers

(b) P fails to be a classical probability:

P (α) 6= Pcl(α)

examples:

– Squeezing: 〈: (∆xϕ)2 :〉 < 0

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

Sought: observable conditions for P (α) 6= Pcl(α)

Problem: P (α) may be strongly singular!

3W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)4L.M. Johansen, Phys. Lett. A 329, 184 (2004)5L. Mandel, Phys. Scr. T 12, 34 (1986)

Nonclassical characteristic functions

Characteristic function of P (α):

Φ(β) =

∫d2αP (α) exp[(αβ∗ − α∗β)]

Theorem (Bochner 1933): 6

P (α) is a probability distribution iff for any smooth functionf (α) with compact support the following expression is non-negative: ∫∫

d2α d2β Φ(α− β) f ∗(α)f (β) > 0

corresponding discrete version:n∑

i,j=1

Φ(βi − βj) ξ∗i ξj ≥ 0,

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

6T. Kawata, Fourier Analysis in Probability Theory, Academic Press, N.Y. 1972

Nonclassical characteristic functions

Define matrix: Φij = Φ(βi − βj)

Theorem: A continuous function Φ(β) with Φ(0) = 1 andΦ∗(β) = Φ(−β) is a classical characteristic function, iff

Dk ≡ Dk(β1, . . . βk) =

∣∣∣∣∣∣∣∣∣1 Φ12 · · · Φ1k

Φ∗12 1 · · · Φ2k

. . . . . . . . . . . . . . . .Φ∗

1k Φ∗2k · · · 1

∣∣∣∣∣∣∣∣∣ ≥ 0

for any order k = 1, . . . ,+∞.

Nonclassicality: 7

P (α) is not a probability iff there exist values of k and βk(k = 2 . . .∞) with

Dk(β1, . . . βk) < 0

7T. 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) ⇔ FT[p(x, ϕ)],

in the ground state:

Φgr = 1 ⇔ Ggr(k) = exp

(−k

2

2

)First-order nonclassicality: 8

D2 < 0 ⇒ |G(k, ϕ)| > Ggr(k)

applies to many nonclassical states:Squeezed, Fock, superpositions of coherent states, . . .

Slow decay of G(k, ϕ) → narrow structures in p(x, ϕ)

8W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)

Nonclassical characteristic functions

Typical examples: 9

Ground (vacuum) state (dotted)

Fock state |n = 4〉 (full lines)

Even coherent state: |α〉+ ∼ (|α〉 + | − α〉) (dashed)

9W. Vogel and D-G. Welsch, Quantum Optics (Wiley-VCH, Berlin, 2006), 3rd edition.

Nonclassical characteristic functions

Sub-Planck structures in phase space: 10

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

-0.55

0.64

10W.H. Zurek, Nature 412, 712 (2001)

Nonclassical characteristic functions

Experiment: 11 ρ = η|1〉〈1| + (1− η)|0〉〈0|

11A.I. Lvovsky and J.H. Shapiro, Phys. Rev. A 65, 033830 (2002)

Nonclassical characteristic functions

Photon-added thermal state: 12 ρ = N a†ρtha

First- (a) and second-order (b) nonclassicality

12Zavatta, Parigi and Bellini, Phys. Rev. A 75, 052106 (2007)

Nonclassical characteristic functions

Direct observation via fluorescence 13

resonance fluorescence

Hamiltonian:

Hint = 12~

(ΩA12 + Ω∗A21

)xϕ

⇒ experimental realization for motion of trapped ion14

13S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995)14P. Haljan, K. Brickman, L. Deslauriers, P. Lee, C. Monroe, Phys. Rev. Lett. 94, 153602 (2005)

Nonclassical moments

Nonclassicality: P -function is not a probability distribution

Equivalent condition:

∃f : 〈: f †f :〉 < 0

chosing

f =

∫d2α f (α) :D(−α) :

⇒ Bochner condition!

Normally-ordered expansions (exists and converges):

using quadratures:15 f =∑n,m

fnm : xnϕpmϕ :

using a†, a:16 f =∑n,m

cnm a†nam

15E. Shchukin, Th. Richter, and W. Vogel, PR A 71, 011802(R) (2005)16E. Shchukin and W. Vogel, Phys. Rev. A, 72, 043808 (2005)

Nonclassical momentsQuadrature expansion: 17

f = f (xϕ, pϕ) =∑n,m

fnm : xnϕpmϕ :

• nonclassicality condition

〈: f †f :〉 ⇒∑n,m,k,l

fnmf∗klMnm,kl(ϕ) < 0

whereMnm,kl(ϕ) = 〈: xn+k

ϕ pm+lϕ :〉

• special case:18

f = f (xϕ) =∑n

fn : xnϕ :

• Conditions: negative minors with quadrature moments

17E. Shchukin, Th. Richter, and W. Vogel, Phys. Rev. A 71, 011802(R) (2005)18G.S. Agarwal, Opt. Comm. 95, 109 (1993)

Nonclassical moments

Annihilation/creation operators: 19

Quadratic form: 〈: f †f :〉 =∑

n,m,k,l

c∗nmckl〈a†m+kan+l〉

Leading principal minors:

dN =

∣∣∣∣∣∣∣∣∣∣∣∣∣

1 〈a〉 〈a†〉 〈a2〉 〈a†a〉 〈a†2〉 . . .

〈a†〉 〈a†a〉 〈a†2〉 〈a†a2〉 〈a†2a〉 〈a†3〉 . . .

〈a〉 〈a2〉 〈a†a〉 〈a3〉 〈a†a2〉 〈a†2a〉 . . .

〈a†2〉 〈a†2a〉 〈a†3〉 〈a†2a2〉 〈a†3a〉 〈a†4〉 . . .

〈a†a〉 〈a†a2〉 〈a†2a〉 〈a†a3〉 〈a†2a2〉 〈a†3a〉 . . .

〈a2〉 〈a3〉 〈a†a2〉 〈a4〉 〈a†a3〉 〈a†2a2〉 . . .

∣∣∣∣∣∣∣∣∣∣∣∣∣Principal minors with rows and columns k1 < · · · < kn:

dk, k = (k1, . . . , kn)

Nonclassicality criterion:

∃k : dk < 0

19E. Shchukin and W. Vogel, Phys. Rev. A, 72, 043808 (2005)

Nonclassical moments

Lowest-order nonclassicality condition:

d3 =

∣∣∣∣∣∣1 〈a†〉 〈a〉〈a〉 〈a†a〉 〈a2〉〈a†〉 〈a†2〉 〈a†a〉

∣∣∣∣∣∣ < 0

Factorization:

d3 =1

4minϕ

⟨:(∆xϕ

)2:⟩

maxϕ

⟨:(∆xϕ

)2:⟩,

where xϕ = ae−iϕ + a†eiϕ.

The condition d3 < 0 is equivalent to ordinary squeezing:

∃ϕ :⟨

:(∆xϕ

)2:⟩< 0

d3 < 0 is optimized with respect to the phase ϕ.

Nonclassical momentsHigher-order squeezing:

k-th power amplitude squeezing 20

∆k =

∣∣∣∣∣∣1 〈a†k〉 〈ak〉〈ak〉 〈a†kak〉 〈a2k〉〈a†k〉 〈a†2k〉 〈a†kak〉

∣∣∣∣∣∣ < 0

Factorization:

∆k =1

4minϕ

⟨:(∆F (k)

ϕ

)2:⟩

maxϕ

⟨:(∆F (k)

ϕ

)2:⟩,

where F (k)ϕ = ake−iϕ + a†keiϕ.

Amplitude-squared squeezing: 21

∆2 =

∣∣∣∣∣∣1 〈a†2〉 〈a2〉〈a2〉 〈a†2a2〉 〈a4〉〈a†2〉 〈a†4〉 〈a†2a2〉

∣∣∣∣∣∣ < 0

20E. Shchukin and W. Vogel, J. Phys: Conference Series 36, 183 (2006)21M. Hillery, Phys. Rev. A 72, 3796 (1987)

Nonclassical moments

Higher order squeezing:

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

Q-function of states with third (left) and fourth (right) orderamplitude squeezing.

Recent experiments

Reconstruction of a nonclassical P function

Single photon: P (α) =(1 + ∂

∂α∂∂α∗

)δ(α)

⇒ Photon on a thermal background: 22

Single-photon added thermal state (SPATS): ρ = N a†ρtha

22Zavatta, Parigi, Bellini, Phys. Rev. A 75, 052106 (2007)

Nonclassical P functions

Easy to measure: quadrature characteristic function

G(k, ϕ) =⟨eikxϕ

⟩=

∫dx p(x, ϕ)eikx ⇒ G =

1

N

N∑j=1

eikxϕ(j)

Characteristic function of P (α) : Φ(ike−iϕ) =⟨eikxϕ

⟩ek

2/2

Nonclassical P functions

Resulting characteristic function Φ:

|β|

Φ(β

)

0 0.5 1 1.5 2 2.5 3−1

−0.8−0.6−0.4−0.2

00.20.40.60.8

1

(b)

(a)

(a) SPATS, for nth ≈ 1.1 and η = 0.6

(b) Mixture of SPATS with 19% thermal state, nth ≈ 3.71

Nonclassical P functions

P function of phase-independent states:

Hankel transform ⇒ P (α) = 2π

∫ |β|c0 bJ0(2b|α|)Φ(b)db

Result, for nth ≈ 1.1:23

23Kiesel, Vogel, Zavatta, Parigi, Bellini, Phys. Rev. A 78, 021804(R) (2008)

Nonclassical P functions

Noise effects:

P(α

)

|α|−5 −4 −3 −2 −1 0 1 2 3 4 5

−0.4

−0.2

0

0.2

0.4

0.6

(b)(a)

10 × P(α)

(a) clear statistical significance, for nth ≈ 1.1

(b) at the limits: SPATS mixed with 19% thermal noise,for nth ≈ 3.71

Characteristic function and moments

Phase-diffused squeezed vacuum state

• Wigner function:

W (α) =

∫f (ϕ) 1

2π√VxVp

exp−Re2(αe−iϕ)

2Vx− Im2(αe−iϕ)

2Vp

dϕ

• uncertainty relation: VxVp ≥ 1

• Gaussian distribution f (ϕ) with variance σ2

• experiment with Vx = 0.36, Vp = 5.2824

• 107 quadrature values (balanced homodyne detection)

• state is squeezed for σ < 22.2

Does nonclassicality remain for larger σ?

Which criteria display nonclassicality under such conditions?

24Kiesel, Vogel, Hage, DiGuglielmo, Samblowski, Schnabel, Phys. Rev. A 79, 150505 (2009)

Characteristic function and moments

Quadrature moments

• Hong-Mandel higher-order squeezing25

q2n =

⟨(∆x)2n

⟩(2n− 1)!!

− 1

• a state is nonclassical if ∃n : q2n < 0

σ/ q2 q4 q6 q8

0.0 −0.6362(1± 0.3%) −0.8667(1± 0.16%) −0.9506(1± 0.12%) −0.9813(1± 0.09%)6.3 −0.5717(1± 0.04%) −0.8090(1± 0.03%) −0.9102(1± 0.03%) −0.9549(1± 0.04%)12.6 −0.4060(1± 0.08%) −0.5509(1± 0.15%) −0.5459(1± 0.60%) −0.3852(1± 4.2%)22.2 0.0196(1± 3.2%) 0.6864(1± 0.53%) 2.982(1± 0.84%) 10.61(1± 1.7%)∞ 1.908(1± 0.09%) 10.68(1± 0.16%) 51.72(1± 0.32%) 249.6(1± 0.65%)

Higher-order squeezing does not reveal nonclassicality be-yond ordinary squeezing

25Hong, Mandel, Phys. Rev. Lett. 54, 323 (1985)

Characteristic function and moments

Normally ordered quadrature moments

• a state is nonclassical if

M (l) =

1 〈: x :〉 . . .

⟨: xl−1 :

⟩〈: x :〉

⟨: x2 :

⟩. . .

⟨: xl :

⟩... ... . . . ...⟨

: xl−1 :⟩ ⟨

: xl :⟩. . .

⟨: x2l−2 :

⟩

is not positive semidefinite26

• check sign of minimum eigenvalue

σ/ 2× 2 Matrix 4× 4 Matrix 6× 6 Matrix 8× 8 Matrix0.0 −0.6362(1± 0.25%) −4.294(1± 0.86%) −104.0(1± 2.5%) −6201(1± 6.1%)6.3 −0.5717(1± 0.03%) −3.337(1± 0.11%) −69.93(1± 0.35%) −3593(1± 0.98%)12.6 −0.4060(1± 0.08%) −2.040(1± 1.1%) −6.728(1± 53%) −107.4(1± 110%)22.2 0.0197(1± 3.0%) −0.2323(1± 1.1%) −0.5358(1± 4.1%) −2.299(1± 71%)∞ 1.0000(1± 0%) 0.7856(1± 1.2%) 0.5493(1± 12%) 10.85(1± 13%)

⇒ Extended range of detection of nonclassicality

26 Agarwal, Opt. Commun. 95, 109 (1993)

Characteristic function and moments

Nonclassicality in the characteristic function

A state is nonclassical if27 ∃β with |Φ(β)| > 1

⇒ Lowest order of a hierarchy of conditons28

σ = 0.0

σ = 6.3

σ = 12.6

σ = 22.2

σ = ∞

1 2 3 4 51

0.5

2

51020

50100200

50010002000 |Φ(β)|

|β|

⇒ Nonclassical for all parameters

27W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)28Richter, Vogel, Phys. Rev. Lett. 89, 283601 (2002)

Characteristic function and moments

Some conclusions from the experiments:

• All phase-randomized squeezed states are nonclassical

• High significance of nonclassical effects in the charac-teristic function Φ(β)

• Fourier transform of Φ(β) does not exist

• P function is highly singular

• When |Φ(β)| 6> 1 → FT may exist

• P function becomes regular (example: SPATS)

Nonclassical Correlation PropertiesFirst demonstration of nonclassical lightPhoton Antibunching: 29

C

Normally- and time-ordered intensity correlations:⟨ I(0)I(τ )

⟩>

⟨: [I(0)]2 :

⟩⇒ Violation of Schwarz inequality!

29Kimble, Dagenais, and Mandel, Phys. Rev. Lett. 39, 691 (1977)

Nonclassical Correlation Properties

Radiation source: resonance fluorescence

atomic beam of low density

⇒ single atom emits separated photons!

Nonclassical Correlation Properties

Experimental results: [Kimble, Dagenais, and Mandel (1977)]

Nonclassical Correlation Properties

Generalization: 30 P function ⇒ P functional

P [E(+)(i)] = 〈 k∏

i=1

δ(E(+)(i)− E(+)(i)) 〉

⇒ normally and time-ordered

Classical Correlations:P [E(+)(i)] is a joint probability density ⇒ non-negative

⇔ ∀f : 〈 f †f 〉 ≥ 0

⇒ General quadratic form:

〈 f †f 〉 =

∑pi,qi,ni,mi〈

[E(−)(1)]n1+q1 . . . [E(−)(k)]nk+qk

× [E(+)(k)]mk+pk . . . [E(+)(1)]m1+p1 〉 c∗pi,qicni,mi

Quantum Correlations ⇔ ∃f : 〈 f †f 〉 < 0

⇔ There exists (at least one) negative principal minor

30W. Vogel, Phys. Rev. Lett. 100, 013605 (2008)

General Quantum Correlations

Lowest-order conditions (minors of second order):∣∣∣⟨ [E(−)(1)]n1+q1 . . . [E(−)(k)]nk+qk[E(+)(1)]m1+p1 . . . [E(+)(k)]mk+pk

⟩∣∣∣2 >⟨ [I(1)]n1+m1 . . . [I(k)]nk+mk

⟩ ⟨ [I(1)]p1+q1 . . . [I(k)]pk+qk

⟩Special cases:

• Photon antibunching (nonstationary):⟨ I(1)I(2)

⟩>

√⟨: [I(1)]2 :

⟩ ⟨: [I(2)]2 :

⟩• Intensity-fieldstrength correlations:∣∣∣⟨

E(−)(1)I(2)

⟩∣∣∣ > √⟨I(1)

⟩ ⟨: [I(2)]2 :

⟩,

∣∣∣⟨ E(1)I(2)

⟩∣∣∣ > √⟨: [E(1)]2 :

⟩ ⟨: [I(2)]2 :

⟩Recent experiment with trapped ions31

31Gerber, Rotter, Slodicka, Eschner, Carmichael, Blatt, Phys. Rev. Lett. 102, 183601 (2009)

Quantifying nonclassicality

General nonclassicality condition:

〈: f †f :〉 < 0

Example: quadrature squeezing

f ≡ ∆xϕ = xϕ − 〈xϕ〉 → 〈: (∆xϕ)2 :〉 < 0

Limit for negativity:32

∆ = 〈: f †f :〉 − 〈f †f〉 → ∆ ≤ 〈: f †f :〉 < 0

Operational relative nonclassicality:

R ≡ 〈: f †f :〉∆

=〈: f †f :〉

〈: f †f :〉 − 〈f †f〉

Perfect situation: ∆ = 〈: f †f :〉 ⇐⇒ 〈f †f〉 = 0

Realized for: f |ψ〉 = 0

32C. Gehrke and W. Vogel, arxiv:0904.3390 [quant-ph]

Quantifying nonclassicality

Example: squeezed vacuum state:

(µa + νa†)|0; ν〉 = 0 , µ2 − |ν|2 = 1

Needed: measurement of f †f with f ≡ µa + νa†

Realization for trapped ion:

Hint =~2Ωf A21 + H.c., ν =

|Ωb||Ω|

ei∆ϕ

Electronic-state dynamics:

p2(t) =1

2

1 + Tr

[ρ(0) cos

(|Ω|t

√f †f + 1

)]For ρ (0) = |0; ν〉〈0; ν| → p2(t) = 1

2 [1 + cos (|Ω|t)]

Quantum-noise free measurement: moderate squeezing!

Quantifying nonclassicality

Experimental realization:

Summary

• Reconstruction of nonclassical P functions

• Nonclassical characteristic functions

• General conditions for nonclassical moments

• Experimental realizations

• General nonclassical correlation properties

• Operational quantification of nonclassicality

• Quantum noise free measurements requires only mod-erate squeezing!