Nonclassical Harmonic Oscillator. Werner Vogel Universität Rostock, Germany

Transcription

1 Nonclassical Harmonic Oscillator Werner Vogel Universität Rostock, Germany

2 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

3 Introduction Squeezing: ( ˆx ϕ ) 2 < ( ˆx ϕ ) 2 gr, ˆx ϕ = âe iϕ + â e iϕ nonclassical? Subtracting ground-state (vacuum) noise: : ( ˆx ϕ ) 2 : < 0 ( x ϕ ) 2 cl 0 observable (sufficient) condition for nonclassicality!

4 Nonclassical phase-space functions P-representation of the density operator: 1 ˆρ = d 2 α P (α) α α resembles classical mixture! Expectation values: : ˆF (â, â) : = d 2 αp (α)f (α, α) Correspondence to classical mean values: (1) subtracting ground-state noise via ˆF : ˆF : (2) P corresponds to classical probability: 2 P (α) P cl (α) 1 E. C. G. Sudarshan, Phys. Rev. Lett. 10, 227 (1963); R. J. Glauber, Phys. Rev. 131, 2766 (1963) 2 U.M. Titulaer and R.J. Glauber, Phys. Rev. 140, B676 (1965)

5 Nonclassical phase-space functions A state is nonclassical, if: 3 (a) ground-state noise is substantial, cf. nonclassicality in weak measurements 4 alternatively: 5 small photon numbers (b) P fails to be a classical probability: examples: P (α) P cl (α) Squeezing: : ( ˆx ϕ ) 2 : < 0 sub-poissonian statistics: : ( ˆn) 2 : < 0 Sought: observable conditions for P (α) P cl (α) Problem: P (α) may be strongly singular! 3 W. Vogel, Phys. Rev. Lett. 84, 1849 (2000) 4 L.M. Johansen, Phys. Lett. A 329, 184 (2004) 5 L. Mandel, Phys. Scr. T 12, 34 (1986)

6 Nonclassical characteristic functions Characteristic function of P (α): Φ(β) = d 2 α P (α) exp[(αβ α β)] Theorem (Bochner 1933): 6 P (α) is a probability distribution iff for any smooth function f(α) with compact support the following expression is nonnegative: d 2 α d 2 β Φ(α β) f (α)f(β) 0 corresponding discrete version: n Φ(β i β j ) ξi ξ j 0, i,j=1 for any integer n and all complex β i, ξ k (i, k = 1... n). 6 T. Kawata, Fourier Analysis in Probability Theory, Academic Press, N.Y. 1972

7 Nonclassical characteristic functions Define matrix: Φ ij = Φ(β i β j ) Theorem: A continuous function Φ(β) with Φ(0) = 1 and Φ (β) = Φ( β) is a classical characteristic function, iff 1 Φ 12 Φ 1k Φ D k D k (β 1,... β k ) = 12 1 Φ 2k Φ 1k Φ 2k 1 for any order k = 1,..., +. Nonclassicality: 7 P (α) is not a probability iff there exist values of k and β k (k = 2... ) with D k (β 1,... β k ) < 0 7 T. Richter and W. Vogel, Phys. Rev. Lett. 89, (2002)

8 Nonclassical characteristic functions Observable characteristic functions of quadratures: G(k, ϕ) = G gr (k) Φ(ike iϕ ) FT[p(x, ϕ)], in the ground state: Φ gr = 1 G gr (k) = exp ( k2 2 ) First-order nonclassicality: 8 D 2 < 0 G(k, ϕ) > G gr (k) applies to many nonclassical states: Squeezed, Fock, superpositions of coherent states,... Slow decay of G(k, ϕ) narrow structures in p(x, ϕ) 8 W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)

9 Nonclassical characteristic functions Typical examples: 9 Ground (vacuum) state (dotted) Fock state n = 4 (full lines) Even coherent state: α + ( α + α ) (dashed) 9 W. Vogel and D-G. Welsch, Quantum Optics (Wiley-VCH, Berlin, 2006), 3 rd edition.

10 Nonclassical characteristic functions Sub-Planck structures in phase space: W.H. Zurek, Nature 412, 712 (2001)

11 Nonclassical characteristic functions Experiment: 11 ˆρ = η (1 η) A.I. Lvovsky and J.H. Shapiro, Phys. Rev. A 65, (2002)

12 Nonclassical characteristic functions Photon-added thermal state: 12 ˆρ = N â ˆρ th â First- (a) and second-order (b) nonclassicality 12 Zavatta, Parigi and Bellini, Phys. Rev. A 75, (2007)

13 Nonclassical characteristic functions Direct observation via fluorescence 13 resonance fluorescence Hamiltonian: ( ) Ĥ int = 1 2 ΩÂ12 + Ω Â 21 ˆx ϕ experimental realization for motion of trapped ion S. Wallentowitz and W. Vogel, Phys. Rev. Lett. 75, 2932 (1995) 14 P. Haljan, K. Brickman, L. Deslauriers, P. Lee, C. Monroe, Phys. Rev. Lett. 94, (2005)

14 Nonclassical moments Nonclassicality: P -function is not a probability distribution Equivalent condition: ˆf : : ˆf ˆf : < 0 chosing ˆf = d 2 α f(α) : ˆD( α): Bochner condition! Normally-ordered expansions (exists and converges): using quadratures: 15 ˆf = f nm : ˆx n ϕ ˆp m ϕ : n,m using â, â: 16 ˆf = c nm â n â m n,m 15 E. Shchukin, Th. Richter, and W. Vogel, PR A 71, (R) (2005) 16 E. Shchukin and W. Vogel, Phys. Rev. A, 72, (2005)

15 Nonclassical moments Quadrature expansion: 17 ˆf = f(ˆx ϕ, ˆp ϕ ) = n,m f nm : ˆx n ϕ ˆp m ϕ : nonclassicality condition : ˆf ˆf : where special case: 18 n,m,k,l M nm,kl (ϕ) = : ˆx n+k ϕ f nm f klm nm,kl (ϕ) < 0 ˆp m+l ϕ : ˆf = f(ˆxϕ) = f n : ˆx n ϕ : Conditions: negative minors with quadrature moments n 17 E. Shchukin, Th. Richter, and W. Vogel, Phys. Rev. A 71, (R) (2005) 18 G.S. Agarwal, Opt. Comm. 95, 109 (1993)

16 Nonclassical moments Annihilation/creation operators: 19 Quadratic form: : ˆf ˆf : = n,m,k,l c nmc kl â m+k â n+l Leading principal minors: 1 â â â 2 â â â 2... â â â â 2 â â 2 â 2 â â 3... â â d N = 2 â â â 3 â â 2 â 2 â... â 2 â 2 â â 3 â 2 â 2 â 3 â â 4... â â â â 2 â 2 â â â 3 â 2 â 2 â 3 â... â 2 â 3 â â 2 â 4 â â 3 â 2 â 2... Principal minors with rows and columns k 1 < < k n : d k, k = (k 1,..., k n ) Nonclassicality criterion: k : d k < 0 19 E. Shchukin and W. Vogel, Phys. Rev. A, 72, (2005)

17 Nonclassical moments Lowest-order nonclassicality condition: 1 â â d 3 = â â â â 2 â â 2 â â < 0 Factorization: d 3 = 1 4 min ϕ where ˆx ϕ = âe iϕ + â e iϕ. : ( ˆx ϕ ) 2 : max ϕ : ( ˆx ϕ ) 2 :, The condition d 3 < 0 is equivalent to ordinary squeezing: ϕ : : ( ) 2 ˆx ϕ : < 0 d 3 < 0 is optimized with respect to the phase ϕ.

18 Nonclassical moments Higher-order squeezing: k-th power amplitude squeezing 20 Factorization: k = k = 1 4 min ϕ 1 â k â k â k â k â k â 2k â k â 2k â k â k : ( ) (k) 2 ˆF ϕ : max ϕ : ( < 0 ) (k) 2 ˆF ϕ :, where ˆF (k) ϕ = â k e iϕ + â k e iϕ. Amplitude-squared squeezing: 21 1 â 2 â 2 2 = â 2 â 2 â 2 â 4 â 2 â 4 â 2 â 2 < 0 20 E. Shchukin and W. Vogel, J. Phys: Conference Series 36, 183 (2006) 21 M. Hillery, Phys. Rev. A 72, 3796 (1987)

19 Nonclassical moments Higher order squeezing: Q-function of states with third (left) and fourth (right) order amplitude squeezing.

20 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): ˆρ = N â ˆρ th â 22 Zavatta, Parigi, Bellini, Phys. Rev. A 75, (2007)

21 Nonclassical P functions Easy to measure: quadrature characteristic function G(k, ϕ) = e ikˆx ϕ = dx p(x, ϕ)e ikx G = 1 N N j=1 e ikx ϕ(j) Characteristic function of P (α) : Φ(ike iϕ ) = e ikˆx ϕ e k2 /2

22 Nonclassical P functions Resulting characteristic function Φ: Φ(β) (a) (b) β (a) SPATS, for n th 1.1 and η = 0.6 (b) Mixture of SPATS with 19% thermal state, n th 3.71

23 Nonclassical P functions P function of phase-independent states: Hankel transform P (α) = 2 π Result, for n th 1.1: 23 β c 0 bj 0 (2b α )Φ(b)db 23 Kiesel, Vogel, Zavatta, Parigi, Bellini, Phys. Rev. A 78, (R) (2008)

24 Noise effects: Nonclassical P functions P(α) P(α) (a) (b) α (a) clear statistical significance, for n th 1.1 (b) at the limits: SPATS mixed with 19% thermal noise, for n th 3.71

25 Characteristic function and moments Phase-diffused squeezed vacuum state Wigner function: W (α) = f(ϕ) 1 2π V x V p exp uncertainty relation: V x V p 1 { } Re2 (αe iϕ ) 2V x Im2 (αe iϕ ) 2V p Gaussian distribution f(ϕ) with variance σ 2 experiment with V x = 0.36, V p = 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? dϕ 24 Kiesel, Vogel, Hage, DiGuglielmo, Samblowski, Schnabel, Phys. Rev. A 79, (2009)

26 Characteristic function and moments Quadrature moments Hong-Mandel higher-order squeezing 25 ( ˆx) 2n q 2n = (2n 1)!! 1 a state is nonclassical if n : q 2n < 0 σ/ q 2 q 4 q 6 q (1 ± 0.3%) (1 ± 0.16%) (1 ± 0.12%) (1 ± 0.09%) (1 ± 0.04%) (1 ± 0.03%) (1 ± 0.03%) (1 ± 0.04%) (1 ± 0.08%) (1 ± 0.15%) (1 ± 0.60%) (1 ± 4.2%) (1 ± 3.2%) (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 beyond ordinary squeezing 25 Hong, Mandel, Phys. Rev. Lett. 54, 323 (1985)

27 Characteristic function and moments Normally ordered quadrature moments a state is nonclassical if 1 : ˆx :... M (l) : ˆx : : ˆx 2 :... =..... : ˆx l 1 : : ˆx l :... : ˆx l 1 : : ˆx l :. : ˆx 2l 2 : is not positive semidefinite 26 check sign of minimum eigenvalue σ/ 2 2 Matrix 4 4 Matrix 6 6 Matrix 8 8 Matrix (1 ± 0.25%) 4.294(1 ± 0.86%) 104.0(1 ± 2.5%) 6201(1 ± 6.1%) (1 ± 0.03%) 3.337(1 ± 0.11%) 69.93(1 ± 0.35%) 3593(1 ± 0.98%) (1 ± 0.08%) 2.040(1 ± 1.1%) 6.728(1 ± 53%) 107.4(1 ± 110%) (1 ± 3.0%) (1 ± 1.1%) (1 ± 4.1%) 2.299(1 ± 71%) (1 ± 0%) (1 ± 1.2%) (1 ± 12%) 10.85(1 ± 13%) Extended range of detection of nonclassicality 26 Agarwal, Opt. Commun. 95, 109 (1993)

28 Characteristic function and moments Nonclassicality in the characteristic function A state is nonclassical if 27 β with Φ(β) > 1 Lowest order of a hierarchy of conditons Φ(β) σ = 0.0 σ = 6.3 σ = 12.6 σ = 22.2 σ = β Nonclassical for all parameters 27 W. Vogel, Phys. Rev. Lett. 84, 1849 (2000) 28 Richter, Vogel, Phys. Rev. Lett. 89, (2002)

29 Characteristic function and moments Some conclusions from the experiments: All phase-randomized squeezed states are nonclassical High significance of nonclassical effects in the characteristic function Φ(β) Fourier transform of Φ(β) does not exist P function is highly singular When Φ(β) 1 FT may exist P function becomes regular (example: SPATS)

30 Nonclassical Correlation Properties First demonstration of nonclassical light Photon Antibunching: 29 Normally- and time-ordered intensity correlations: Î(0)Î(τ) > : [Î(0)]2 : Violation of Schwarz inequality! C 29 Kimble, Dagenais, and Mandel, Phys. Rev. Lett. 39, 691 (1977)

31 Nonclassical Correlation Properties Radiation source: resonance fluorescence atomic beam of low density single atom emits separated photons!

32 Nonclassical Correlation Properties Experimental results: [Kimble, Dagenais, and Mandel (1977)]

33 Nonclassical Correlation Properties Generalization: 30 P function P functional P [{E (+) (i)}] = k ˆδ(Ê(+) (i) E (+) (i)) i=1 normally and time-ordered Classical Correlations: P [{E (+) (i)}] is a joint probability density non-negative General quadratic form: ˆf : ˆf ˆf 0 ˆf ˆf = {p i,q i,n i,m i } [Ê( ) (1)] n 1+q 1... [Ê( ) (k)] n k+q k [Ê(+) (k)] m k+p k... [Ê(+) (1)] m 1+p 1 c {p i,q i } c {n i,m i } Quantum Correlations ˆf : ˆf ˆf < 0 There exists (at least one) negative principal minor 30 W. Vogel, Phys. Rev. Lett. 100, (2008)

34 General Quantum Correlations Lowest-order conditions (minors of second order): [Ê( ) (1)] n 1+q 1... [Ê( ) (k)] n k+q k [Ê(+) (1)] m 1+p 1... [Ê(+) (k)] m k+p k 2 > [Î(1)]n 1+m 1... [Î(k)]n k+m k [Î(1)]p 1+q 1... [Î(k)]p k+q k Special cases: Photon antibunching (nonstationary): Î(1)Î(2) > : [Î(1)]2 : : [Î(2)]2 : Intensity-fieldstrength correlations: Ê( ) (1)Î(2) > Î(1) : [Î(2)]2 :, Ê(1)Î(2) > : [Ê(1)]2 : : [Î(2)]2 : Recent experiment with trapped ions Gerber, Rotter, Slodicka, Eschner, Carmichael, Blatt, Phys. Rev. Lett. 102, (2009)

35 Quantifying nonclassicality General nonclassicality condition: : ˆf ˆf : < 0 Example: quadrature squeezing ˆf ˆx ϕ = ˆx ϕ ˆx ϕ : ( ˆx ϕ ) 2 : < 0 Limit for negativity: 32 = : ˆf ˆf : ˆf ˆf : ˆf ˆf : < 0 Operational relative nonclassicality: R : ˆf ˆf : = : ˆf ˆf : : ˆf ˆf : ˆf ˆf Perfect situation: = : ˆf ˆf : ˆf ˆf = 0 Realized for: ˆf ψ = 0 32 C. Gehrke and W. Vogel, arxiv: [quant-ph]

36 Quantifying nonclassicality Example: squeezed vacuum state: (µâ + νâ ) 0; ν = 0, µ 2 ν 2 = 1 Needed: measurement of ˆf ˆf with ˆf µâ + νâ Realization for trapped ion: p 2 (t) = 1 2 Ĥ int = 2 Ω ˆfÂ21 + H.c., ν = Ω b Ω ei ϕ Electronic-state dynamics: { 1 + Tr [ ˆρ(0) cos ( )]} Ω t ˆf ˆf + 1 For ˆρ (0) = 0; ν 0; ν p 2 (t) = 1 2 [1 + cos ( Ω t)] Quantum-noise free measurement: moderate squeezing!

37 Quantifying nonclassicality Experimental realization:

38 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 moderate squeezing!