FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS

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1 FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Matter Optics and Shelving Effect Bassano Vacchini DIPARTIMENTO DI FISICA - UNIVERSITÀ DI MILANO ISTITUZIONI DI FISICA TEORICA 30 MAGGIO 2003 FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.1

2 Contents Statistical nature of experiments in quantum mechanics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

3 Contents Statistical nature of experiments in quantum mechanics Matter optics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

4 Contents Statistical nature of experiments in quantum mechanics Matter optics electron optics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

5 Contents Statistical nature of experiments in quantum mechanics Matter optics electron optics neutron optics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

6 Contents Statistical nature of experiments in quantum mechanics Matter optics electron optics neutron optics atom optics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

7 Contents Statistical nature of experiments in quantum mechanics Matter optics electron optics neutron optics atom optics molecular optics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

8 Contents Statistical nature of experiments in quantum mechanics Matter optics electron optics neutron optics atom optics molecular optics Continual measurement in quantum mechanics FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

9 Contents Statistical nature of experiments in quantum mechanics Matter optics electron optics neutron optics atom optics molecular optics Continual measurement in quantum mechanics shelving effect FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.2

10 Order of magnitudes m(g) λ db (pm) v(m/s) realization electrons , 5X neutrons atoms molecules Bohr radius: a 0 0,5 Å visible light: λ 0,5 µm X rays: λ 1Å 1 Å= 100 pm=10 10 m 1 µm=10 4 Å=10 6 m FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.3

11 Traditional axiomatic of quantum mechanics One particle system: ψ normalized vector in a Hilbert space H FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.4

12 Traditional axiomatic of quantum mechanics One particle system: ψ normalized vector in a Hilbert space H Statistical interpretation: e event associated to a subspace S e H p e = ˆP e ψ 2 = ψ ˆP e ψ e t time translated event p e (t) = ˆP e (t)ψ 2 = ψ t ˆP e ψ t ψ t = e i Ĥt ψ FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.4

13 Traditional axiomatic of quantum mechanics One particle system: ψ normalized vector in a Hilbert space H Statistical interpretation: e event associated to a subspace S e H p e = ˆP e ψ 2 = ψ ˆP e ψ e t time translated event p e (t) = ˆP e (t)ψ 2 = ψ t ˆP e ψ t ψ t = e i Ĥt ψ Observable quantity: Â self-adjoint operator in H Â = n a ˆP n n Â t = n a np en (t) = n a n ˆP en (t)ψ 2 family of events e n corresponding to the property: the quantity A has the value a n FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.4

14 Statistical nature of experiments General Scheme of single particle experiment: SOURCE preparation procedure direct interaction DETECTOR registration procedure FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.5

15 Statistical nature of experiments General Scheme of single particle experiment: SOURCE preparation procedure direct interaction DETECTOR registration procedure Available data: description of source and detector, relative frequencies (reproducible) FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.5

16 Statistical nature of experiments General Scheme of single particle experiment: SOURCE preparation procedure direct interaction DETECTOR registration procedure Available data: description of source and detector, relative frequencies (reproducible) Preparation of the system: ψ normalized vector in a Hilbert space H FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.5

17 Statistical nature of experiments General Scheme of single particle experiment: SOURCE preparation procedure direct interaction DETECTOR registration procedure Available data: description of source and detector, relative frequencies (reproducible) Preparation of the system: ψ normalized vector in a Hilbert space H Observed data: probability that the quantity A associated to the self-adjoint operator Â in H takes values in a given interval M of the real line FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.5

18 Matter optics Linearity of the Schrödinger equation: i d ψ dt t = Ĥ ψ t ψ 1t = e iφt ψ 1t, ψ 2t = e iχt ψ 2t stationary solutions ψ 1t + ψ 2t solution ψ 1t + ψ 2t 2 = ψ 1t 2 + ψ 2t ψ 1t ψ 2t cos[(φ χ)t] FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.6

19 Matter optics Linearity of the Schrödinger equation: i d dt ψ t = Ĥ ψ t ψ 1t = e iφt ψ 1t, ψ 1t + ψ 2t solution ψ 2t = e iχt ψ 2t stationary solutions ψ 1t + ψ 2t 2 = ψ 1t 2 + ψ 2t ψ 1t ψ 2t cos[(φ χ)t] Equation for the stationary states: Ĥ ψ = E ψ ψ(x) + 2m [E V (x)]ψ(x) = 0 2 V (x) optical potential (also complex, macroscopic quantity) 2m k(x) = 1 [E V (x)]ê n(x) = k(x) 2 wave vector and refraction index k 0 = 1 V (x) E FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.6

20 Matter optics Linearity of the Schrödinger equation: i d dt ψ t = Ĥ ψ t ψ 1t = e iφt ψ 1t, ψ 1t + ψ 2t solution ψ 2t = e iχt ψ 2t stationary solutions ψ 1t + ψ 2t 2 = ψ 1t 2 + ψ 2t ψ 1t ψ 2t cos[(φ χ)t] Equation for the stationary states: Ĥ ψ = E ψ ψ(x) + 2m [E V (x)]ψ(x) = 0 2 V (x) optical potential (also complex, macroscopic quantity) 2m k(x) = 1 [E V (x)]ê n(x) = k(x) 2 wave vector and refraction index k 0 = 1 V (x) E Helmholtz equation: [ + k 2 (x)]ψ(x) = 0 FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.6

Electron optics Biprism Source: electron gun Phase: interaction with electrostatic field Values: m 10 27

21 Electron optics Biprism Source: electron gun Phase: interaction with electrostatic field Values: m g λ db 5pm Single particle experiment FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.7

Experimental results Difficulties in producing a sufficiently coherent beam (monochromatic and well collimated) Developments in electronic

22 Experimental results Difficulties in producing a sufficiently coherent beam (monochromatic and well collimated) Developments in electronic microscopy Detection efficiency close to one Experiments last approximately 30 minutes FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.8

Neutron optics I Single and double slit Source: nuclear reactor Values: m 10 24 g λ db 2000pm

23 Neutron optics I Single and double slit Source: nuclear reactor Values: m g λ db 2000pm Single particle experiment FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.9

24 Experimental results Slit width 90µm Slit width 23µm Single slit obtained through glass plates coated with strong absorbers (gadolinium, borum) Double slit obtained inserting in the middle a thin line of borum Double slit Experiments last approximately 300 hours FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.10

a sample of homogeneous material (Al) Values: m 10 24 g λ db 200pm Single

25 Neutron optics II Mach-Zender interferometer (Bragg reflection from a monolithic silicon crystal) Source: nuclear reactor Phase: interaction with a sample of homogeneous material (Al) Values: m g λ db 200pm Single particle experiment FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.11

$Experimental results Optic potential V = 2π 2 m ρ sample density ρb Refractive index n = 1 2π 2 p 2 ρb e i p x e i p (x ) = 2π 2 p 2 ρbl Al Good visibility also at high$

26 Experimental results Optic potential V = 2π 2 m ρ sample density ρb Refractive index n = 1 2π 2 p 2 ρb e i p x e i p (x ) = 2π 2 p 2 ρbl Al Good visibility also at high interference order Macroscopic dimensions of apparatus (centimeters) and related possibility to interact with the beam FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.12

27 Interference of Gaussian wave-packets Preparation given by a Gaussian minimum uncertainty wave-packet: ( ) 3/4 ( ) 1 ψ in (x) = 2πσ exp 1 x 2 + i p x 2 4σx 2 0 x ( ) 3/4 ( ) 1 ψ in (p) = 2πσ exp 1 (p p p 2 4σp 2 0 ) 2 σ x σ p = 2 FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.13

28 Interference of Gaussian wave-packets Preparation given by a Gaussian minimum uncertainty wave-packet: ( ) 3/4 ( ) 1 ψ in (x) = 2πσ exp 1 x 2 + i p x 2 4σx 2 0 x ( ) 3/4 ( ) 1 ψ in (p) = 2πσ exp 1 (p p p 2 4σp 2 0 ) 2 σ x σ p = 2 Output state: ψ out (x) = 1 2 ψ out (p) = [ψ in (x) + ψ in (x )] [ 1 ψin 2 (p) + ψ ] in (p)e i p relative phase due to interaction with the aluminum sample in one of the two paths of the interferometer The free evolution does not influence the interference pattern FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.13

29 Related probability distributions Position and momentum probability distributions: I(x) = ψ out (x) 2 ( = πσ 2 x ) 3/2 [e 1 2σx 2 x 2 + e 1 2σx 2 (x ) 2 + 2e 2 σp cos ( p 0 ] ) e 1 2σx 2 (x 2 )2 I(p) = ψ out (p) 2 ( = πσ 2 p ) 3/2 e 1 σp 2 (p p 0 ) 2 [ ( 1 + cos p )] FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.14

30 Related probability distributions Position and momentum probability distributions: I(x) = ψ out (x) 2 ( = πσ 2 x I(p) = ψ out (p) 2 ( = πσ 2 p ) 3/2 [e 1 2σx 2 x 2 + e 1 2σx 2 (x ) 2 + 2e 2 σp 2 ) 3/2 e 1 σp 2 (p p 0 ) 2 [ ( 1 + cos p Total fraction of outgoing neutrons: I = R 3 d 3 x ψ out (x) 2 = R 3 d 3 p ψ out (p) 2 = 1 4 )] I measured quantity as a function of Interference pattern lost if 2 σ 2 p σ 2 x cos ( p 0 ] ) e 1 2σx 2 (x 2 )2 [ 1 + e 2 σ 2 p 2 2 cos ( p 0 ) ] FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.14

31 Loss of interference pattern With increasing relative phase the interference pattern is more and more suppressed I osc e 2 σ 2 p 2 2 cos ( p 0 ) FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.15

32 Recover of interference pattern I osc e 2 σ 2 p 2 2 cos ( p 0 ) By momentum selection σ p is decreased and the interference pattern is recovered Obvious reduction of average countings FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.16

33 Interference: x versus p I osc e versus I osc (p) e 1 2σ 2 p σ 2 x cos ( p 0 ) (p p 0 ) 2 cos ( ) p setting χ = = 2π 2 ρbl p 2 Al 0 one has I osc e σx 2 cos (χp 0 ) versus I osc (p) e 1 2σp 2 (p p 0 ) ) 2 p cos (χp 0 0 p The spectrum of outgoing neutrons is measured FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.17

34 Interference: x versus p I osc e versus I osc (p) e 1 2σ 2 p σ 2 x cos ( p 0 ) (p p 0 ) 2 cos ( ) p setting χ = = 2π 2 ρbl p 2 Al 0 one has I osc e σx 2 cos (χp 0 ) versus I osc (p) e 1 2σp 2 (p p 0 ) ) 2 p cos (χp 0 0 p The spectrum of outgoing neutrons is measured FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.17

35 Absorption in one path: amplitude versus probability Sample of absorbing material in one path: ψ out (x) = [ψ in (x) + aψ in (x )] a transmission probability, a = e σabsnl 2, 0 a 1 [ I = a + 2 ae 2 σ 2 p cos ( p 0 ) ] FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.18

36 Absorption in one path: amplitude versus probability Sample of absorbing material in one path: ψ out (x) = [ψ in (x) + aψ in (x )] a transmission probability, a = e σabsnl 2, 0 a 1 [ I = a + 2 ae 2 σ 2 p cos ( p 0 ) ] Turning sawtooth wheel of absorbing material in one path: a transmission probability, a = t open t open +t close, 0 a 1 I = a 1 d 3 x 4 R [ψ in (x) + ψ in (x )] 2 + (1 a) 1 d 4 R 3 x ψ 3 in (x) 2 [ = a + 2ae 2 σ 2 p cos ( p 0 ) ] FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.18

Experimental results Experiments performed with different absorbers Experiment feasible due to the macroscopic separation between the paths Different visibility of the interference

37 Experimental results Experiments performed with different absorbers Experiment feasible due to the macroscopic separation between the paths Different visibility of the interference pattern despite the same number of detected neutrons Complementary informations: path knowledge or fringes visibility FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.19

38 Light versus matter Optics: optical elements made up of matter (slits, gratings, mirrors, dispersive materials) FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.20

39 Light versus matter Optics: optical elements made up of matter (slits, gratings, mirrors, dispersive materials) Matter optics: optical elements realized through external fields, due to interaction with both matter and electromagnetic waves FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.20

40 Light versus matter Optics: optical elements made up of matter (slits, gratings, mirrors, dispersive materials) Matter optics: optical elements realized through external fields, due to interaction with both matter and electromagnetic waves Kapitza Dirac effect: diffraction of matter by light gratings, stationary electromagnetic wave obtaining through two counterpropagating laser beams FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.20

41 Atom optics I Diffraction by a light grating Sodium atoms Source: high temperature oven Phase: dipole interaction with the electric field of the stationary wave Values: m g λ db 20pm Single particle experiment FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.21

42 Experimental results Similar experiments performed with matter gratings Realization of the Kapitza Dirac effect Complex description of light matter interaction: correlation between internal and translational degrees of freedom FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.22

43 Atom optics II Mach-Zender interferometer realized with light gratings Argon atoms Source: high temperature oven Phase: dipole interaction with the electric field of the stationary wave Values: m g λ db 12pm FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.23

44 Experimental results Real spatial separation between the two paths of the interferometer Higher accuracy in the determination of the grating period with respect to material gratings Similar experiments performed with fullerene molecules FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.24

45 Atom interferometry Wide variety of atom species, with many internal degrees of freedom (addressable through interaction with electromagnetic fields) Higher measurement accuracy: λ light λ db (pm... µm, room temperature... laser cooling) Sensitivity to gravitational effects Cheap sources Efficient detectors FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.25

46 Molecular optics Diffraction by a material grating Fullerene molecule C60 Fullerene molecules: C60 e C70 Source: high temperature oven Experimental apparatus Values: m g λdb 3pm Single particle experiment FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.26

47 Experimental results v/v 60% Most massive objects considered up to now v/v 17% Very complex internal structure Study of the transition between quantum and classical regime FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.27

48 Paul ion trap Ion in an electromagnetic trap stored and cooled with laser techniques 3 level system, coupled to 2 lasers τ 1 τ 2 strong and weak transition Monitoring in time of the strong transition fluorescence FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.28

49 Shelving effect Single trapped ion Intermittent fluorescence Bohr jumps Shelving effect FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.29

50 Shelving effect Single trapped ion Intermittent fluorescence Bohr jumps Shelving effect FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.29

51 Experimental observation Effect observed with different atoms Amplification scheme for the weak transition Precision spectroscopy Telegraphic signal observed in time FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.30

52 Theory Quantum description Intertime law for the countings p(t): probability density to detect the next photon at time t given a counting at time t = 0 p(t) Ae t τ L + Be t τ B τ B τ L T (fluorescence) = τ B N T (dark) = τl Description of counting statistics System evolution conditioned upon the performed measurements FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.31

53 Bibliography Electron optics A. Tonomura, The quantum world unveiled by electron waves (World Scientific, 1998). A. Tonomura et al., Am. J. Phys. 57, 117 (1989). Neutron optics S. A. Werner and H. Rauch, Neutron Interferometry (Oxford University Press, Oxford, 2000). V. F. Sears, Neutron Optics (Oxford University Press, Oxford, 1989). R. Gähler and A. Zeilinger, Am. J. Phys. 59, 316 (1991). Atom optics P. Meystre, Atom Optics (Springer, Berlin, 2001). C. S. Adams, M. Siegel, J. Mlynek, Phys. Rep. 240, 143 (1994). Molecular optics O. Nairz, M. Arndt and A. Zeilinger, Am. J. Phys. 71, 319 (2003). Shelving effect H. Dehmelt, Rev. Mod. Phys. 62, 525 (1990). M. B. Plenio and P. Knight, Rev. Mod. Phys. 70, 101 (1998). FOUNDATIONAL EXPERIMENTS IN QUANTUM MECHANICS Bassano Vacchini p.32

Fisica Generale 3. Bassano Vacchini.

Fisica Generale 3. Bassano Vacchini. Fisica Generale 3 Bassano Vacchini bassano.vacchini@mi.infn.it Overview Ø Interference experiments Ø Diffraction and interference with light Ø Interference with light in the limit of low intensity Ø The