Weak values of electron spin in a double quantum dot

Transcription

1 Weak values of electron spin in a double quantum dot Alessandro Romito collaborators: Yuval Gefen Yaroslav Blanter Oded Zilberberg Chernogolovka, Semptember 2010

2

3 weak values A = χ 0  χ 0 = f χ f χ 0 2 χ f  χ 0 χ f χ 0 = f  w 0 probability to obtain the post-selected state preselection preselection: prepare the system in the state χ 0 weak measurement of  postselection post-selection: by a strong measurement of ˆB project the system onto the state χ f time

4 weak values P (I) weak measurement: strongly overlapping wavefunctions strong projective measurement: separated peaks P (I) I L I R I I L I R I

5 developments of weak values weak values in solid state? quantum optics: weak values of photon polarization (exp.) foundations of quantum mechanics: interpretation of Hardy paradox and counterfactual statements in quantum mechanics Pryde et al., PRL (2005) Aharonov et al., Phys Lett. A(2002) enhancement of a small effect: observation of photon spin-hall effect Hosten et al., Science (2008) short decoherence time find non-commuting observables... First proposal for WV in solid state systems weak value tomography role of fluctuations and decoherence Many-body weak values Romito et al., PRL (2008) Romito et al., Phys. E (2010) Shpitalnik et al., PRL (2008)

6 manipulating electron spin J. R. Petta et al. Science 309 (2005) V L V R energy S e S L T 0 T 0,± V T T 0 T + B nl B nr ɛ B 0 ɛ A S g S R S g control the voltage difference:! ɛ B 0 (1,1) (0,2) ɛ A

7 manipulating electron spin Ĥ = T 0 S g (ɛ) ( J(ɛ) HN H N 0 ) T 0 S g (ɛ) energy S e T 0 S L T 0,± nuclear spins induce transitions S g (ɛ B ) S g S R T 0 (1,1) (0,2) prepare any spin state with Sz=0

8 spin to charge conversion energy S e T 0 S L (1,1) T 0,± charge sensors singlet and triplet with similar charges are weakly distinguishable S g S R (1,1) charge sensing (0,2) induces a strong spin measurement

9 spin weak value Romito, Gefen, Blanter, PRL 100 (2008) energy S L S e T 0,± T 0 S g S R P (I) weak measurement P (I) I time evolution strong measurement postselection spin weak value I f I 0 = I 0 + (2e 2 V/h)2Re{ f  (W ) 0 t 0δt}

10 weak measurement weak detector H QPC = H (1,1) +(H (0,2) H (1,1) ) (0, 2) (0, 2) k[a L,k a L,k + a R,k a R,k] k U i,j (k, h)a i,k a j,h i,j=l,r k,h V H int = H (1,1) + J(ɛ)/ (ɛ) H (1I Ŝ2 /2) change in the scattering matrix of the QPC H(1,1) describes scattering of electrons in the QPC: weakly measured operator in φ = t 0 t + r 0 r!h: in φ = (t 0 + δt(ɛ)) t + (r 0 + δr(ɛ)) r = φ + φ 1+t 0 δt + r 0 δr N 1 is the condition for weak measurement

11 results "#$! control on pre and postselectrion γ " availability of exerimental ingredient!"#$!!"#$! " "#$! α angles defining the time evolution of the spin f A (w) 0!"!# $ # " % f I 0 = I 0 + (2e 2 V/h)2Re{ f  (W ) 0 t 0δt}

12 charge sensing weak values with proper post-selection give large values of the population in the right dot. can one use weak values protocols to sense very small populations in the right dot?

13 signal amplification ψ(x) e ikˆx ψ(x) ˆp = k k is a (small) shift in the momentum we want to detect x IR IR screen IR Piezo Driven Mirror incident laser beam P. B. Dixon et al. PRL 102 (2009)

14 signal amplification H: horizontal polarization V: vertical polarization To Oscilloscope or Lock-In Amplifier CCD Camera 50/50 H to V polarization convertor HWP Piezo Driven Mirror SBC SBC (extra phase for horizontal polarization) Quadrant Detector x Polarizing IR H 10x Objective Fiber incident laser beam HWP QWP

15 signal amplification H: horizontal polarization V: vertical polarization To Oscilloscope or Lock-In Amplifier CCD Camera 50/50 H to V polarization convertor HWP Piezo Driven Mirror SBC SBC (extra phase for horizontal polarization) Quadrant Detector x Polarizing 10x Objective Fiber ψ(x) 1/ 2( + ) HWP QWP

16 signal amplification H: horizontal polarization V: vertical polarization preselection To Oscilloscope or Lock-In Amplifier CCD Camera 50/50 H to V polarization convertor HWP Piezo Driven Mirror SBC SBC (extra phase for horizontal polarization) Quadrant Detector x Polarizing 10x Objective Fiber ψ(x) 1/ 2(e iφ + ) HWP QWP

17 signal amplification H: horizontal polarization V: vertical polarization To Oscilloscope or Lock-In Amplifier  = CCD Camera 50/50 H to V polarization convertor HWP Piezo Driven Mirror SBC SBC (extra phase for horizontal polarization) Quadrant Detector x Polarizing 10x Objective Fiber e ikâˆx ψ(x) 1/ 2(e iφ + ) HWP QWP

18 signal amplification H: horizontal polarization V: vertical polarization To Oscilloscope or Lock-In Amplifier  = CCD Camera IR 50/50 H to V polarization convertor HWP Piezo Driven Mirror SBC SBC (extra phase for horizontal polarization) post-selection Quadrant Detector ˆp = A w k ψ (x) e i A wkˆx ψ(x) x Polarizing HWP QWP 10x Objective Fiber

19 signal amplification experimental detection of 400 frad P. B. Dixon et al. PRL 102 (2009) CCD Camera H to V polarization convertor HWP Piezo Driven Mirror To Oscilloscope or Lock-In Amplifier IR 50/50 SBC (extra phase for horizontal polarization) Quadrant Detector x Polarizing 10x Objective Fiber HWP QWP ˆp = k ˆp = A w k ψ (x) e i A wkˆx ψ(x) ~0.5!m

20 charge sensing amplification B Th rew O. Zielberberg, A. Romito, Y. Gefen, in preparation spin polarized electrons are injected from a ferromagnetic source the ferromagnetic drain acts as a spin detector n L FIG. 1: (a) (Color online) A sketch of the setup. An Aharonov Bohm FIG. FIG. 1: FIG. ring (a) 1: (Color with 1: (a) (a) spin orbit (Color online) (Coloronline) coupled online) A sketch AAto sketch of half metalic sketch theof of setup. the thesetup An left and FIG. 1: (a) (Color online) with A sketch of the setup. Aharonov Bohm right Aharonov Bohm leads with spin ring orientation with ringspin orbit with ˆn spin orbit L, ˆn coupled R, respectively. coupled to half metalic to to The length Aharonov Bohm left and right ring leads with with spin orbit spin coupled ˆn to L ˆn half me left and ofleft the FIG. right and upper(lower) 1: leads right (a) with leads (Color arm spin with is online) orientation Lspin 1 (Lorientation 2 ). AInside sketch ˆn L, ˆn the R ˆn, of L respectively ring, ˆn the R R,, setup left and The The right length length leads of of the FIG. the with upper(lower) spin orientation 1: (a) (Color armisis ˆn online) L L 1, ˆn (L 1 (L R, 2 ). 2 ). respectiv The length Aharonov Bohm of the upper(lower) ring with armspin orbit is L 1 (L 2 ). coupled the electrons AInside sketc The are length subject of the to an upper(lower) effective magnetic arm is LfieldInside t the electrons Aharonov Bohm are subject to an effective ring 1 (L ˆ Bz to thehalf ring wi with 2 ). Inside see left and right leads with spin orientation ˆn the Fig. 1. the A classical electrons spin orbit field c the the electrons electrons charge are subject qare changes subject to anweakly effective to effective themagnetic geometry L, ˆn magnetic field of R, ˆ Bz respe an field see The length Fig. 1. A are of classical subject the leftupper(lower) and charge toright an effective qleads arm changes with is magnetic L the upper weakly spin orientatio Fig. 1. A classical charge q changes weakly the the field geom geom ˆ Bz Fig. arm1. inducing A classical a weak charge coupling q changes between weakly the electrons 1 (L 2 ). Inside tht the geometry o spin orbit Fig. 1. A upper classical armthe charge inducing length qchernogolovka, a changes of weak thecoupling upper(lower) weakly September the geometr thephase upper arm 2010 the is el the theand upper upper arm electrons its inducing arm which path are inducing asubject weaka degree coupling weak to of coupling effective freedom. res between between magnetic the electron the field ele Fig. spin orbit 1. arm A classical inducing phase the electrons and charge a weak its which path qcoupling between are changes subject weakly degree to anthe of effective freedo elect geom or { I n R e igµ BBL/v F ˆσ z n L 2 n L e iαqˆσ z n L n L + iqαˆσ z n L n L q wh de res FIG. 1: (a) (Color online) A sket Aharonov Bohmring with spin orbi left andright leads with spin orient e lengthoftheupper(lower) ar rons are subject to an e ssical charge q c ucing a wea its w

21 wh de res B Th rew The tra rewritte φ i = t e iφ AB t 2 2 φ f = amplitude = φ f φ i FIG. 1: (a) (Color online) A sketch of the setup. An Aharonov Bohm FIG. FIG. 1: FIG. ring (a) 1: (Color with 1: (a) (a) spin orbit (Color online) (Coloronline) online) coupled A sketch AAto sketch of sketch half metalic theof of setup. the thesetup An FIG. 1: (a) (Color n online) left and with A sketch of the setup. Aharonov Bohm right Aharonov Bohm leads with spin ring orientation with ringspin orbit L n with spin orbit ˆn L, ˆn coupled L R + iqαˆσ, respectively. coupled todenoted half metalic z n L The length Aharonov Bohm left and ring with with spin orbit spin coupled ˆn to L ˆn half met left and ofleft the right and upper(lower) leads rightwith leadsarm spin with isorientation spin L 1 (Lorientation 2 ). Inside ˆn L, ˆn the R ˆn, L respectively, ring R, left and The right length leads ˆn R, The length of of the FIG. the with upper(lower) spin orientation 1: (a) (Color arm armisis ˆn online) L L, ˆn 1 (L 1 (L R, 2 ). 2 ). respectiv The length of the upper(lower) arm is L 1 (L 2 ). InsideAInside the sketc ring wi the electrons tht The are length subject of the to an upper(lower) effective magnetic arm is Lfield the electrons Aharonov Bohm are subject to an effective ring 1 (L ˆ Bz with 2 ). Inside see n the Fig. 1. the A classical electrons spin orbit fieldc the the electrons electrons charge are subject qare changes subject to anweakly effective to L an effective themagnetic geometry magnetic field of ˆ Bz field see Fig. 1. A are classical subject left and charge toright an effective qleads changes with magnetic weakly spin orientatio Fig. 1. A classical charge q changes weakly the the field geom geom th the upper ˆ Bz Fig. arm1. inducing A classical a weak charge coupling q changes between weakly the electrons the geometry o spin orbit Fig. 1. the A upper classical armthe charge inducing length q a changes of weak thecoupling upper(lower) weakly the geometr thephase upper armthe isel the upper upper and armits inducing arm which path inducing a weaka degree coupling weak coupling of freedom. res between between the electrons the spin orbit spin orbit arm inducing phase phase the and electrons and a weak its its which path coupling between are subject degree degree to an of of the effective freedom electr spin orbit phase and its which path degree of freedom. ori spin orbit phase and its which path degree of freedom.  = 1 1 Fig. 1. A classical charge q changes w p σ p θ (σ), are given by the upper arm inducing a weak coupling p σ p σ p θ (σ), spin orbit are given by phase and its which path de p σ p θ (σ), are given by σ θ (σ), are given by p θ (σ), are given by FIG. 1: (a) (Color online) A sketch of the setup. An Aharonov Bohm ring with spin orbit coupled to half metalic left and right leads with spin orientation ˆn L, ˆn R, respectively. The length of the upper(lower) arm is L 1 (L 2 ). Inside the ring φ i n L eiαqâˆσ z φ i n L n L n L + iαq A wˆσ z n L FIG. 1: (a) (Color online) A sketc Aharonov Bohmring with spin orbi left andright leads with spin orient Thelengthoftheupper(lower) ar the electrons are subject to an e Fig. 1. A classical charge q ch theupper arm inducing a wea spin orbit phase and its wh ( p σ = p ( ) σ = ) 2 ( 2 p 2r ˆσ ± + p 2 0 2r + mb σ = p ( ( ) 2 σ p θ (σ), are given by z ˆσ, p 2r ˆσ ± p 2 0 2r + mb z ˆσ, σ = ) 2 p σ = ) 2 2r ˆσ ± + p 2 2r ˆσ ± 0 2r + mb 2r ˆσ ± + p + 2 z ˆσ, (4) 0 2r + p 2mB 0 2r + mb with z ˆσ, z ˆσ, j(4) wh where p 0 = [2m(E F E 0 )] 1/2 arm, and σ = ( where p 0 = [2m(E F p E 0 )] Chernogolovka, September 1/2 σ = ±1 = and (, ) ) 2, and σ j where p 0 = [2m(E F E 0 )] 1/2, ±1 where p the spin-state, 2r along ˆσ and ± len is the where spin-state, p 0 = orientated [2m(E σ = ±1 + = p 0 = [2m(E F along E F 0 )] E 1/2 the, 0 )] 1/2 and ẑ axis. σ = This ±1 is = (, ), and σ the = 2010 ±1 2r = ( readilyisgeneralized the is isspin-state, the the spin-state, to other orientated wire orientated shapes, alongwith the rẑ axis. the varying the ẑ axis. This left ma is T readily readily spin-state, generalized orientated to other other along wire wire the shapes, ẑ axis. with with Thi sp along the readily wire generalized coordinate, to x, (r( x) otherto wire be kept shapes, large). withthe r varying rr v the electrons are subject to an effective magnetic field ˆ Bz see Fig. 1. A classical charge q changes weakly the geometry of the upper arm inducing a weak coupling between the electrons where in p σ p θ (σ), are given p σ = ere p 0 = spin

22 (signal-to-noise ratio)spin+interferometer (signal-to-noise ratio)spin is amplified current response to the extra charge t n L n L n L + iαq A wˆσ z n L

23 summary Weak measurements with post-selection lead to weak values steaming from quantum mechanical correlations. Weak values can be experimentally accessed in solid state systems. Weak values measurement can be used to amplify a signal interactions... weak to strong crossover... conditional quantities...