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Quantum fluctuations and the Casimir Effect in meso- and macro- systems Yoseph Imry 1 QUANTUM-NOISE-05 I. Noise in the Quantum and Nonequilibrium Realm, What is Measured? Quantum Amplifier Noise. ----------------------------work with: Uri Gavish, Weizmann (ENS) Yehoshua Levinson, Weizmann B. Yurke, Lucent 2 Thanks: E. Conforti, C. Glattli, M. Heiblum, R. de Picciotto, M. Reznikov, U. Sivan QUANTUM-NOISE-05 II. Sensitivity of Quantum Fluctuations to the volume: Casimir Effect Y. Imry, Weizmann Inst. Thanks: M. Aizenman, A. Aharony, O. Entin, U. Gavish Y. Levinson, M. Milgrom, S. Rubin, A. Schwimmer, A. Stern, Z. Vager, W. Kohn. 3 QUANTUM-NOISE-05 Quantum, zero-point fluctuations Nothing comes out of a ground state system, but: Renormalization, Lamb shift, Casimir force, etc. No dephasing by zero-point fluctuations! How to observe the quantum-noise? (Must “tickle” the system). 4 QUANTUM-NOISE-05 Outline: • Quantum noise, Physics of Power Spectrum, dependence on full state of system • Fluctuation-Dissipation Theorem, in steady state • Application: Heisenberg Constraints on Quantum Amps’ • Casimir Forces. 5 QUANTUM-NOISE-05 Understanding The Physics of Noise-Correlators, and relationship to DISSIPATION: 6 QUANTUM-NOISE-05 Classical measurement of time-dependent quantity, x(t), in a stationary state. x(t) C(t’-t)=<x(t) x(t’)> t 7 QUANTUM-NOISE-05 Classical measurement of a time-dependent quantity, x(t), in a stationary state. x(t) C(t’-t)=<x(t) x(t’)> t Quantum measurement of the expectation value, <xop(t)>, in a stationary state. <x(t)> C(t)=? t 8 QUANTUM-NOISE-05 The crux of the matter: ------ From Landau and Lifshitz,Statistical Physics, ’59 (translated by Peierls and Peierls). 10 QUANTUM-NOISE-05 Van Hove (1954), EXACT: 11 QUANTUM-NOISE-05 12 QUANTUM-NOISE-05 Emission = S(ω) ≠ S(-ω) = Absorption, (in general) From field with Nω photons, net absorption (Lesovik-Loosen, Gavish et al): Nω S(-ω) - (Nω + 1) S(ω) For classical field (Nω >>> 1): CONDUCTANCE [ S(-ω) - S(ω)] / ω 13 QUANTUM-NOISE-05 This is the Kubo formula (cf AA ’82)! Fluctuation-Dissipation Theorem (FDT) Valid in a nonequilibrium steady state!! Dynamical conductance - response to “tickling”ac field, (on top of whatever nonequilibrium state). Given by S(-ω) - S(ω) = F.T. of the commutator of the temporal current correlator 14 QUANTUM-NOISE-05 Nonequilibrium FDT • Need just a STEADY STATE SYSTEM: Density-matrix diagonal in the energy representation. “States |i> with probabilities Pi , no coherencies” • Pi -- not necessarily thermal, T does not appear in this version of the FDT (only ω)! 15 QUANTUM-NOISE-05 Partial Conclusions • The noise power is the ability of the system to emit/absorb (depending on sign of ω). FDT: NET absorption from classical field. (Valid also in steady nonequilibrium States) • Nothing is emitted from a T = 0 sample, but it may absorb… • Noise power depends on final state filling. • Exp confirmation: deBlock et al, Science 2003, (TLS with SIS detector). 16 QUANTUM-NOISE-05 A recent motivation How can we observe fractional charge (FQHE, superconductors) if current is collected in normal leads? Do we really measure current fluctuations in normal leads? ANSWER: NO!!! THE EM FIELDS ARE MEASURED. (i.e. the radiation produced by I(t)!) 17 QUANTUM-NOISE-05 Important Topic: Fundamental Limitations Imposed by the Heisenberg Principle on Noise and Back-Action in Nanoscopic Transistors. Will use our generalized FDT for this! 18 QUANTUM-NOISE-05 A Linear Amplifier Must Add Noise (E.g., C.M. Caves, 1979) Input (“signal”) Amplifier x s , ps X a XGx Pa Gs 1 a s,GxsPa, Gp s Gp [ X a , Pa ] i [ xs , ps ] i x p s s G2 2G 2 2 Heisenberg principle is violated. 19 Detector Xa , Pa Linear Amplifier: But then Output QUANTUM-NOISE-05 A Linear Amplifier Must Add Noise (E.g., C.M. Caves) Input (“signal”) Amplifier x s , ps X a XGx Pa Gs 1 a s,GxsPa, Gp s Gp [ X a , Pa ] i [ xs , ps ] i x p s s G2 2G 2 2 Heisenberg principle is violated. A Linear Amplifier does not exist ! 20 Detector Xa , Pa Linear Amplifier: But then Output QUANTUM-NOISE-05 A Linear Amplifier Must Add Noise (E.g., C.M. Caves, 1979) Input (“signal”) x s , ps Amplifier Output Detector Xa , Pa In order to keep the linear input-output relation, with a large gain, the amplifier must add noise Xs N ,PN Pa Gps PN X a Gxs X N X, a PGx a s Gp 21 QUANTUM-NOISE-05 A Linear Amplifier Must Add Noise (E.g., C.M. Caves, 1979) Input (“signal”) x s , ps Amplifier Output Detector Xa , Pa In order to keep the linear input-output relation, with a large gain, the amplifier must add noise Xs N ,PN Pa Gps PN X a Gxs X N X, a PGx a s Gp choose [ X N , PN ] - (G 2 1)i X N , PN act on the amplifier state then [ X a , Pa ] [ X N , PN ] [ xs , ps ] G 2i - (G 2 1)i i 22 QUANTUM-NOISE-05 Cosine and sine components of any current filtered with window-width 23 QUANTUM-NOISE-05 For phase insensitive linear amp: gL and gS are load and signal conductances (matched to those of the amplifier). G2 = power gain. 24 QUANTUM-NOISE-05 For Current Comm-s we Used Our Generalized Kubo: S () S () 2 g where g is the differential conductance, leads to: 25 QUANTUM-NOISE-05 , Average noise-power delivered to the load (one-half in one direction) 26 QUANTUM-NOISE-05 A molecular or a mesoscopic amplifier Resonant barrier coupled capacitively to an input signal Ia()= I0()+G Is() +back-action noise, In B Is() input siganl Cs Ls 29 QUANTUM-NOISE-05 A new constraint on transistor-type amplifiers Coupling to signal = γ Noise is sum of original shot-noise I0~ γ0 and “amplified back-action noise” In~ γ2 30 QUANTUM-NOISE-05 General Conclusion: one should try and keep the ratio between old shot-noise and the amplified signal constant, and not much smaller than unity. In this way the new shot-noise, the one that appears due to the coupling with the signal, will be of the same order of the old shot-noise and the amplified signal and not much larger. 31 QUANTUM-NOISE-05 Amp noise summary • Mesoscopic or molecular linear amplifiers must add noise to the signal to comply with Heisenberg principle. • This noise is due to the original shot-noise, that is, before coupling to the signal, and the new one arising due to this coupling. • Full analysis shows how to optimize these noises. 32 QUANTUM-NOISE-05 The Casimir effect in meso- and macro- systems 34 QUANTUM-NOISE-05 Even at T=0, we are sorrounded by huge g.s. energy of various fields. No energy is given to us (& no dephasing!). But: various renormalizations, Lamb-shift… Casimir: If g.s. energy of sorrounding fields depends on system parameters (e.g. distances…) – a real force follows! This force was measured, It is interesting and important. Will explain & discuss some new features. 35 QUANTUM-NOISE-05 The Casimir Effect The attractive force between two surfaces in a vacuum - first predicted by Hendrik Casimir over 50 years ago - could affect everything from micromachines to unified theories of nature. (from Lambrecht, Physics Web, 2002) 36 QUANTUM-NOISE-05 Buks and Roukes, Nature 2002 (Effect relavant to micromechanical devices) 39 From: QUANTUM-NOISE-05 Why interesting? • (Changes of) HUGE vacuum energy—relevant • Intermolecular forces, electrolytes. • Changes of Newtonian gravitation at submicron scales? Due to high dimensions. • Cosmological constant. • “Vacuum friction”; Dynamic effect. • “Stiction” of nanomechanical devices… • Artificial phases, soft C-M Physics. 40 QUANTUM-NOISE-05 Casimir’s attractive force between conducting plates i) (c)= Soft cutoff at p ii) E 0' (d ) E 0 (d ) - E 0 ( ) 41 QUANTUM-NOISE-05 42 QUANTUM-NOISE-05 Subtracted quantity E 0 () / d is radiation pressure of the vacuum outside (Casimir, Debye, Gonzalez, Milonni et al, Hushwater), What is it (for volume V)? Milonni et al (kinetic theory): momentum delivered to the wall/unit time. For every photon, momentum/unit time = E 0 ()/ d , same for many photons. Pressure || z in k state: 43 QUANTUM-NOISE-05 Total pressure: Replacing sum by integral, integrating over angles and changing from k to ω, . with ω=: Defining: 44 QUANTUM-NOISE-05 A “thermodynamic” calculation: D(ω) is photon DOS D(ω) extensive and >0 P0 is same order of magnitude, but NEGATIVE??? 45 QUANTUM-NOISE-05 Why kinetics and thermodynamics don’t agree? M. Milgrom: ‘Thermodynamic’ calculation valid for closed system. But states are added (below cutoff) with increasing V ! 1 cutoff 46 Increasing V QUANTUM-NOISE-05 Allowed k’s Result for P0 is non-universal Depends on: Cutoff p and details of cutoff function), Nature of slab, Dielectric function, ( ) , of medium. Can be used! 47 QUANTUM-NOISE-05 Effect of dielectric on one side “Macroscopic Casimir Effect” With ( ): P0 P0 (1) (c ) 6 2c 3 3 3/ 2 d ( ) , P0 ( ) ( ) 1 P0 ( ) Larger than for 1 ! 1 F 48 () Further possibilities QUANTUM-NOISE-05 Net Force on slab between different dielectrics P0 ~ 0.1 N/cm2 , for =1, p =10eV. Typical differences ~ 10-5 of that. Force balanced by elasticity/surface tension of materials. Force and slab's position depend on 's and on slab material, differences measurable, in principle. 49 QUANTUM-NOISE-05 Effect of dielectric inside on the “Mesoscopic Casimir Effect” With (): Will change the sign of the Casimir Force at large () With (): enough separations, Depending on ( ) Interesting in static limit: 50 QUANTUM-NOISE-05 d << c/ p Quasistationary (E. Lifshitz, 56) regime Length scale d << c/ p no retardation Can use electrostatics (van Kampen et al, 68) Casimir force becomes (no c!): p /d 51 3 QUANTUM-NOISE-05 Vacuum pressure on thin metal film Quasistationary: d<<c/ωp Surface plasmons on the two edges Even-odd combinations: d 52 QUANTUM-NOISE-05 Dispersion of thin-film plasmons For d<<c/ωp, ω/ωp light-line ω=ck 1 is very steep-full 0.8 0.6 EM effects don’t 0.4 0.2 1 2 3 4 kd Matter-- quasi stationary appr. Note: opposite dependence of 2 branches on d 53 QUANTUM-NOISE-05 Casimir pressure on the film, from derivative of total zero-pt plasmon energy.: Large positive pressures on very thin metallic films, approaching eV/A3 scales for atomic thicknesses. 54 QUANTUM-NOISE-05 Conclusions • EM Vacuum pressure is positive, like kinetic calculation result. It is the Physical subtraction in Casimir’s calculation. Depends on properties of surface! • Effects due to dielectrics in both macro- and mesoregimes. Some sign control. • Large positive vacuum pressure due to surface plasmons, on thin metallic films. 55 QUANTUM-NOISE-05 END, Thanks for attention! 56 QUANTUM-NOISE-05