Passivity breaking of a local vacuum state in a quantum Hall system Go Yusa,∗ Wataru Izumida, and Masahiro Hotta Department of Physics, Tohoku University, Sendai 980-8578, Japan (Dated: January 17, 2011) We propose an experimental method for extracting the zero-point energy from a local vacuum state by performing local operations and classical communication (LOCC). We model a quantum Halledgechannelasaquantumentangledmany-bodychannelandthezero-pointfluctuationinthe charge density of the channel as thevacuum state. We estimate the order of theenergy gain using reasonableexperimentalparameters. Suchaquantumfeedbacksystembreaksthepassivityofalocal vacuumstate. It can beusedto demonstrateMaxwell’s demon orquantumenergyteleportation in which no physical entity transports energy. 1 1 PACSnumbers: 03.67.-a,73.43.-f 0 2 According to quantum mechanics, a many-body sys- n tem in the vacuum state possesses a zero-point energy. a J However, this must be considered a non-available re- 4 source since it is not possible to extract the zero-point 1 energy from a vacuum by performing local operations ; this is known as the passivity of the vacuum state [1]. ] h However, recent theoretical studies have suggested that p the passivity can be broken locally by performing local - measurements followed by local operations and classical t n communication (LOCC) locally[2]. This scheme can be a interpreted in terms of information thermodynamics as u a quantum version of Maxwell’s demon [3]; specifically, q [ two demons cooperatively extract energy from a local FIG. 1. (color online). Schematic diagram of the quantum 1 vacuum state. We consider a quantum entangled many- Hall system used in this study. Edge channels S and U are v body system in the vacuum state. As mentioned above, formed at the boundaries of separate quantum Hall bulk re- 6 the demons cannot extract energy from the system by gionsS andU,respectively. Redandbluearrowsindicatethe 6 performing local operations. Here, we define two sub- directions of propagating waves. WU is a wavepacket excited 7 systems A and B that are separated by an appropriate in region U. 2 distance. Since local quantum fluctuations are entan- . 1 gled in the vacuum state [4], demon A can obtain a cer- 0 tain amount of information on quantum fluctuations at beformedinatwo-dimensionalelectronsysteminasemi- 1 B by performing local measurements at A. However, in conductor subjected to a strong perpendicular magnetic 1 : exchange for this information, demon A has to pay an field. Such a system is suitable because the zero resis- v energy EA to his/her own subsystem. Immediately after tance of the quantum Hall effect means that the system i X demon A informs demon B of the information (i.e., the isdissipationlessandquasi-one-dimensionalchannelsap- r measurement result), demon B can extract energy EB pearattheboundaryofthebulkincompressibleregionof a even though his/her subsystem remains in a local vac- a quantum Hall system; such an edge channel is consid- uum state. This is passivity breaking by LOCC. This eredtobehaveasachiralLuttingerliquid[7]. Power-law scheme is called quantum energy teleportation (QET) behaviors have been experimentally demonstrated, sug- because no actual carriers transfer energy from A to B gestingthatthesystemhasnocertaindecaylength[8,9]. butenergycanbegainedataremotelocationB[2]. This Furthermore,sincethevacuumisazero-pointfluctuation type of quantum feedback is also relevant to black hole in the charge density, the Coulomb interaction (i.e., ca- entropy, whose origin has often been discussed in string pacitive coupling), can be used as a sensitive probe for theory [5], because energy extraction from a black hole detecting the vacuum. In addition, semiconductor nan- reduces the horizon area (i.e., the entropy of the black otechnology can be used to design LOCC. hole [6]). In this study, we discuss possible passivity breaking in a quantum Hall system and estimate the order of the To verify this theory in a realistic system requires a energy gain at B by employing reasonable experimental dissipationlessquantum-entangledchannelwithamacro- scopiccorrelationlength,adetectionschemeforthe vac- parameters. uum state, and a suitable implementation of LOCC. We consider the system depicted schematically in QuantumHallsystemsarepromisingsystems. They can Fig. 1. Experiments should be performed at low tem- 2 peratures of the order of millikelvins (mK) to produce H 0 =0. Taking region A for x [a , a ] we adopt S S − + | i ∈ these vacuum states in the edge channels. Two regions, the RC-circuit-detector model proposed by F`eve et al. A and B, are defined on the left-going edge channel S. [14] to measure the voltage induced by the zero-point We describe S by a chiralLuttinger boson field with the fluctuation of S. The charge fluctuation at A is esti- trajectoryparameterizedby a spatialcoordinate x along mated as the channel. Here, we consider the left-going chiral field ∞ whoseelectronnumberdensity̺S(x+vgt)deviatesfrom QS(t)=e ̺S(x+vgt)wA(x)dx, (1) that of the vacuum state equilibrium. vg is the group Z−∞ velocity of a charge density wave propagating along the with a window function w (x), which we assume to be A channel (i.e., a magnetoplasmon [10] in the zero-energy a Gaussian function with its maximum value and width limit) and t is the elapsed time. The zero-point fluctu- being of the order of unity and l, respectively. In this ation in the charge density in region A can be experi- model[14],the voltagebetweenthedetectorcontactand mentally measured by an RC circuit that consists of the S is given by V(t) = 1 [Q (t) Q(t)], where Q(t) is C S − input resistanceRofanamplifier andthe capacitanceC the charge on the capacitor. The quantum noise of the between S and a local metal gate fabricated on S. For voltage V(t= 0) is described by the operator Vˆ − positiveoperatorvaluedmeasure(POVM)measurements ~ [9],the RC circuit(i.e., the detector)canbe switchedon Vˆ = only during the measurement. t = 0 is defined as the − πRC2 r time when the switch is turned on. During the measure- ∞ √ω √ω dω a (ω)+ a† (ω) , ment, the detector excites S and injects energy EA into × ω 1 in ω+ 1 in S. The measured voltage signal υ is amplified and then Z0 (cid:20) − iRC iRC (cid:21) communicated to region U to excite a wave packet WU whereain(ω)(ain(ω)†)istheannihilation(creation)oper- on the other edge channel U. We describe the trajec- atorofexcitationofachargedensitywaveinthedetector tory of U by a right-going chiral boson field ̺U(y−vgt) circuit satisfying ain(ω), ain(ω′)† =δ(ω−ω′). Before parameterizedbyaspatialcoordinateywithidenticalve- the measurement (i.e., the signal input from S to the locity vg, which assumes that S and U are formedin the detector), V(t =(cid:2) 0) equals Vˆ. U(cid:3)sing the fast detector − samemanner. Thus,WU travelsalongU carryingenergy condition (RC l/vg), the voltage after the measure- ≪ E1 towardregion B, where the two edge channels S and ment is computed as U approach each other. These channels are capacitively V(t=+0)=Vˆ +RQ˙ (0), (2) coupled only at B, where W interacts with the zero- S U point fluctuation of S. After the interaction, the energy where RQ˙ (0) denotes the voltage shift induced by the S carriedby WU changesfromE1 to E2. If no information signalandthedotofQ˙ (0)indicatesthetimederivative. S about υ is communicated (i.e., WU is created indepen- Theamplitude∆V ofVˆ inthevacuumstate 0 ofthe RC dently of signal υ) W will inject energy to S due to the | i U passivity of the vacuum state [1]. Thus, EB = E2−E1 RCcircuitcanbeestimatedas∆V = h0RC|Vˆ2|0RCi= will be negative. However, in our system, since WU ex- O ~ 10 µV. The root mqean square of the plicitly depends on υ, the passivity is broken and EB is RC2 ∼ positive; WU gains positive energy from the zero-point (cid:18)q (cid:19) 2 fluctuationofS. Inthe following,weprovethis factthe- voltage shift, 0S RQ˙S(0) 0S , is estimated to be h | | i r oretically and estimate EB by setting the experimental O(100 µV) from Eq.(cid:16)(1), so qu(cid:17)antum fluctuations of the parameters vg 106 m/s [11, 12], R 10 kΩ [13], and edge current will be detectable. ∼ ∼ C 10 fF. The length b of regions U and B and the To estimate the energy injected into A after the mea- ∼ length of region A are approximated by a typical length surement, we reduce the measurement operators [9] to scale of l 10 µm. the pointer basis of von Neumann [15]. By regarding Vˆ ∼ We start the detailed discussion by modeling the edge as a preamplified quantum pointer operator, the instan- channel S. The chiral field operators ̺S(x) satisfy the taneous shift in Eq. (2) can be reproduced by the mea- commutation relation [̺S(x), ̺S(x′)] = i2νπ∂xδ(x−x′). surement Hamiltonian, Hm(t) = δ(t)RQ˙S(0)PVˆ, where The energy density operator of ̺S(x) is written as P is the conjugate momentum operator of Vˆ. Using Vˆ π~v the eigenvalue υ of Vˆ (Vˆ υ = υ υ ), we can assume ε (x)= g :̺ (x)2 :, | i | i S S the initial wavefunction of the quantum pointer in the ν S υ representation to be Ψ (υ)∝exp 1 υ2 , whereas where ν is the Landau level filling factor of S and i −4∆V2 S the wavefunction after the measurement is translated as :: denotes normal ordering, which causes the expecta- (cid:2) 2 (cid:3) Ψ (υ) ∝ exp 1 υ RQ˙ (0) . After turning tion value of εS(x) to vanish for the vacuum state 0S ; f −4∆V2 − S 0 ε (x)0 = 0. The free Hamiltonian of S is g|ivein the measurem(cid:20)ent inter(cid:16)action on, we(cid:17)p(cid:21)erform a projec- S S S hby H| =| ∞i ε (x)dx. The eigenvalue of 0 vanishes, tive measurement of Vˆ to obtain an eigenvalue υ of Vˆ. S −∞ S | Si R 3 Thisreductionanalysisprovesthatthemeasurementop- U. Then, W evolves into regionB by H . The average U U erator M [16] is Ψ (υ). The corresponding POVM is energy of W is denoted by E (=Tr[H ρ ]), which is υ f U 1 B SU represented by the operator Π = M†M , which satis- calculated as υ υ υ ∞ fies the standard sum rule, Π dυ = I , where I −∞ υ S S π~v ∞ 1 is the identity operator of the Hilbert space of S. The E = g (∂ λ (y))2dy 0 G2 0 + , R 1 ν y B h S| S| Si 4 probability density of the result υ is p(υ) = 0S Πυ 0S . U Z−∞ (cid:20) (cid:21) h | | i The post-measurement state of S corresponding to the where G = evgR ∞ ̺ (x)∂ w (x)dx. E is esti- resultυ iscomputedtobe Mυ 0S upto anormalization S −2∆V −∞ S x A 1 | i mated to be of the order of 10 meV when ν and ν are constant. Hence, the average state of S right after the R S U respectively 3and 6. InregionB,W and̺ (x)in- measurement is given by ∼ ∼ U S teract with each other via the Coulomb interaction such ∞ that ρ = M 0 0 M†dυ. 1 υ| Sih S| υ Z−∞ e2 b+ b+ H = dx dy̺ (x)f(x,y)̺ (y). int S U The injected energy EA is calculated to be 4πǫZb− Zb− ∞ Here, ǫ is 10ǫ for the host semiconductor (e.g., gallium E = 0 M†H M 0 dυ 0 A Z−∞h S| υ S υ| Si Tarhseenfiudnec)t,iwonhefr(exǫ,0y)isisthgeivdeinelebcytric con1stant.ofHvearec,uudmis. = ~vgνS evgR 2 ∞ dx ∂2w (x) 2. √(x−y)2+d2 4π 2∆V x A the separation length between the two edge channels at (cid:18) (cid:19) Z−∞ B and is approximated by l. After the interaction,the (cid:0) (cid:1) Usingtheexperimentalparametersmentionedabove,EA energyof WU becomes E2. The energy gain, EB =E2 − is estimated to be of the order of 1 meV for ν 3. E , is estimated by lowest-order perturbation theory in S 1 ∼ We now considerthe edge channel U. The free Hamil- terms of Hint as follows. tonian of U is H = π~v ∞ : ̺ (y)2 : dy. The mea- U νU −∞ U e2v ∞ b+ b+ surementresultυisamplifiedandtransferredtoregionU g E = i dz dx dy f(x ,y ) R B B B B B asanelectricalsignalanditthengeneratesawavepacket − 4ǫνS Z−∞ Zb− Zb− W of ̺ (y) (i.e., a localized right-goingcoherentstate) ∞ ∞ U U dt dυ 0 M′†̺ (x +v t)M′ 0 in a region with y [b− L, b+ L]. This process can × h S| υ S B g υ| Si be expressed by th∈e υ-d−ependent−unitary operation U Z−∞ Z−∞ υ to the vacuum state |0Ui of ̺U(y); |υUi=Uυ|0Ui. Uυ is ×hυU′ | ̺U(z−vgtf)2, ̺U(yB−vgt) |υU′ i, given by h i where M′ = U (t T)†M U (t T) and υ′ = U (t )† υυ . HeSre,iU−(T) =υexSp(i −iTH ). t |aUndi t U i U U ~ U i f πiυ b+−L are resp|ectiively the start and end t−imes of the interac- U =exp λ (y)̺ (y)dy , υ B U νU∆V Zb−−L ! tion between S and U. By substituting the commuta- whereL isthe distancebetweenregionsU andB andν tion relation given by ̺U(z)2, ̺U(yB) =−iνπU∂δ(z− U is the filling factor of U. Here, the length b b = b yB)̺U(z) and integrahting with respectito z, we obtain + − − the following relation: is approximated by l. We assume that λ (y) is a Gaus- B sianfunctionwhosemaximum,λB(b/2−L),givestheor- e2vg b+ b+ der of the total number of excess and deficient electrons EB = dxB dyBf(xB,yB) 4πǫ from the vacuum-state equilibrium in [b− L, b+ L], Zb− Zb− − − ∞ which are excited by the amplified voltage. We take the dt∂2λ (y v (t t )) B B g i amplitude of λ O(10), which can be experimentally × − − B ∼ Z−∞ done by tuning the amplifier gain appropriately. By us- ∞ ionfgW[̺U(isy)c,om̺Up(uyt′e)d] =as−υiν2Uπ̺∂y(δy(y)υ− y′=), tυh∂eλwa(vye).foBrmy ×Z−∞υh0S|Mυ′†̺S(xB +vgt)Mυ′|0Sidυ. U U U U y B h | | i defining T asthe elapsedtime whenW has beengener- Note that the last integral is computed as U ated, the composite state of S and U at T is calculated as ∞ υ 0 M′†̺ (x +v t)M′ 0 dυ ρSU =Z−∞∞dυUSMυ|0Sih0S|Mυ†US† ⊗|υUihυU|, Z=−∞eυhRS| υ∞ dSx¯ ∂Bw (x¯g ) υ| Si A A A −4∆V whereU (T)=exp( iTH ). Thisstateisthescattering Z−∞ S ~ S − ∆(x¯ x v (t+T t ))+c.c., input state for the Coulomb interaction between S and A B g i × − − − 4 where in a negative averagequantumenergy density aroundB. ν ∞ This generation of a negative energy density is attained S ∆(x)= dkkexp( ikx). by squeezing the amplitude of the zero-point fluctuation 4π2 − Z0 less than that of the vacuum state during the interac- To integrate E with respect to t, we take the Fourier tion [17]. E and E will run ”chirally” on the edge B B A − transform of ∂2λB in EB and obtain ∂2λB(y) = toward the downstream electrical ground with identical −21π −∞∞k′2λ˜B(k′)eik′ydk′. Using −∞∞dtexp[−i(k′ ± velocitiesofvg. Becauseofthis chirality,evenaftermea- k)v t]= 2πδ(k′ k), E is estimated as surementatregionA, SaroundregionB will remainin a gR vg ± B R local vacuum state with zero energy density. 3e3vRν ∞ b+ b+ b+ UnlikequantumHallsystems,severalsuccessfulexper- S E = dx¯ dy¯ dx dy B 4π3ǫ∆V A B B B imental studies have been conducted in quantum optics Z−∞ Zb− Zb− Zb− by introducing LOCC including quantum teleportation 1 [18,19]. Lightisamasslesselectromagneticfieldwithan × (x y )2+d2 infinite correlation length. However, it propagates three B B − q w (x¯ )λ (y¯ L) dimensionallysothattheenergygaindecaysrapidlywith A A B B − , increasing distance between A and B. In addition, it is × (x +y x¯ y¯ +L+v T)5 B B− A− B g currently difficult to measure the vacuum state experi- where v T = O(10−2L). The parameter L + v T(= mentally due to the lack of an appropriate interaction g g such as the Coulomb interaction in quantum Hall sys- O(L)) corresponds to the distance between A and B. tems. Thus, our quantum Hall system is considered to Thus, the energy output E is estimated to be B beverysuitablefordemonstratinglocalvacuumpassivity e2λ ev R l 5 breaking. B g E =O . (3) B 4πǫl l∆V L Inconclusion,wehavetheoreticallydemonstratedthat (cid:18) (cid:19) ! the passivity of the vacuum can be locally broken by Since the function λ (y¯ +L) is positive, E must also electrical LOCC in a realistic system using a quantum B B B be positive. Eq. (3) shows that increasing L will rapidly Hall edge channel as a many-body quantum channel. reduce the magnitude of E (e.g., E 1 µeV for L The authors gratefully acknowledge K. Akiba and T. B B ∼ ∼ 4l). Nevertheless, for L 2l, E will be of the order Yugeforfruitfuldiscussions. G.Y.,W.I.,andM.H.are B ∼ of 100µeV. This is muchlargerthan the thermal energy supportedby Grants-in-AidforScientific Research(Nos. 1 µeV at a temperature of 10 mK, which is the 21241024, 22740191, and 21244007, respectively) from ∼ ∼ temperature at which quantum Hall effect experiments the Ministry of Education, Culture, Sports, Science and are generally performed using a dilution refrigerator. Technology(MEXT), Japan. W. I.and M.H. are partly To observe E experimentally, we measure the cur- supportedbytheGlobalCOEProgramofMEXT,Japan. B rent passing through the edge channel U. The relation, G. Y. is partly supported by the Sumitomo Foundation. ε= π~ j2, between the energy density ε and the cur- νUe2vg rent j gives an energy density of 10-µeV/µm, which cor- responds to a current of 10-nA. This can be detected experimentallyusingstate-of-the-artelectronics. To ver- ∗ [email protected] ify that energy is extracted at B, a single-shot current [1] W. Pusz and S. L. Woronowicz, Commun. Math. 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