Passivity breaking of a local vacuum state in a quantum Hall system
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 Hall edge channel as a quantum entangled many-body channel and the zero-point fluctuation in the charge density of the channel as the vacuum state. We estimate the order of the energy gain using reasonable experimental parameters. Such a quantum feedback system breaks the passivity of a local vacuum state. It can be used to demonstrate Maxwell’s demon or quantum energy teleportation in which no physical entity transports energy.
According to quantum mechanics, a many-body system in the vacuum state possesses a zero-point energy. However, this must be considered a non-available resource since it is not possible to extract the zero-point energy from a vacuum by performing local operations ; this is known as the passivity of the vacuum state passivity (). However, recent theoretical studies have suggested that the passivity can be broken locally by performing local measurements followed by local operations and classical communication (LOCC) locally1 (). This scheme can be interpreted in terms of information thermodynamics as a quantum version of Maxwell’s demon demon (); specifically, two demons cooperatively extract energy from a local vacuum state. We consider a quantum entangled many-body system in the vacuum state. As mentioned above, the demons cannot extract energy from the system by performing local operations. Here, we define two subsystems A and B that are separated by an appropriate distance. Since local quantum fluctuations are entangled in the vacuum state R (), demon A can obtain a certain amount of information on quantum fluctuations at B by performing local measurements at A. However, in exchange for this information, demon A has to pay an energy to his/her own subsystem. Immediately after demon A informs demon B of the information (i.e., the measurement result), demon B can extract energy even though his/her subsystem remains in a local vacuum state. This is passivity breaking by LOCC. This scheme is called quantum energy teleportation (QET) because no actual carriers transfer energy from A to B but energy can be gained at a remote location B 1 (). This type of quantum feedback is also relevant to black hole entropy, whose origin has often been discussed in string theory strominger (), because energy extraction from a black hole reduces the horizon area (i.e., the entropy of the black hole bh ()).
To verify this theory in a realistic system requires a dissipationless quantum-entangled channel with a macroscopic correlation length, a detection scheme for the vacuum state, and a suitable implementation of LOCC. Quantum Hall systems are promising systems. They can be formed in a two-dimensional electron system in a semiconductor subjected to a strong perpendicular magnetic field. Such a system is suitable because the zero resistance of the quantum Hall effect means that the system is dissipationless and quasi-one-dimensional channels appear at the boundary of the bulk incompressible region of a quantum Hall system; such an edge channel is considered to behave as a chiral Luttinger liquid wen (). Power-law behaviors have been experimentally demonstrated, suggesting that the system has no certain decay length chang (); grayson (). Furthermore, since the vacuum is a zero-point fluctuation in the charge density, the Coulomb interaction (i.e., capacitive coupling), can be used as a sensitive probe for detecting the vacuum. In addition, semiconductor nanotechnology can be used to design LOCC.
In this study, we discuss possible passivity breaking in a quantum Hall system and estimate the order of the energy gain at B by employing reasonable experimental parameters.
We consider the system depicted schematically in Fig. 1. Experiments should be performed at low temperatures of the order of millikelvins (mK) to produce these vacuum states in the edge channels. Two regions, A and B, are defined on the left-going edge channel . We describe by a chiral Luttinger boson field with the trajectory parameterized by a spatial coordinate along the channel. Here, we consider the left-going chiral field whose electron number density deviates from that of the vacuum state equilibrium. is the group velocity of a charge density wave propagating along the channel (i.e., a magnetoplasmon allen () in the zero-energy limit) and is the elapsed time. The zero-point fluctuation in the charge density in region A can be experimentally measured by an RC circuit that consists of the input resistance of an amplifier and the capacitance between and a local metal gate fabricated on . For positive operator valued measure (POVM) measurements chang (), the RC circuit (i.e., the detector) can be switched on only during the measurement. is defined as the time when the switch is turned on. During the measurement, the detector excites and injects energy into . The measured voltage signal is amplified and then communicated to region U to excite a wave packet on the other edge channel . We describe the trajectory of by a right-going chiral boson field parameterized by a spatial coordinate with identical velocity , which assumes that and are formed in the same manner. Thus, travels along carrying energy toward region , where the two edge channels and approach each other. These channels are capacitively coupled only at , where interacts with the zero-point fluctuation of . After the interaction, the energy carried by changes from to . If no information about is communicated (i.e., is created independently of signal ) will inject energy to due to the passivity of the vacuum state passivity (). Thus, will be negative. However, in our system, since explicitly depends on , the passivity is broken and is positive; gains positive energy from the zero-point fluctuation of . In the following, we prove this fact theoretically and estimate by setting the experimental parameters m/s ashoori (); kamata (), k impedance (), and fF. The length of regions U and B and the length of region A are approximated by a typical length scale of m.
We start the detailed discussion by modeling the edge channel . The chiral field operators satisfy the commutation relation . The energy density operator of is written as
where is the Landau level filling factor of and denotes normal ordering, which causes the expectation value of to vanish for the vacuum state ; . The free Hamiltonian of is given by . The eigenvalue of vanishes, . Taking region A for we adopt the RC-circuit-detector model proposed by Fève et al. Feve () to measure the voltage induced by the zero-point fluctuation of . The charge fluctuation at A is estimated as
with a window function , which we assume to be a Gaussian function with its maximum value and width being of the order of unity and , respectively. In this model Feve (), the voltage between the detector contact and is given by , where is the charge on the capacitor. The quantum noise of the voltage is described by the operator
where () is the annihilation (creation) operator of excitation of a charge density wave in the detector circuit satisfying . Before the measurement (i.e., the signal input from to the detector), equals . Using the fast detector condition (), the voltage after the measurement is computed as
where denotes the voltage shift induced by the signal and the dot of indicates the time derivative. The amplitude of in the vacuum state of the RC circuit can be estimated as V. The root mean square of the voltage shift, , is estimated to be V from Eq. (1), so quantum fluctuations of the edge current will be detectable.
To estimate the energy injected into A after the measurement, we reduce the measurement operators chang () to the pointer basis of von Neumann vN (). By regarding as a preamplified quantum pointer operator, the instantaneous shift in Eq. (2) can be reproduced by the measurement Hamiltonian, , where is the conjugate momentum operator of . Using the eigenvalue of (), we can assume the initial wavefunction of the quantum pointer in the representation to be , whereas the wavefunction after the measurement is translated as . After turning the measurement interaction on, we perform a projective measurement of to obtain an eigenvalue of . This reduction analysis proves that the measurement operator nc () is . The corresponding POVM is represented by the operator , which satisfies the standard sum rule, , where is the identity operator of the Hilbert space of . The probability density of the result is . The post-measurement state of corresponding to the result is computed to be up to a normalization constant. Hence, the average state of right after the measurement is given by
The injected energy is calculated to be
Using the experimental parameters mentioned above, is estimated to be of the order of meV for .
We now consider the edge channel . The free Hamiltonian of is . The measurement result is amplified and transferred to region U as an electrical signal and it then generates a wave packet of (i.e., a localized right-going coherent state) in a region with . This process can be expressed by the -dependent unitary operation to the vacuum state of ; . is given by
where is the distance between regions U and B and is the filling factor of . Here, the length is approximated by . We assume that is a Gaussian function whose maximum, , gives the order of the total number of excess and deficient electrons from the vacuum-state equilibrium in , which are excited by the amplified voltage. We take the amplitude of , which can be experimentally done by tuning the amplifier gain appropriately. By using , the wave form of is computed as . By defining as the elapsed time when has been generated, the composite state of and at is calculated as
where . This state is the scattering input state for the Coulomb interaction between and . Then, evolves into region B by . The average energy of is denoted by (), which is calculated as
where . is estimated to be of the order of meV when and are respectively and . In region B, and interact with each other via the Coulomb interaction such that
Here, is for the host semiconductor (e.g., gallium arsenide), where is the dielectric constant of vacuum. The function is given by . Here, is the separation length between the two edge channels at B and is approximated by . After the interaction, the energy of becomes . The energy gain, , is estimated by lowest-order perturbation theory in terms of as follows.
where and . Here, . and are respectively the start and end times of the interaction between and . By substituting the commutation relation given by and integrating with respect to , we obtain the following relation:
Note that the last integral is computed as
To integrate with respect to , we take the Fourier transform of in and obtain . Using , is estimated as
where . The parameter corresponds to the distance between A and B. Thus, the energy output is estimated to be
Since the function is positive, must also be positive. Eq. (3) shows that increasing will rapidly reduce the magnitude of (e.g., 1eV for ). Nevertheless, for , will be of the order of eV. This is much larger than the thermal energy eV at a temperature of mK, which is the temperature at which quantum Hall effect experiments are generally performed using a dilution refrigerator.
To observe experimentally, we measure the current passing through the edge channel . The relation, , between the energy density and the current gives an energy density of -, which corresponds to a current of -. This can be detected experimentally using state-of-the-art electronics. To verify that energy is extracted at B, a single-shot current measurement should be repeated by switching the circuit on and off to perform POVM measurements a sufficient number of times to generate meaningful statistics. In this process, electrical noise, which can be introduced in the classical channel, is averaged out and thus does not affect .
We now examine energy conservation and dynamics in the system. As we have shown, extracting from the local vacuum state requires measurement (energy injection) at A. What is the source of ? We consider a POVM measurement, so that switching the RC circuit injects an energy into . Therefore, a battery may provide to drive the switching device if the switch is electrically operated. After extracting , the total energy of the system will be non-negative, as it should be because . Due to local energy conservation laws, energy transfer of from to will result in a negative average quantum energy density around B. This generation of a negative energy density is attained by squeezing the amplitude of the zero-point fluctuation less than that of the vacuum state during the interaction negativeE (). and will run ”chirally” on the edge toward the downstream electrical ground with identical velocities of . Because of this chirality, even after measurement at region A, around region B will remain in a local vacuum state with zero energy density.
Unlike quantum Hall systems, several successful experimental studies have been conducted in quantum optics by introducing LOCC including quantum teleportation QT (); exp-QT (). Light is a massless electromagnetic field with an infinite correlation length. However, it propagates three dimensionally so that the energy gain decays rapidly with increasing distance between A and B. In addition, it is currently difficult to measure the vacuum state experimentally due to the lack of an appropriate interaction such as the Coulomb interaction in quantum Hall systems. Thus, our quantum Hall system is considered to be very suitable for demonstrating local vacuum passivity breaking.
In conclusion, we have theoretically demonstrated that the passivity of the vacuum can be locally broken by electrical LOCC in a realistic system using a quantum Hall edge channel as a many-body quantum channel.
Acknowledgements.The authors gratefully acknowledge K. Akiba and T. Yuge for fruitful discussions. G. Y., W. I., and M. H. are supported by Grants-in-Aid for Scientific Research (Nos. 21241024, 22740191, and 21244007, respectively) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. W. I. and M. H. are partly supported by the Global COE Program of MEXT, Japan. G. Y. is partly supported by the Sumitomo Foundation.
- (1) W. Pusz and S. L. Woronowicz, Commun. Math. Phys. 58, 273 (1978).
- (2) M. Hotta, Phys. Lett. A 372, 5671 (2008); M. Hotta, Phys. Rev. D 78, 045006 (2008); M. Hotta, J. Phys. Soc. Jap. 78, 034001 (2009).
- (3) W. H. Zurek, in G. T. Moore and M. O. Scully, Frontiers of Nonequilibrium Statistical Physics (Plenum Press, New York), 151, (1984); S. Lloyd, Phys. Rev. A 56, 3374 (1997); T. Sagawa and M. Ueda, Phys. Rev. Lett. 100, 080403 (2008).
- (4) B. Reznik, Found. Phys. 33, 167 (2003); J. Silman and B. Reznik, Phys. Rev. A 71, 054301 (2005).
- (5) A. Strominger and C. Vafa, Phys. Lett. B 379, 99 (1996); A. Sen, Gen. Rel. Grav. 40, 2249 (2008).
- (6) M. Hotta, Phys. Rev. D 81, 044025 (2010).
- (7) X. G. Wen, Phys. Rev. B 43, 11025 (1991).
- (8) M. Grayson et al., Phys. Rev. Lett. 80, 1062 (1998).
- (9) A. M. Chang, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 77, 2538 (1996).
- (10) S. J. Allen, Jr., H. L. Störmer, and J. C. M. Hwang, Phys. Rev. B 28, 4875 (1983).
- (11) R. C. Ashoori et al., Phys. Rev. B 45, 3894 (1992).
- (12) H. Kamata et al., Phys. Rev. B 81, 085329 (2010).
- (13) The characteristic impedance of wires is matched to .
- (14) G. Fève, P. Degiovanni, and Th. Jolicoeur, Phys. Rev. B 77, 035308 (2008).
- (15) J. von Neumann, ”Mathematical Foundations of Quantum Mechanics”, Princeton University Press, (1955).
- (16) M. A. Nielsen and I. L. Chuang, ”Quantum Computation and Quantum Information”, Cambridge University Press, Cambridge, (2000).
- (17) Such an emergence of negative energy density has been widely known in quantum field theory. For example, one of the most simple cases can be given with linear superposition of the vacuum state and multi-particle states of a quantum field Ford ().
- (18) C.H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993).
- (19) D. Bouwmeester et al., Nature 390, 575 (1997); A. Furusawa et al., Science 282, 706 (1998).
- (20) L. H. Ford, Proc. R. Soc. London A 364, 227 (1978).