Effects of a quantum measurement on the electric conductivity: Application to graphene

(1)

http://www.diva-portal.org

This is the published version of a paper published in Physical Review B. Condensed Matter and Materials Physics.

Citation for the original published paper (version of record):

Bernad, J., Jääskeläinen, M., Zulicke, U. (2010)

Effects of a quantum measurement on the electric conductivity: Application to graphene.

Physical Review B. Condensed Matter and Materials Physics, 81(7): 073403 http://dx.doi.org/10.1103/PhysRevB.81.073403

Access to the published version may require subscription.

N.B. When citing this work, cite the original published paper.

Permanent link to this version:

http://urn.kb.se/resolve?urn=urn:nbn:se:du-11202

(2)

Effects of a quantum measurement on the electric conductivity: Application to graphene

J. Z. Bernád,^1,*M. Jääskeläinen,¹and U. Zülicke^1,2

1Institute of Fundamental Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Massey University, Manawatu Campus, Private Bag 11 222, Palmerston North 4442, New Zealand

2Centre for Theoretical Chemistry and Physics, Massey University, Albany Campus, Private Bag 102904, North Shore MSC, Auckland 0745, New Zealand

共Received 9 December 2009; published 8 February 2010兲

We generalize the standard linear-response共Kubo兲 theory to obtain the conductivity of a system that is subject to a quantum measurement of the current. Our approach can be used to specifically elucidate how back-action inherent to quantum measurements affects electronic transport. To illustrate the utility of our general formalism, we calculate the frequency-dependent conductivity of graphene and discuss the effect of measurement-induced decoherence on its value in the dc limit. We are able to resolve an ambiguity related to the parametric dependence of the minimal conductivity.

DOI:10.1103/PhysRevB.81.073403 PACS number共s兲: 72.10.Bg, 03.65.Ta, 73.23.⫺b, 81.05.U⫺

The fact that measurements exert a back-action on the measured object has attracted a lot of attention,^1–3partly due to its relevance for the foundations of quantum physics, but also because of implications for metrology⁴and the design of solid-state devices.⁵ Fundamental considerations necessitate a distinction between selective and nonselective descriptions of quantum measurements.⁶ Selective descriptions use sto- chastic differential equations,^7,8or restricted path integrals,⁹ and result in conditional quantum dynamics when the measurement results are recorded. Some properties of selective measurements have been verified experimentally for areas as diverse as cavity QED共Ref. 10兲 and superconducting phase qubits.¹¹Nonselective descriptions represent the evolution of the measured system irrespective of the measurement result.

This description takes into account all possible readouts, and the actual readout is assumed not to be known.^6,8Quantum- mechanical back-action on the unsharply measured system causes loss of coherence between eigenstates of the measured quantity. In this work, we discuss measurement back- action theoretically within the nonselective framework. This approach makes it possible to determine how a macroscopic observable such as the conductivity of a system is affected when the current is detected in an unsharp-measurement sce- nario. We derive a generalized Kubo formula where the measurement back-action provides a natural damping mecha- nism. We demonstrate the power of the developed general formalism by calculating the frequency-dependent conductivity ␴共␻兲 of graphene,¹² a promising candidate for future microelectronics and nanoelectronics¹³ and also a low- energy laboratory of relativistic physics.^14,15 Electronic- transport properties of graphene were analyzed in several previous studies using different methods, e.g., the Landauer- Büttiker formalism,^16,17 the linear-response Kubo formula,^17–20 and the Boltzmann equation.²¹ It was found that different parametric dependences of the dc conductivity can result from different limiting procedures applied to ordi- nary Kubo formulas.²⁰Within our generalized Kubo formalism, we obtain physical conditions for when these results apply. Two regimes can be distinguished by comparison of the energy scaleប⌫ that quantifies measurement-induced decoherence with the greater one among the thermal energy kBT and the chemical potential␮共measured from the Dirac point兲. Weak back-action 共ប⌫⬍max兵kBT ,␮其兲 results in

Drude-type behavior ␴共0兲⬀1/⌫. In the opposite 共strong- back-action兲 limit, a mixing of intraband and interband contributions occurs that changes the parametric dependence of the dc conductivity such that␴共0兲⬀⌫.

We employ the linear-response 共Kubo兲 formalism²² and divide the system’s Hamiltonian into the part Hˆ

0, which gov- erns the free evolution, and␦^Hˆ , the perturbation associated with an external electric field E. For simplicity, we take the latter to be constant in space and assume the field to be applied between t = −⬁ and t=0. The perturbation Hamil- tonian is␦^H^{ˆ =−eE·rˆe}ⁱ^␻t. In a nonselective description,^2,3the dynamics of the density matrix, when the current is measured, is governed by a master equation with the back-action caused by a term of Lindblad form²³

d␳^ˆ dt = − i

ប关Hˆ,␳^ˆ兴 −␥

8†jˆ,关jˆ,␳^ˆ兴‡ = L␳^{ˆ −} ⁱ

ប关␦^H^{ˆ ,}␳^ˆ兴, 共1兲 where L␳^{ˆ = −}_បⁱ关Hˆ0,␳^ˆ兴−^␥₈[jˆ ,关jˆ,␳^ˆ兴] with the current operator jˆ = i_ប^e关Hˆ0, rˆ兴 being our measured observable.²⁴ We note that the measured observable is arbitrarily chosen to be the current, as it is relevant for the conductance calculation at hand.

The main parameter of an unsharp quantum measurement device, as described by Eq.共1兲, is the detection performance

␥⁼共⌬t兲⁻¹共⌬j兲⁻², where⌬t is the time resolution or, equiva- lently, the inverse bandwidth of the detector and ⌬j the sta- tistical error characterizing unsharp detection of the average current. Within linear-response theory, we can linearize ␳^ˆ

=␳^ˆ0+␦␳^{ˆ , where} ␳^ˆ0 is the system’s equilibrium density matrix. Keeping only linear terms in Eq.共1兲, we get

d␦␳^ˆ

dt =L␦␳⁻_បⁱ关␦^H^{ˆ ,}␳^ˆ0兴 −␥

8†jˆ,关jˆ,␳^ˆ0兴‡, 共2兲 assuming that the unsharp detection does not affect the equilibrium and using 关Hˆ0,␳^ˆ0兴=0 as well as 关␦^H^{ˆ ,}␦␳^ˆ兴⯝0. Intro- ducing⌬␳^{ˆ = e}⁻^Lt␦␳^{ˆ yields}

d⌬␳^ˆ

dt = e^−Lt

再

⁻^បⁱ^关^␦^H^{ˆ ,}^␳^ˆ⁰^{兴 −}^␥⁸^{†jˆ,关jˆ,}^␳^ˆ⁰^兴‡

冎

^. ^共3兲

Note that ⌬␳^{ˆ and}␦␳ˆ have the same value at t = 0, and both vanish at t = −⬁. Integration yields

(3)

␦␳^ˆ共t = 0兲 =冕_−⬁⁰ ^dte^−Lt

再

⁻^បⁱ^关^␦^H^{ˆ ,}^␳^ˆ⁰^{兴 −}^␥⁸^{†jˆ,关jˆ,}^␳^ˆ⁰^兴‡

冎

^. ^共4兲

The exponential factor in Eq.共4兲 ensures convergence of the time integral, making it unnecessary to introduce the phe- nomenological adiabatic damping parameter employed in conventional linear-response theory.²² Inserting Eq. 共4兲 into the expectation value for the current density jˆ and dividing by兩E兩 yields the optical conductivity

␴_␮␯共␻兲 =冕−⬁ 0

关K_␮␯共t兲eⁱ^␻t+ K_␮␯

⬘

共t兲兴dt 共5兲

with the kernels

K_␮␯共t兲 = − 1

iបTr兵jˆ_␮e⁻^L共jˆ^␮^兲t共关erˆ_␯,␳^ˆ0兴兲其, 共6兲

K_␮␯

⬘

共t兲 = Tr

冉

^E^jˆ^␮␯e⁻^L共jˆ^␮^兲t

再

⁻^␥⁸^†jˆ^␮^,关jˆ^␮^,^␳^ˆ⁰^兴‡

冎 ^冊

^. ^共7兲

Equations 共6兲 and 共7兲 are the general result for the response to an unsharp measurement and could, in principle, be applied to any quantum system. As an instructive application, we now use Eq. 共5兲 to calculate the frequency- dependent conductivity of single-layer graphene within the continuum model for quasiparticles close to the K point in the Brillouin zone. In Eq.共1兲, the Hamiltonian Hˆ0 could, in principle, include Coulomb interactions, impurity scattering, density dependencies, etc. In the present work, we just use the free-quasiparticle Hamiltonian for graphene in plane- wave representation,

Hˆ

0共k兲 = បv共␴xkx+␴yky兲, 共8兲 where kxand kyare the Cartesian components of wave vector k, ␴i denote Pauli matrices acting in the sublattice-related pseudospin space, and v is the Fermi velocity, which has a value ⯝10⁶ m/s. With the position operator rˆ being the wave-vector gradient, the current operator is jˆ_␮

=^ie_ប关Hˆ0共k兲,rˆ_␮兴=_ប^e^⳵Hˆ_⳵k⁰^共k兲_␮ . Single-particle eigenstates of clean graphene can be written as a direct product of a plane wave in configuration space with a spinor 兩n典=兩k典丢兩␴典k. Here␴ labels the electron and hole bands, respectively, and the spinor wave function depends on wave vector k. The current operators with the Hamiltonian 共8兲 in the spinor space are jˆx= ev␴x, and jˆy= ev␴y.²⁵ From the definition of the equilibrium density matrix in the spinor space we find ␳^ˆ0兩␴典k

= f共ប⑀_k,␴兲兩␴典k, and Hˆ

0兩␴典k=ប⑀_k,␴兩␴典k, where f is the Fermi- Dirac distribution function and ⑀_k,⫾=⫾兩k兩. 共For simplicity, the speed v has been absorbed into k.兲 Together with the antisymmetry in momentum space, this implies K_␮␯

⬘

^{= 0. The} calculation of K_␮␯ is straightforward, and using the Laplace transform to solve for the dynamics, we find

K_␮␯共t兲 =e²

ប冕共2^d␲²^k兲²Res兵K˜_␮␯共k,␥,z兲e^zt其, 共9兲 where Res stands for the sum of residues of the integrand. It can be seen that K_xy= K_yx= 0. The remaining two conductivi- ties are identical, as a change in variables kx↔kyin the ex- pression for Kxxyields Kyy. The choice of conductivity measured will decide what kind of back-action will influence the system. Here we take jˆ_x, the current along the x direction, which breaks the isotropy of the problem and we find

K˜

xx共k,⌫,z兲 =

k_x²兩k兩共16⌫²− 8z⌫ + 4兩k兩²+ z²兲兺_␴

冉

⁻

冏

^df共ប^d^⑀^⑀^兲

冏

_⑀=⑀_k,_␴

冊

^{+ k}^y²^{共z − 4⌫兲}²^关f共ប^⑀^k,−^{兲 − f共ប}^⑀^k,+^兲兴

兩k兩³关16z⌫²− 8共2ky2+ z²兲⌫ + z共4兩k兩²+ z²兲兴 , 共10兲

where the parameter ⌫=␥^e²^v²/8 was introduced. The cubic factor in the denominator of Eq. 共10兲 has three roots zi, which give the poles in Eq.共9兲.

For small ⌫, the roots are to lowest order z1= 0 and z_2,3

=⫾2i兩k兩. Using this and performing the time integration in the limit ⌫→0, the known intraband and interband contri- butions

␴^共intra兲

␴0

= ␲ 2␦共␻兲冕0

⬁

x

冋

^␴=⫾^兺 ^冉⁻^冏^df共ប^d^⑀^⑀^兲^冏^⑀=␴x^冊

册

^dx

= ␲

2␦冉^k^បB^␻T冊冋2 log共1 + e^␮/k^B^T兲 − ␮

kBT册^, ^共11兲

␴^共inter兲

␴0

=␲ 8

sinh冉2k^ប^␻BT冊

cosh冉^k^␮BT冊^{+ cosh}冉^2k^ប^␻BT冊 ^共12兲

to the conductivity of clean graphene are found. The scale factor␴0= 4e²/h accounts for spin and pseudospin degenera- cies.

In the following, the conductivity is calculated numeri- cally for finite values of ⌫ from Eq. 共5兲 with Eqs. 共9兲 and 共10兲. We use kBT as unit of energy. Figures1and2show the ac共optical兲 conductivity,²⁷whereas Fig.3shows the dc conductivity that is measured, e.g., in mesoscopic transport experiments.

BRIEF REPORTS PHYSICAL REVIEW B 81, 073403共2010兲

073403-2

(4)

In Fig. 1, the conductivity is shown as function of frequency for different values of the coupling strength when the chemical potential remains fixed. For high frequencies, the conductivity saturates to the universal value ␲/8, indicated by a dashed line. The detailed shape of the crossover to saturation depends on ⌫ with higher values pushing it to higher frequencies.

In the limit of small⌫, measurement-induced decoherence appears to simulate the effect of life-time broadening due to inelastic scattering,^19,20,26but a closer look reveals that it is fundamentally different. Technically, the effect of⌫ goes beyond merely broadening of distribution functions, it also moves the position of their peaks in energy, thus changing the resonance condition. For clean graphene, there is only a delta function peak for the intraband contribution, whereas here the intraband and interband contributions to the conductivity become mixed. The existence of such a mixing has been inferred from recent experiments.²⁷The dependence on the chemical potential is illustrated in Fig. 2. Generally, increasing the chemical potential shifts the frequency beyond which the conductivity attains its universal saturation to

higher values. This behavior is as expected theoretically²⁶ and observed in experiments.²⁷At frequencies␻smaller than a few times the chemical potential, there is a significant de- parture from the universal conductance plateau. For fixed␮ and k_BT the saturation point in scattering models is indepen- dent of the scattering parameter, whereas in our work it strongly depends on the value of⌫. For ␮ⲏប⌫, the saturation occurs as in clean graphene, whereas the opposite case gives saturation for larger frequencies with increasing ⌫, as seen in Fig.1.

When the effect of disorder is modeled conventionally by a life-time broadening due to inelastic scattering,^19,20a Drude peak is found for ប␻/kBT⬃0, with a height inversely proportional to the inelastic-scattering rate. Since the master equation describing the effect of quantum measurements is formally identical to certain models of decoherence, we ex- pect that␴共0兲/␴0⬃1/⌫. This turns out to be correct only for ប⌫⬍max兵␮^{, k}BT其, as can be seen in Fig. 3共b兲. In the opposite limit, the parametric dependence of the dc conductivity is changed; it then grows linearly with ⌫. A similar anomalous behavior was obtained in Ref.20by applying an uncon- ventional limiting procedure within the Kubo formalism.

Here we are able to readily identify the regimes where ordi- nary Drude behavior or anomalous band-mixing behavior will be exhibited. Clearly, observing the latter should be easi- est when the chemical potential is at the Dirac point and the temperature is low enough to satisfyប⌫⬎kBT, which corre- sponds to the minimal conductivity regime for graphene.

Figure 3共a兲 shows how the nonmonotonic⌫ dependence of the conductivity would be manifested in a typical transport experiment where the chemical potential共i.e., the density兲 is varied.

10⁻² 10⁻¹ 10⁰ 10¹ 10²

10⁻¹ 10⁰ 10¹

¯ hω/kBT σ (ω )

σ0

¯hΓ kBT= 10² 10¹

10⁰ 10⁻¹ 10⁻²

FIG. 1. Frequency dependence of the conductivity of clean single-layer graphene when the current is unsharply quantum measured. The chemical potential is fixed at␮/kBT = 1, and the value for measurement-induced decoherence assumed for each curve is indicated. The dashed line is at␲/8. As the coupling to the measurement device is decreased, the curves more closely resemble the result found for clean graphene.

0 5 10 15 20 25

0 0.2 0.4 0.6

¯hω /kBT σ (ω )

σ0

µ kBT= 1

5

10

FIG. 2. ac conductivity of single-layer graphene, for fixed ប⌫/kBT = 1 and different values of chemical potential␮ 共measured from the Dirac point兲.

10⁻¹ 10⁰ 10¹ 10²

10² 10³ 10⁴ 10⁵

µ /kBT σ(0)

σ₀ ^¯hΓ/k^B^{T = 0.01}

0.1 1

10 a)

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³

10² 10³ 10⁴ 10⁵

¯hΓ/kBT σ(0)

σ0 µ/kBT = 1

10 100 b)

FIG. 3. 共a兲 dc conductivity plotted as a function of chemical potential for different values of⌫. The conductivity is linear in the chemical potential for large values of ␮/kBT. 共b兲 dc conductivity shown as a function ofប⌫/kBT for different values of the chemical potential. For ␮⬎kBT and ប⌫⬍␮, the conductivity is inversely proportional to the effective rate of decoherence, in analogy with the behavior expected from inelastic impurity scattering. In contrast, for ប⌫⬎␮ there is direct proportionality. This behavior was exhibited as long as ␮ⲏkBT, whereas curves for ␮ⱕkBT practi- cally coincide.

(5)

The theory presented here is based on an unsharp measurement of the current density. To estimate the magnitude⌫ of measurement-induced decoherence, we must consider the two situations most closely related to our result, optical conductivity measurements, and mesoscopic transport measurements. For the latter case, it can be easily seen that e²v²/共⌬j兲² is the signal-to-noise ratio. Using typical values

⌬␯ⲏ10⁶ Hz for the bandwidth and ⌬I/I⬇10⁻³ for the signal-to-noise ratio, we find that values up to ប⌫/kBT

⬇10⁻¹– 10²can be achieved for T⬇1–300 K. For measurements of the optical conductivity, the situation is more com- plex, as the measured signal is induced by the currents gen- erated by the applied optical field. As a result, additional uncertainties such as geometrical factors and detector effi- ciencies become important, possibly bringing down the detection performance, but this can, in principle, be compen- sated by the increase in bandwidth offered by optical detectors,⌬␯ⲏ10⁹ Hz.

In conclusion, we derived a new Kubo formula to study the effect of measurement-induced back-action on the conductivity. The back-action naturally introduces a source of damping and thus makes the converged adiabaticity parameter frequently used in Kubo formula calculations superflu- ous. We applied this approach to calculate the electric conductivity of single-layer graphene. Mixing of the intraband and interband contributions to the dc conductivity strongly affect its parametric dependence on the detector performance

⌫. The regime of weak coupling to the measuring device models a standard Drude-type behavior, whereas in the opposite limit of strong back-action, we find that measuring a current in graphene will actually enhance the conductivity.

This work was supported by the Massey University Re- search Fund and by the Marsden Fund Council共Contract No.

MAU0702兲 from Government funding, administered by the Royal Society of New Zealand.

*j.z.bernad@massey.ac.nz

1V. B. Braginsky and F. Ya. Khalili, Quantum Measurement 共Cambridge University Press, Cambridge, 1992兲.

2L. Diósi, in Encyclopedia of Mathematical Physics, edited by J.-P. Françoise, G. L. Naber, and S. T. Tsou共Academic, Oxford, 2006兲, Vol. 4, p. 276.

3K. Jacobs and D. A. Steck, Contemp. Phys. 47, 279共2006兲.

4J. M. Geremia et al., Phys. Rev. Lett. 91, 250801共2003兲.

5J. Z. Bernad et al., Phys. Rev. B 77, 073311共2008兲.

6M. B. Menskii, Phys. Usp. 41, 923共1998兲; 46, 1163 共2003兲.

7N. Gisin, Phys. Rev. Lett. 52, 1657共1984兲; A. Barchielli, Phys.

Rev. A 34, 1642 共1986兲; L. Diosi, Phys. Lett. A 129, 419 共1988兲.

8C. M. Caves and G. J. Milburn, Phys. Rev. A 36, 5543共1987兲.

9M. B. Mensky, Phys. Rev. D 20, 384共1979兲; C. M. Caves, ibid.

33, 1643共1986兲.

10C. Guerlin et al., Nature共London兲 448, 889 共2007兲.

11N. Katz et al., Science 312, 1498共2006兲.

12K. S. Novoselov et al., Science 306, 666 共2004兲; K. S. No- voselov et al.,Proc. Natl. Acad. Sci. U.S.A. 102, 10451共2005兲.

13P. Avouris et al., Nat. Nanotechnol. 2, 605共2007兲; Z. Chen et al., Physica E 40, 228 共2007兲; J. B. Oostinga et al., Nature Mater. 7, 151共2007兲.

14Y. Zhang et al., Phys. Rev. Lett. 94, 176803共2005兲; Y. Zhang et al., Nature共London兲 438, 201 共2005兲.

15K. S. Novoselov et al., Nature共London兲 438, 197 共2005兲.

16M. I. Katsnelson, Eur. Phys. J. B 51, 157共2006兲; J. Tworzydlo et al., Phys. Rev. Lett. 96, 246802共2006兲.

17S. Ryu et al., Phys. Rev. B 75, 205344共2007兲.

18A. W. W. Ludwig et al., Phys. Rev. B 50, 7526共1994兲; E. V.

Gorbar et al., ibid. 66, 045108共2002兲; V. P. Gusynin and S. G.

Sharapov, Phys. Rev. Lett. 95, 146801共2005兲; Phys. Rev. B 73,

245411共2006兲; N. M. R. Peres et al.,ibid. 73, 125411 共2006兲; P.

M. Ostrovsky et al., ibid. 74, 235443共2006兲; J. Cserti, ibid. 75, 033405共2007兲.

19K. Ziegler, Phys. Rev. Lett. 97, 266802共2006兲.

20K. Ziegler, Phys. Rev. B 75, 233407共2007兲.

21K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602 共2006兲; 98, 076602 共2007兲; N. M. R. Peres et al., Phys. Rev. B 76, 073412共2007兲; E. H. Hwang et al., Phys. Rev. Lett. 98, 186806共2007兲.

22O. Madelung, Introduction to Solid-State Theory共Springer, Ber- lin, 1978兲.

23Starting from positive operator-valued measure and using the standard Gaussian model we arrive in the selective measurement case to stochastic master equation and in the nonselective case at a master equation with the double commutator.

24For other types of experiments, the corresponding measured operator will occur in the Lindbladian term and, thus, induce back- action on the system. Although the formalism is, in its mathematical structure, identical to decoherence, the effects on the dynamics depend on what kind of measurement is performed, which is a matter of choice. In contrast, environmentally induced decoherence that is determined by interaction with an ever-present reservoir cannot be manipulated at will. The choice of measured observable thus decides the type of decoherence present in the system through experimental design.

25The Hilbert space is decomposed into irreducible components.

Each pseudospin sector is a superselection sector, preserved by the current operator.

26N. M. R. Peres et al., EPL 84, 38002共2008兲.

27K. F. Mak et al., Phys. Rev. Lett. 101, 196405共2008兲; F. Wang et al., Science 320, 206共2008兲; Z. Q. Li et al., Nat. Phys. 4, 532共2008兲.