Quantum electromechanical systems

Available online at www.sciencedirect.com

Physics Reports 395 (2004) 159 – 222 www.elsevier.com/locate/physrep

Quantum electromechanical systems Miles Blencowe∗ Department of Physics and Astronomy, 6127 Wilder Laboratory, Dartmouth College, Hanover, NH 03755, USA Accepted 23 December 2003 editor: C.W.J. Beenakker

Abstract Quantum electromechanical systems are nano-to-micrometer (micron) scale mechanical resonators coupled to electronic devices of comparable dimensions, such that the mechanical resonator behaves in a manifestly quantum manner. We review progress towards realising quantum electromechanical systems, beginning with the phononic quantum of thermal conductance for suspended dielectric wires. We then describe e1orts to reach the quantum zero-point displacement uncertainty detection limit for (sub)micron-scale mechanical resonators using the single electron transistor as displacement transducer. A scheme employing the Cooper-pair box as coherent control device to generate and detect quantum superpositions of distinct position states is then described. Finally, we outline several possible schemes to demonstrate various other quantum e1ects in (sub)micron mechanical resonators, including single phonon detection, quantum squeezed states and quantum tunnelling of mechanical degrees of freedom. c 2003 Elsevier B.V. All rights reserved.
PACS: 85.85.+j; 85.35.Gv; 03.65.Ta; 03.65.Ud; 03.65.Xp; 03.65.Yz Keywords: Quantum electromechanical systems; Quantized thermal conductance; Quantum-limited displacement detection; Macroscopic quantum mechanical e1ects

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The quantum of thermal conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Derivation of the Landauer formula for the thermal conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Observation of thermal conductance quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Universal nature of thermal conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Information capacity of single quantum channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Ultrasensitive displacement detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Tel.: +1-603-646-2969; fax: +1-603-646-1446. E-mail address: [email protected] (M. Blencowe).

c 2003 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2003.12.005

160 164 165 170 175 179 181

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3.1. Overview of various displacement detection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Analysis of the fundamental noise limits on SET-based displacement detection . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Experimental progress towards quantum-limited SET-based displacement detection . . . . . . . . . . . . . . . . . . . . . . 4. Macroscopic mechanical superposition states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Background and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Analysis of the Cooper-pair box-based mechanical superposition scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Quantifying the Cooper box-mechanical resonator entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Other quantum electromechanical e1ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Single phonon detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Quantum squeezing of mechanical motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Quantum tunnelling of mechanical degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 184 192 195 195 198 205 206 207 209 212 216 217 217

1. Introduction Quantum electromechanical systems [1–3], or QEMS for short, is an emerging branch of mesoscopic physics [4] made possible by recent advances in microfabrication technology. A QEM device typically comprises a nano-to-micron scale mechanical resonator, such as a cantilever (suspended beam which is clamped at one end) or bridge (suspended beam clamped at both ends), which is electrostatically coupled to an electronic device of comparable dimensions, such as a single electron transistor (SET) [5]. While a mechanical resonator comprises three times as many normal vibrational modes as it does atoms, only the lowest, few Jexural modes will strongly couple to the electronic device. Provided that the quality factors of these lowest modes are very large, then for small amplitudes the mechanical resonator behaves e1ectively like a few independent damped harmonic oscillators. The quality factor for a given oscillator is deKned as Q = !, where ! is the oscillator angular frequency and  is the energy decay time-constant, i.e., the time taken for the energy stored in the oscillator to decay by a factor e from its initial value. Measured quality factors of the lowest modes of nano-to-micron scale mechanical resonators in moderate vacuum are typically in the range 103 –104 [6]. With the appropriate temperature and vacuum conditions, and device operating parameters, these oscillators will behave in a manifestly quantum manner, as indirectly evidenced through the electronic device behaviour. An important rule-of-thumb for observing quantum behaviour is hm & kB T ;

(1)

where m is the resonator’s lowest, Jexural fundamental mode frequency (the “m” subscript denotes “mechanical”), T is the resonator temperature and h and kB are Planck’s and Boltzmann’s constants, respectively. The lowest, typical achievable temperature using a dilution refrigerator is about 30 mK which gives m & 600 MHz, in the radio frequency regime. For the example of a cantilever with length l, width w, thickness t, mass density and Young’s modulus E, the frequency m is (see, e.g., pp. 119 –120 of Ref. [7])
E t : (2) m = 0:56 2 l 12

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161

Assuming, for example, the bulk material values for silicon (Si): E =1:5×1011 N m−2 and =2:33× 103 kg m−3 , and expressing the size dimensions in microns and frequency in GHz, (2) becomes t (3) m = 1:3 2 : l For example, a 1
m long and 0:1
m thick Si cantilever has a fundamental frequency of 130 MHz. Thus, to observe quantum e1ects, submicron scale mechanical resonators must be employed. Note, however, as we shall learn in this review, condition (1) is not always strictly necessary in order to observe quantum behaviour. For example, it should be possible to demonstrate quantum entanglement at T = 30 mK for a resonator with m = 50 MHz. Decoherence due to the interaction of the resonator with its environment governs the quantum-classical border in this case (Section 4). Nevertheless, the lower temperature limits to which mechanical resonators can be cooled essentially deKne the size scale for which quantum e1ects can be observed; it is therefore extremely important to develop alternative methods of cooling suited to mechanical devices if quantum e1ects are to be demonstrated at larger-than-micron scales [8–10]. Despite the relatively small sizes of the (sub)micron scale mechanical devices, they comprise up to ∼ 1010 –1011 atoms and so in some sense their quantum behaviour would be deemed macroscopic [11,12], certainly substantially pushing the quantum-classical divide out of the microscopic and into the macroscopic realm for mechanical systems (see Section 4). Another important quantum mechanical scale is the zero-point displacement uncertainty in the fundamental mode, which for the cantilever is
˝ Sxzp = ; (4) 2me1 !m where me1 = m=4 is the cantilever’s e1ective motional mass, with m the physical mass. Again expressing the cantilever dimensions in microns and the displacement uncertainty in angstroms, and using the material parameters for Si, (4) becomes √ l −5 √ : Sxzp = 3:3 × 10 (5) t w For the example of a radio frequency, micron-scale Si cantilever with dimensions 1
m × 0:1
m × T As we shall see (Section 3), such small displacements may be 0:1
m, we have Sxzp ≈ 10−3 A. resolvable using a SET. One of the Krst demonstrations of a radio frequency, micron-scale mechanical resonator with both displacement actuation and detection in the frequency range about the fundamental Jexural mode, was a single crystal Si beam with length 7:7
m, width 0:33
m, thickness 0:8
m, and measured fundamental frequency 70:72 MHz (Fig. 1) [13]. A few years later, a Si beam resonator with dimensions 2
m × 0:2
m × 0:1
m and with measured fundamental frequency 380 MHz was realized [14]. And very recently, a silicon carbide (SiC) beam resonator with dimensions 1:1
m × 0:12
m × 0:075
m and measured fundamental frequency 1:029 GHz was demonstrated [15]. Other materials which have been used for radio frequency, micron-scale mechanical resonators include gallium arsenide (GaAs) [16,17], silicon nitride (Six Ny ) [18], aluminum nitride (AlN) [19] and diamond [20,21]. The choice of material depends on several factors. Withthe fundamental Jexural frequency depending on Young’s modulus E and the mass density as E= , and the zero-point uncertainty depending on these parameters as (E )−1=4 , it is clear that the quantum limit favours

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Fig. 1. Scanning electron microscope (SEM) micrograph of Si beam. (Reproduced from Ref. [13].)

materials which are both strong (large E) and light (small ). In this respect, diamond, SiC, SiN and AlN are preferable to Si and GaAs. Furthermore, diamond, SiC and AlN have good chemical stability, suggesting the possibility to increase the quality factors of micron-scale resonators fashioned from these materials through certain surface treatments. AlN is also piezoelectrically active, with the possible advantage of more straightforward displacement actuation and detection [19]. The fabrication procedure for a suspended, micron-scale structure involves several steps which we will now outline for the example of a SiN structure [18]; the fabrication procedures for other materials are similar (see, e.g., Ref. [22] for a comprehensive review of device fabrication). The process starts with a single crystal Si (100) wafer on which is grown a few hundred nanometer thick sacriKcial layer of silicon dioxide, followed by the deposition of a silicon nitride layer of comparable thickness and Knally a bilayer of poly-methylmethacrylate (PMMA) resist (Fig. 2a). The geometry of the suspended structure is then patterned on the resist using electron-beam lithography and the resist developed (Fig. 2b). A chromium (Cr) layer is evaporated onto the surface and lifted-o1 with the undeveloped PMMA resist (Fig. 2c); because Cr adheres less well to the PMMA than to the SiN layer, the remaining Cr after lift-o1 covers only the previously developed areas, forming a mask for the subsequent etching of the SiN layer. The SiN area is etched using a CH4 -H2 plasma process, transferring the Cr mask pattern to the SiN layer beneath it (Fig. 2d). An O2 plasma is used to etch away the Cr mask and a chemically selective wet etch (hydroJuoric acid) is used to remove the silicon dioxide layer, hence releasing the structure from the substrate (Fig. 2e). Final, metallization of the structure may be required, so as to enable displacement actuation and detection using, for example, the magnetomotive method (discussed in the beginning of Section 3). This can be achieved through the evaporation of a few nm thick adhesion layer of Cr, followed by a thicker layer of gold. A nice example of a micron scale electromechanical device is the mechanical single electron shuttle shown in Fig. 3 [23]. By driving the cantilever close to one of its lowest resonant frequencies, it can shuttle electrons from the source to drain electrode via the small metallic island at the tip of the cantilever. The cantilever resonant frequencies involved are . 100 MHz, while the lowest quoted

M. Blencowe / Physics Reports 395 (2004) 159 – 222

163

PMMA SiN SiO2 Si (b)

(a)

Cr

(c)

(d)

(e)

Fig. 2. The various steps in the fabrication of a SiN suspended beam structure shown in cross section.

Fig. 3. SEM micrograph of micron scale, mechanical single electron shuttle. The cantilever (C) is driven at its fundamental frequency by an ac voltage applied to gates G1 and G2. Electron transport occurs from source (S) to drain (D) via the metallic island at the end of the cantilever. (Reproduced from Ref. [23].)

temperature at which the device is operated is 4:2 K; the cantilever behaves as a classical oscillator. However, by scaling down the cantilever a bit so as to increase its lowest resonant frequencies and in addition cooling the device down to a few tens of mK, it should be possible to observe in the source–drain I –V characteristics the quantum signatures of single-to-few phonon absorption

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M. Blencowe / Physics Reports 395 (2004) 159 – 222

and emission from these vibrational modes when driven by the shuttling electron current alone (i.e., without the external ac driving of the cantilever) [24–26]. In the following sections, we review our current knowledge of QEMS as deKned above. Section 2 describes the quantized heat Jow properties of suspended, dielectric wires of submicron cross sectional dimensions at subKelvin temperatures, such that the heat Jow is e1ectively one dimensional. The quantum of thermal conductance is derived and the recent experiments which measured this conductance quantum are reviewed. A discussion is also given of the universal, statistics independent nature of the thermal conductance quantum, as well as its connections to the classical information capacity of one-dimensional quantum channels. Section 3 begins with an overview of various sensitive displacement detection techniques for (sub)micron scale mechanical resonators and then focuses on a SET-based displacement detection scheme, giving a sensitivity analysis as well as discussing the status of experimental e1orts to realize such a device. A scheme to generate quantum superpositions of position states for micron scale mechanical resonators, as well as measure their decoherence, is described in Section 4. Section 5 gives an overview of various proposed schemes to detect single phonons, generate and detect quantum squeezed states and generate and detect quantum tunnelling of mechanical degrees of freedom. Concluding remarks are given in Section 6. This review places an emphasis on addressing the Krst stage of the realisation of QEMS and their relevant quantum mechanical behaviour, as constrained by existing possibilities in microfabrication technology. The essential next stage will involve investigating the expected rich, coupled quantum electromechanical dynamics of the various described systems, with a view to Knding common principles in their behaviour and to deepening our understanding of the quantum-classical divide. Many open problems are discussed and several of the proposed schemes to demonstrate certain types of quantum phenomena have yet to be fully worked out. This is particularly the case for the schemes described in Section 5. Other reviews complementary to the present one which emphasize di1erent aspects of (nano)electromechanical systems are given in Refs. [2,22,6,27–29]. Practically no mention is made in this review of work in the complementary Keld of quantum optomechanical systems. To learn more about this area of research, the reader may wish to consult, for example, Refs. [30,31]. 2. The quantum of thermal conductance One of the original pioneering experiments in mesoscopic physics was the measurement of the conductance of point contacts in a two-dimensional electron gas [32,33]. The electronic conductance was found to be quantized in steps of universal magnitude e2 =˝, explained by the classic theory of Landauer [34]. In the year 2000, the phononic quantized thermal conductance counterpart, kB2 T=6˝, was measured for the Krst time for suspended, dielectric wires of submicron cross sectional dimensions [35]. The experiment was a tour-de-force in microfabrication techniques and metrology and can be viewed as the Krst demonstration of an actual QEM device. Sections 2.1 and 2.2, which are taken essentially without change from Refs. [36,37], give a reasonably complete account of the theory and experiment for the quantum of thermal conductance. Sections 2.3 and 2.4 attempt to understand the thermal conductance quantum at a more fundamental level and put it into a broader context by addressing its universal nature and also its connections to the classical information capacity of quantum communication channels. The existence of the universal quantum of thermal conductance is

M. Blencowe / Physics Reports 395 (2004) 159 – 222

165

scattering region Reservoir T1

lead 1

lead 2

Reservoir T2

Fig. 4. Schematic diagram of the model wire. The left and right reservoirs are at temperatures T1 and T2 , respectively.

relevant to the problem of cooling nanomechanical resonators down to the quantum regime [8–10], since it provides an upper limit to the rate at which heat can be removed from the resonator. 2.1. Derivation of the Landauer formula for the thermal conductance In the following we shall derive the Landauer formula for the phononic thermal conductance of a suspended, dielectric wire [38,39,36], analogous to the electrical conductance formula [34]. No assumptions are made about the elastic moduli of the wire material. As we shall see, all the material properties dependent on the wire’s mass density and elastic moduli will eventually drop out to yield the universal thermal conductance quantum. Readers not interested in the details of the derivation may skip directly to the Landauer formula (24). The model wire structure on which the calculations are based is shown in Fig. 4. Two very long, perfect leads (i.e., crystalline and with uniform cross section) join a central section in which the phonon scattering occurs. The scattering may be caused by any combination of the following: a changing cross section, surface roughness, or various internal defects. The only restriction we place on the scattering is that it be elastic; phonon–phonon interactions are also neglected. The other ends of the leads are connected to heat reservoirs with Bose–Einstein distributions for the phonons. Perfect adiabatic reservoir-lead connections are assumed so that no scattering occurs at these connections. The wire, comprising leads and central scattering section, must be free-standing so that the phonons do not “leak” out, except to the reservoirs at the ends. Furthermore, the wire is electrically insulating, so that only phonons transport heat in the wire. Our point of departure is the classical equations of motion for the lattice dynamics of a perfect wire (i.e., no scattering) and also the expression for the classical energy current Jowing in the wire. At Kelvin or lower reservoir temperatures, phonon wavelengths typically exceed several hundred angstroms, and thus the continuum approximation can be used, so that the equation of motion is just the wave equation (see, e.g., Ref. [40, p. 446]):

9 2 ui  92 ul − c =0 ; ijkl 9t 2 9xj 9xk jkl

(6)

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where is the mass density, ui denotes the ith component of the displacement Keld and cijkl is the elastic modulus tensor of the wire material. For a free-standing wire the stress normal to the surface vanishes, so that we have the following boundary condition at the wire surface:    9ul  cijkl nj k  = 0 ; (7) 9x  jkl S

where nj is the jth component of the unit vector normal to the wire surface S. In terms of the displacement Keld and elastic modulus tensor, the energy current at a given location x is (the x coordinate runs along the length of the wire):  9uj 9ul I (x; t) = −cxjkl dy d z ; (8) 9t 9xk A jkl

where the integral is over the cross sectional surface A at x. In order to quantize Eq. (6), we require a complete set of normal mode solutions. For a perfect, inKnitely long wire, these solutions can be written in the following form: 1 (9) un; q; i (r; t) = √ e−i(!n; q t −qx) n; q; i (y; z) ; 2 where n; q; i denotes the transverse modes, q is the longitudinal wave vector along the wire axis and n is the subband label. In the presence of scattering, we can still construct solutions in the leads using the perfect wire solutions (9) as follows:   un; q; i + un
; −q
; i tn11
n (!) lead 1 ;    n
un;1 q; i =  (10)  21 

u t (!) lead 2 ;
n ;q ;i n n  n


and

 un
; −q
; i tn12
n (!)   
n un;2 q; i =    un
; q
; i tn22
n (!)  un; −q; i +

lead 1 ; lead 2 ;

(11)

n


where q; q ¿ 0. The solutions un;1 q; i describe waves propagating from lead 1 to lead 2, while solutions un;2 q; i propagate from lead 2 to lead 1. The absolute value of the scattering matrix element tnba
n (!) gives the fraction of the incident wave in lead a, with frequency ! and subband label n, which is transmitted/reJected into lead b and subband n . In the sum over n , the frequency ! is kept Kxed, while q is treated as a function of n and ! through the condition !n
; q
= !n; q = !. From energy conservation, the time average of the energy current I (x; t) should be independent of the position x. Substituting into the deKnition for the energy current (8) an arbitrary linear combination of solutions (10) and (11) and demanding that the time averaged energy currents in leads 1 and 2 be the same, we obtain the following conditions on the scattering matrix elements:   vn

; q

tn11

n (!)tn11

∗n
(!) + vn

; q

tn21

n (!)tn21

∗n
(!) = vn; q %nn
; (12) n



n



M. Blencowe / Physics Reports 395 (2004) 159 – 222



vn

; q

tn22

n (!)tn22

∗n
(!) +

n



and



167

vn

; q

tn12

n (!)tn12

∗n
(!) = vn; q %nn


(13)

vn

; q

tn21

n (!)tn22

∗n
(!) = 0 ;

(14)

n





vn

; q

tn11

n (!)tn12

∗n
(!) +

n



 n



where vn; q = 9!n; q =9q is the group velocity. In the derivation of these conditions, we require the following very useful relation between the group velocity and displacement Keld: 

  9un∗
; q
; l 9un; q; l ∗ = !n; q vn; q %nn
; i cxijl dy d z un; q; i − u n
; q
; i (15) 9xj 9xj A ijl

where !n
; q
= !n; q . This relation follows from the equations of motion (6). We are now ready to quantize. In the wire leads, the displacement Keld operator has the solution
 ∞ ˝ dq [aˆ&n; q un;& q; i (r; t) + aˆ&n;†q un;&∗q; i (r; t)] ; (16) uˆ i (r; t) = 2 ! n; q 0 n;& where the phonon creation and annihilation operators satisfy the commutation relations [aˆ&n; q ; aˆ&n
;†q
] = %&&
%nn
%(q − q ) :


Substituting the Keld operator solution (16) into the energy current operator 

 9uˆ j 9uˆl 1 9uˆl 9uˆj Iˆ = − cxjkl dy d z + 2 9t 9xk 9xk 9t A and then taking the expectation value of Iˆ at any location x in leads 1 or 2, we obtain  ∞ 1 Iˆ = d! ˝! vn;−q1 vn
; q
tn21
n (!)tn21
n∗ (!)[f1 (!) − f2 (!)] ; 2 n; n
!n;0

(17) (18)

(19)

where

1 ; (20) e˝!=kB T& − 1 with T& the temperature of the reservoir at the end of lead &. In the derivation of Eq. (19), use is made of relation (15) and conditions (12)–(14). We also use the following creation/annihilation operator expectation values: f& (!) =

aˆ&n;†q aˆ&n
; q
 = f& (!n; q )%&&
%nn
%(q − q ) :


(21)

DeKning Tn21
n (E) = vn;−q1 vn
; q
tn21
n (!)tn21
n∗ (!) ; where E = ˝!, we can rewrite (19) as follows:  1  ∞ Iˆ = dE ETn21
n (E)[f1 (E) − f2 (E)] : 2˝ n; n
En;0

(22) (23)

This is our key expression for the mean energy current. From the form of this expression and condition (12), we see that the matrix Tn21
n (E) is naturally interpreted as the probability for a phonon with energy E in subband n of lead 1 to be transmitted into subband n of lead 2.

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M. Blencowe / Physics Reports 395 (2004) 159 – 222

When the temperature di1erence between the reservoirs is small, i.e., |T1 − T2 |T1 ; T2 , we can expand Eq. (23) to obtain the wire thermal conductance:  kB2 TV  ∞ Iˆ = (= dj g(j)Tn21
n (jkB TV ) ; |T1 − T2 | 6˝ n; n
En;0V

(24)

kB T

where TV is the average temperature and g(j) =

3j2 ej : 2 (ej − 1)2

(25)

Eq. (24) relates the thermal conductance to the single phonon transmission probability and thus we call this the Landauer expression ∞ for the phonon thermal conductance. The function g(j) satisKes 0 dj g(j)=1. Therefore, in the absence of scattering, a given subband n contributes to the reduced conductance (= TV the universal quantum kB2 =6˝ ≈ 9:465 × 10−13 W K −2 in the limit En; 0 =kB TV → 0. In Fig. 5, we show the temperature dependence of the reduced thermal conductance for perfect GaAs wires with uniform, rectangular cross sections of various dimensions comparable to those in the experiments of Tighe et al. [41] and Schwab et al. [35]. The only GaAs wire characteristics which are needed in order to determine the conductance are the zone-centre frequencies !n; 0 . These can be calculated using the elegant numerical method developed in Ref. [42]. In contrast to the quantized conductance of electronic point contacts [32,33], there are no steplike features, a consequence of the broad nature of the Bose–Einstein distribution; at a given temperature, the exponential tails of many subband distributions contribute to the conductance, washing out the steps. There is, however, a plateau for TV → 0 where only phonons in the lowest subband with En; 0 = 0 contribute. The plateau has the value four in universal quantum units, a consequence of there being four basic transverse mode types: dilatational, torsional and two types of Jexural mode [42]. The existence of this lowest plateau is a consequence of the stress-free boundary conditions at the wire surface. With hard-wall boundary conditions on the other hand, the reduced conductance would drop to zero as TV → 0 and there would be no plateau. Of course, whether or not this lowest, universal plateau can be resolved as TV → 0 depends on the extent to which phonon scattering in the wire can be controlled. Scattering due to surface roughness and other wire impurities will reduce the thermal conductance below its universal value [43–45]. However, with the practical impossibility of realizing a perfectly adiabatic reservoir–wire connection, the main obstruction to observing the plateau is in fact the backscattering of reservoir phonons incident on the wire. In classical wave optics and acoustics, the same reJection phenomenon occurs for waves traveling in narrowing waveguides and is called “di1ractional blocking”. Working with a scalar wave model, Rego and Kirczenow [39] explored the e1ect of various reservoir–wire geometries on the thermal conductance plateau. A catenoidal geometry of the form y(x)=w cosh2 (x=L), where w is the minimum width and L is the characteristic length of the catenoid, was found to give a distinct plateau over a wide temperature range, whereas a wedge-shaped reservoir–wire junction gave no resolvable plateau (Fig. 6).

M. Blencowe / Physics Reports 395 (2004) 159 – 222

169

Fig. 5. Reduced thermal conductance versus temperature for perfect GaAs wires with uniform rectangular cross section 200 nm×400 nm (solid line), 200 nm×300 nm (dashed line) and 200 nm×100 nm (dotted line). The reduced conductance is given in units kB2 =6˝ ≈ 9:465 × 10−13 W K −2 . (Reproduced from Ref. [36].)

Fig. 6. Left scale: Reduced thermal conductance of a quantum wire with ideal contacts. Right scale: single scalar mode thermal conductance for various contact shapes; inKnite catenoid for L = 4:6
m (solid line), Knite catenoid for L = 4:6
m (dot-dashed line), Knite catenoid with L = 0:86
m (long-dashed line), and wedge-shape with wedge angle ==6 (solid line with circles). (Reproduced from Ref. [39].)

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Fig. 7. Schematic layout of device with freely suspended Si3 N4 wires for one-dimensional phonon transport studies. (Reproduced from Ref. [46].)

2.2. Observation of thermal conductance quantum The pioneering work of Lee et al. [46] represented the Krst attempt to probe quantized phononic heat transport in suspended, electrically insulating wires of ultrasmall cross section. The motivation came from a theoretical investigation by Kelly [47] of thermal phonon analogues of electrical transport properties of long, thin wires. However, while the possibility of a phonon waveguide is mentioned in Ref. [46], there is no discussion of the universal thermal conductance quantum kB2 T=6˝. Particular attention is instead paid to thermal analogues of quantum corrections to electrical resistance due to scattering from wire impurities. The layout of the structure is shown in Fig. 7. The suspended wires were fashioned out of an amorphous silicon nitride (Si3 N4 ) membrane deposited on a Si substrate. Amorphous Si3 N4 was chosen for its high strength. A large array of wires was fabricated in order to improve the signal/noise ratio. The cross sectional dimensions of the resulting suspended wires were determined by the membrane thickness and the lithography resolution. Fig. 8 shows a micrograph of part of the wire array. The wires are approximately 0:16
m wide, 0:14
m thick, 300
m long, and have a 5% linewidth variation. With such a cross section, one would expect to observe the lowest plateau region in the reduced thermal conductance (=T at temperatures of a few tenths of a Kelvin and below, neglecting phonon scattering. However, at the time, probe techniques were simply not sensitive enough to resolve the extremely small amount of power associated with the thermal conductance quantum: less than a picowatt below one Kelvin. More than a decade went by before another attempt was made to probe low-dimensional phonon transport by Roukes and co-workers [41]. An improvement over the device of Lee et al. [46] was the elimination of all parasitic phonon pathways by making one of the heat reservoirs a suspended cavity, materially connected to the other, external reservoir through the suspended wires only (Fig. 9). The

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171

Fig. 8. SEM micrograph of part of an array of freely suspended Si3 N4 wires projecting over the Si support and the anisotropically etched hole. The wires are about 1
m apart. (Reproduced from Ref. [46].)

Fig. 9. SEM micrograph of a suspended, monocrystalline device for thermal conductance measurements. The centre image, which is an enlargement of the central region of the image on the left, shows a GaAs reservoir cavity with area ≈ 3
m2 which is suspended by four GaAs bridges with length ≈ 5:5
m and cross section ≈ 200 nm × 450 nm. Deposited on top of the reservoir are two meandering Si-doped GaAs conductors, one of which serves as a heater and the other as a thermometer. The image on the right presents an edge view of the device which is approximately 300 nm thick and suspended about 1
m above the substrate. (Reproduced from Ref. [41].)

device was patterned from a GaAs heterostructure comprising three molecular beam epitaxially-grown layers atop an undoped GaAs substrate. The topmost epilayer was heavily Si doped n+ GaAs, out of which two meandering electrical-wire transducers were patterned, one serving as a thermal phonon reservoir through Joule heating of the central cavity, and the other functioning as a thermometer by exploiting the well-characterized temperature-dependence of the weak-localization and electron– electron interaction corrections to the electrical resistance (Ref. [4, p. 34]). The second epilayer was undoped GaAs, out of which the phonon wires and isolated reservoir were patterned. The removal of the third, sacriKcial AlAs epilayer by chemical etching enabled the suspension of the phonon wire/cavity structure with integrated transducers. Fig. 10 shows the measured thermal conductance versus temperature. At the high temperature end, the thermal conductance approaches a T 3 dependence. The kinetic formula for the bulk, 3D thermal conductance of a single wire is (Ref. [48, p. 288]): V A C s; (26) ( 3D = 3L where C ∼ T 3 is the Debye heat capacity, sV the averaged phonon velocity, - the phonon mean-free path, and A and L are the wire cross sectional area and length, respectively. The data is therefore

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Fig. 10. (a) Phononic thermal conductance from the parallel measurement of four 200 nm × 300 nm GaAs bridges. The dashed line represents the predictions of the Debye model for a Kxed boundary-limited mean-free path of 0:51
m. (b) E1ective mean free path deduced from the experimental data. (Reproduced from Ref. [41].)

consistent with bulk heat Jow limited by temperature-independent surface-roughness scattering. At temperatures of a few Kelvin, the dominant thermal phonon wavelength is considerably smaller than the transverse dimensions of the suspended wires (300 nm thick by 200 nm wide), and thus we expect to be in the regime of bulk transport. Substituting into Eq. (26) the measured conductance, wire dimensions and speciKc heat and averaged phonon velocity values for GaAs, we deduce a mean-free path - ∼ 0:5
m, about one-tenth the wire-length. The deviation from T 3 with decreasing temperature does not signify reduced dimensionality for phonon transport (the temperature is still too high), but rather the increasing importance of the parasitic electronic pathways for heat transport relative to the phonons of the doped GaAs sensor wires. Because of this problem, it was not possible to probe low-dimensional phonon transport with this device. Roukes and co-workers eliminated the parasitic electronic thermal conduction problem in a subsequent device (Fig. 11) by employing superconducting Nb Klms on top of the suspended phonon wires to contact the heater and thermometer transducers on top of the central phonon cavity [35,49]. The device incorporated several other improvements as well, all essential for resolving the lowest, thermal conductance quantum plateau. The wires were fashioned from a 60 nm-thick silicon nitride membrane, the same material used in the original device of Lee et al. [46], and patterned according to the catenoidal geometry y(x) = w cosh2 (x=L) with L = 1:0
m and w ≈ 200 nm. As discussed above, for this geometry the plateau is more pronounced than for a wedge-shaped reservoir–wire junction [39]. A sti1er material, the dominant thermal phonon wavelength in silicon nitride is larger than in GaAs at a given temperature: assuming the average sound speed sV ∼ 6000 ms−1 for silicon nitride gives a dominant thermal wavelength .th = hs=k V B T ∼ 300 nm=T . Thus, the transition from bulk to low-dimensional thermal phonon transport occurs at a higher temperature than in the GaAs

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173

Fig. 11. (a) Suspended mesoscopic phonon device comprising a 4 × 4
m phonon cavity (centre) patterned from a 60 nm thick silicon nitride membrane. The “C” shaped objects on the cavity are thin Klm Cr/Au resistors, whereas in the dark areas the membrane has been completely removed. The resistors are connected to thin Klm Nb leads that run atop the four phonon wires and ultimately terminate at the wirebond pads. (b) Close-up of one of the freely suspended catenoidal phonon wires. The narrowest region necks down to ¡ 200 nm width. (Reproduced from Ref. [49].)

device. For the heater and thermometer transducers, two Cr/Au thin Klm resistors were employed. Because of the extremely weak coupling between the Cr/Au electron gas and cavity phonons at low temperatures, it was essential that the power dissipated by the external thermometer circuitry into the Cr/Au resistor be suXciently low so as not to heat the electron gas signiKcantly above the cavity phonons. Otherwise, the thermometer would not give a suXciently accurate reading of the cavity phonon temperature. For example, dissipating only 10−16 W into the electron gas with volume 0:1
m3 heats the electrons by as much as 50 mK above the cavity phonons [49]. Similarly, given the very small volume and hence heat capacity of the central phonon cavity, the DC current supplied to the heater and Johnson noise from the room temperature electronics must be sufKciently small so as to be able to heat controllably the central reservoir by temperature increments of

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Fig. 12. The measured reduced thermal conductance versus temperature. (Reproduced from Ref. [35].)

a few tens of milliKelvin above the dilution refrigerator temperature. Schwab et al. [49] were able to meet these conditions by employing DC SQUID-based noise thermometry, together with extensive Kltering of the noise from the room temperature electronics. The temperature of the Cr/Au resistor is obtained by measuring its Nyquist current noise using a SQUID. The power dissipated in the other, Cr/Au heater resistor is determined by employing another SQUID to measure the voltage across the resistor for given DC current. From Eq. (24), knowing both the power supplied to the reservoir and its temperature allows us to determine the parallel thermal conductance of the four wires. Fig. 12 shows the measured reduced thermal conductance versus temperature. At the high temperature end, the data shows the expected T 3 dependence for bulk transport, consistent with a mean free path ∼ 0:9
m. As the temperature decreases, there is a crossover to a plateau-like region at about 0:8 K. The plateau value is close to 16 in units of the universal quantum. This is precisely the value we expect in the one-dimensional regime with four parallel wires and four lowest, independent transverse modes per wire. The crossover to the one-dimensional regime should occur where the dominant thermal wavelength .th exceeds the spacing between the lowest lying modes S. ∼ 2w. For w = 200 nm, this estimate gives 0:8 K, in agreement with experiment. Within the experimental error, the measured thermal conductance never exceeds the universal conductance quantum in the one-dimensional regime. Again this is consistent with theory, which predicts that the conductance quantum is the maximum possible conductance, achievable for ballistic (i.e., no scattering) transport only. The observed dip in the conductance and the recovery towards the universal upper limit at the lowest temperatures can be attributed to surface roughness-dominated scattering. (Note that, at these low temperatures the amorphous silicon nitride appears largely homogeneous to the long wavelength thermal phonons.) Santamore and Cross [43–45] calculated the e1ects of surface roughness scattering on the thermal conductance and were able to provide a good Kt to the data for roughness amplitude and correlation length values of approximately % = 0:2w and a = 5:5w, respectively (Fig. 13). Electron micrographs of the actual wires show surface roughness on comparable scales. Decreasing the temperature further, the reduced thermal conductance should eventually drop due to the loss of adiabatic coupling between the reservoirs and wire [39]. However, it was not possible to probe this

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175

Fig. 13. Thermal conductance per mode in units of the universal conductance versus temperature. Solid line: Kt using roughness parameters a=w = 5:5 and %=w = 0:2, circles: data from Ref. [35]. The dotted line shows the ideal conductance with no scattering. (Reproduced from Ref. [45].)

e1ect in the experiment; the SQUID noise thermometer saturated at about 80 mK, preventing the accurate measurement of lower temperatures. Subsequently to the successful measurement of the thermal conductance quantum by Schwab et al. [35,49], there have been several related experiments by other groups. Yung et al. [50] measured the quantum of thermal conductance using a device comprising a GaAs suspended island with six, rather than four supporting legs, and with the thermometry performed using superconductor–insulator– normal metal tunnel junctions instead of SQUIDs. In another direction, some progress has been made towards measuring the thermal conductance through suspended carbon nanotube wires. With much smaller cross sections than the patterned wires, it should be possible to observe the lowest thermal conductance quantum plateau at higher temperatures, thus simplifying the thermometry. Evidence for 1D behaviour in the phonon density of states has been observed in carbon nanotube bundles [51] and a device for thermal conductance measurements of single nanotubes has been demonstrated [52]. 2.3. Universal nature of thermal conductance In Section 2.1 we saw that, in common with the quantum limits for other single channel, linear transport coeXcients, such as the electronic conductance quantum e2 =h [32], the thermal conductance quantum does not depend on the form of the !(k) dispersion relations, a consequence of the cancellation of the group velocity and density of states factors in the formula for the one-dimensional heat current. Wires made from di1erent insulating materials and with di1erent cross section geometries will therefore all have the same limiting single channel thermal conductance value for ballistic transport at low temperatures. For this reason, the conductance quantum is often termed “universal”. The thermal conductance is in fact universal in a much wider sense. For a single channel connecting two particle/heat reservoirs with (quasi)particles obeying fractional statistics according to Haldane’s deKnition (which generalizes Bose and Fermi statistics) [53], the thermal conductance quantum in the degenerate limit and in the absence of scattering is independent of the particle

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statistics as well [54,55]. For example, in the case of an ideal electron gas, the single channel thermal conductance coincides with the thermal conductance quantum for phonons, kB2 TV =6˝. While dimensional analysis would lead us to expect the same factor kB2 TV =˝ to occur independently of the statistics, there is no a priori reason to expect the same numerical factor =6 as well, given that the latter results from integrating with respect to the energy the expansion to Krst order in small temperature di1erences of the thermal reservoir distributions, which have qualitatively di1erent forms for particles obeying di1erent statistics. This remarkable property is unique to the thermal conductance: all other single channel transport coeXcients depend on the particle statistics. We now outline the derivation of the single channel thermal conductance for particles obeying Haldane’s statistics [54]. In the absence of scattering, the single-channel heat current is [cf. the energy current expression for phonons, Eq. (23)]  1 ∞ IQ = dE (E − 0)(fg; 1 − fg; 2 ) ; (27) h 0 where we choose the energy origin such that E(k = 0) = 0 and the reservoir chemical potentials satisfy 01 = 02 = 0. The function fg; i is the thermal equilibrium distribution describing particles in reservoir i at temperature Ti , i = 1; 2, and which obey Haldane’s statistics with rational statistics parameter g ¿ 0 [56]:  −1



E−0 +g fg = w ; (28) kB T where the function w(x) satisKes w(x)g [1 + w(x)]1−g = ex :

(29)

From these equations, we can see immediately that g = 0 describes bosons and g = 1 fermions. For small temperature di1erences 0 ¡ T1 − T2 TV = (T1 + T2 )=2, we can expand about the average temperature TV and, performing the change of variables E → x = 1(E − 0) → w [54], the thermal conductance can be written as  k 2 TV ∞ [g ln w + (1 − g) ln(1 + w)]2 (= B dw : (30) h wg (−0=kB TV ) (w + g)2 In the degenerate limit, 1 we have for the g = 0 case (i.e., bosons), −0=kB T → 0+ and wg=0 (0) = 0, while for g ¿ 0, −0=kB T → −∞ and wg¿0 (−∞) = 0, the same limit as for the bosonic case. Numerically evaluating the integral in (30) in the degenerate limit for di1erent values of the statistics parameter g, we Knd ( = kB2 TV =6˝, independent of g. This is a remarkable result: that the integral in (30) does not in fact depend on g in the degenerate limit is not obvious. In the following, we shall derive this statistics-independence in a di1erent way via the entropy current, the motivation being to achieve a deeper understanding of this independence. The thermal conductance determines the rate of heat Jow in the linear response regime. Given that heat is related to entropy through the second law of thermodynamics, it is natural to expect that the single channel entropy current will be similarly independent of the particle statistics. In fact, the net entropy production is quadratic in the reservoir temperature and chemical potential di1erences and 1

In order to have a non-zero conductance current, the degenerate limit should be taken by increasing the reservoir particle densities (by varying the chemical potential 0), while keeping the reservoir temperatures Kxed.

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177

thus, within the linear response approximation, we can equate the entropy and heat currents: IS =IQ = TV , where TV is the average temperature of the reservoirs [57]. As we will now show, this universality occurs at an even more basic level for the individual left- and right-propagating components of the entropy current. In the absence of scattering we have, e.g., for the right-propagating component of the entropy current [57]:  ∞ dk R IS = v(k)sg [11 (E(k) − 01 )] ; (31) 2 0 where v(k) = ˝−1 9E(k)=9k is the group velocity, 11 = 1=kB T1 and sg = −fg ln fg + (1 − gfg ) ln(1 − gfg ) − [1 + (1 − g)fg ] ln[1 + (1 − g)fg ]

(32)

is the entropy density of particles obeying fractional statistics with parameter g and distributed according to the non-equilibrium function fg [55]. Changing integration variables from k to w, the right-propagating entropy current (31) can be rewritten as follows:  k 2 T1 ∞ ISR = B dw[ln(1 + w)=w − ln w=(1 + w)] ; (33) h wg (−01 =kB T1 ) where we choose the energy origin such that E(0) = 0. Notice that, as with other types of current in 1D, there is no dependence on the form of the band dispersion E(k). Notice also that the statistics dependence appears only in the lower integration limit. In the degenerate limit the entropy current is given by the following rather beautiful expression:  kB2 T1 ∞ R IS = dw[ln(1 + w)=w − ln w=(1 + w)] ; (34) h 0 where the statistics dependence has dropped out as well. The deKnite integral can be evaluated by reexpressing it in terms of dilogarithms [58]:  y lim dw[ln(1 + w)=w − ln w=(1 + w)] y→∞ x→0

x

= lim [Li2 (y) − Li2 (−y) + i ln y] − lim [Li2 (1 + x) − Li(−x) − i ln(1 + x)] ; y→∞

where

x→0



y

(35)

ln(1 − z) : (36) z 0 Substituting in the following large-y asymptotic approximations 2 1 2 Li2 (y) ∼ − ln (y) − i ln y ; (37) 3 2 2 1 2 − ln (y) Li2 (−y) ∼ (38) 6 2 and using the identity Li2 (1) = 2 =6, we Knd: k 2 T1 (39) ISR = B : 6˝ The physical implication of result (39) is the following. Consider, for example, an optical Kbre with one end attached to a black-body oven at a given temperature T1 and the other end to an Li2 (y) = −

dz

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oven at a temperature T2 T1 . Alternatively, consider the phonon thermal conductance set-up of Schwab et al. [35] with the temperature of the outer reservoir much lower than that of the central cavity. Or, consider a split-gate quantum wire opening out at each end to 2DEG reservoirs with electrochemical potentials satisfying 01 02 ; kB T1 . Then, provided scattering can be neglected, in each of these examples the entropy current per transverse channel is the same, given by Eq. (39). The statistics-independent thermal conductance quantum follows from Eq. (39): k 2 TV ( = IQ =(T1 − T2 ) = (ISR − ISL )TV =(T1 − T2 ) = B : 6˝ Thus, the statistics-independence of the unidirectional entropy current can be viewed as more fundamental than the statistics-independence of the thermal conductance. In particular, any attempt to understand the statistics-independence at a deeper level should be directed towards the entropy current. One relevant question is whether this independence extends to other classes which interpolate between Bose and Fermi statistics. Either Eq. (39) applies only to particles obeying Haldane’s fractional exclusion statistics [53], or there are other interpolating classes which satisfy (39). Both of these possible alternative scenarios have interesting consequences. The former possibility would suggest a new way to motivate/derive Haldane’s statistics by imposing the condition that the entropy current in the degenerate limit is given by the invariant (39). On the other hand, the latter possibility might lead to new classes of interpolating statistics, all identiKed by the common constraint (39). In the following, we will go a little way towards answering the above question by calculating the unidirectional entropy current for another type of interpolating statistics, the so-called Gentile’s statistics [59,60]. As we shall see, the resulting degenerate entropy current does not satisfy (39), except for the Bose and Fermi cases (as of course it must). Thus, we can at least conclude that (39) does not hold for every interpolating statistics class. Let us suppose then that we have a 1D quantum channel joining two reservoirs which support particles obeying Gentile’s statistics, characterized by statistical parameter a = 1; 2; 3; : : : ; the maximum occupation number of particles in a single-particle state, with a = 1 describing fermions and a = ∞ bosons. Note that this is not the same as Haldane’s statistics, which involves additional constraints on the single-particle state occupation numbers beyond just having maximum occupation number equal to 1=g [61]. The distribution function for right-propagating particles with wavevector k ¿ 0 is [59,60] a+1 1 fa = 11 (E(k)−01 ) − 11 (a+1)(E(k)−01 ) : (40) e −1 e −1 Note that for fa=∞ , the second term drops out and we obtain the Bose distribution, while a little algebra shows that fa=1 gives the Fermi distribution. For a Knite, we have fa = a for E(k) ¡ 01 and fa = 0 for E(k) ¿ 01 in the degenerate limit. The entropy current can be expressed in terms of the distribution function (40) via the Krst law of thermodynamics:

   ∞ kB ∞ R   9fa dE(E − 01 ) d1 1 : (41) IS = − h 0 91 01 11 Carrying out the 1 integral, we have  kB ∞ ISR = dE[sbose (11 (E(k) − 01 )) − sbose (11 (a + 1)(E(k) − 01 ))] ; h 0

(42)

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179

where sbose is the entropy density for bosons (i.e., Eq. (32) for g = 1=a = 0). Notice that the entropy current for particles obeying Gentile’s statistics with parameter a is just the di1erence between the entropy currents for bosons with reservoir temperatures T1 and T1 =(a + 1). Making the change of variables E → x = 1(E − 0) → w = ex − 1, Eq. (42) becomes  ∞ kB2 T1 R dw[ln(1 + w)=w − ln w=(1 + w)] IS = h w(−01 =kB T1 )   ∞ 1 dw[ln(1 + w)=w − ln w=(1 + w)] : (43) − 1 + a w(−01 (a+1)=kB T1 ) For bosons, the second term vanishes since w(−01 (a + 1)=kB T1 ) → ∞ for a → ∞ with 01 ¡ 0, while the Krst term coincides with Eq. (34), hence giving ISR = kB2 T1 =6˝ in the degenerate limit. For Knite statistics parameter a, the lower integration limits in Eq. (43) tend to −1 in the degenerate limit and using Eq. (35) for x → −1, we Knd ISR =

akB2 T1 : 3(1 + a)˝

(44)

Thus, we see from Eq. (44) that the degenerate, unidirectional entropy current is statistics-dependent, with only the Fermi (a = 1) and Bose cases coinciding. At a given temperature the degenerate entropy current for particles obeying Gentile’s statistics exceeds that for particles obeying Haldane’s statistics (except, of course, for the Bose and Fermi cases where they coincide). This may be related to the additional constraints on the occupation numbers for Haldane’s statistics [61]. It would be interesting to determine whether Haldane’s exclusion statistics is the only interpolating class for which the degenerate, unidirectional entropy current (and hence also the thermal conductance) is given by the statistics-independent expression IS = kB2 T=6˝. 2.4. Information capacity of single quantum channels In a completely separate area of research which predated the thermal conductance quantum work, several investigations were carried out by a number of di1erent groups to determine the theoretical limits quantum mechanics places on the information carrying capacity of a single, wideband communication channel with a sender at one end of the channel and a receiver at the other end. In particular, drawing on earlier work [62–65] Caves and Drummond [66] showed that, for given input signal power P, the optimum capacity of a single, wideband bosonic quantum channel is   2P C= bits=s : (45) ln 2 3h The basic form of (45) can be motivated through the following handwaving argument [66]. Consider, for example, a single wideband channel of an optical Kbre formed by all, say, right-propagating, electromagnetic modes having the same given transverse mode label. As another example, consider a suspended dielectric wire, with all lattice vibrational modes having the same transverse mode number furnishing a single wideband acoustic channel [cf., Eq. (9)]. Clearly, there is no di1erence in the physics between these communication channel examples and the 1D channels considered above in the thermal conductance discussions. These right-propagating modes form a band describing the frequency versus longitudinal wavevector dependence. Suppose the maximum frequency of the band

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is f. Divide the band into N non-overlapping frequency intervals (narrowband channels), each with bandwidth b ∼ f=N . Within each narrowband channel, suppose the information is transmitted in binary, with a “one” encoded in a single energy quantum (e.g., a photon or phonon) and a “zero” encoded in the absence of a quantum (vacuum state). Note that the use of more quanta to denote a “one” or “zero” would clearly require more energy to send the same information and hence would reduce the narrowband capacity. Furthermore, the maximum narrowband bit rate cannot obviously exceed the bandwidth b. Thus, neglecting numerical factors, we have for the wideband channel capacity C ∼ bN ∼ f, while for thepower, we have P ∼ hfbN . Eliminating f by combining these two expressions, we Knd that C ∼ P=h. In light of the universal nature of the single channel entropy current and given the close connection between information and entropy [67], a natural question to consider is whether the single-channel, optimum information capacity is universal [68] and, if not, whether it can be modiKed so as to be made so. Eq. (45) is indeed independent of the band dispersion [66], in common with other 1D transport quantities such as the various currents and conductances. But of greater interest is whether the optimum capacity is universal in the wider, statistics-independent sense. For example, does a 1D wire transmitting signals via electrons have the same optimum capacity as (45)? The short answer is “No”. The optimum capacity √ of a single, wideband fermionic channel is smaller than the bosonic capacity (45) by a factor of 2 [65,68]. The reason for the non-universality of the optimum capacity is twofold: both the unidirectional energy (or power) and entropy currents are dependent on the particle statistics for zero chemical potential. The right-hand side of (45) results from maximizing the right-propagating entropy current (which bounds the information Jow rate), subject only to the constraint of Kxed right-propagating energy current [66]. In particular, there is no independent constraint on the particle number current, so that the chemical potential 01 is set to zero in the entropy current expression, Eq. (33), giving a statistics-dependent lower integration limit wg (0). For example, from (29) we see that w0 (0) = 0 (bosons) and w1 (0) = 1 (fermions), so that the integral in (33) will be di1erent in each case. However, even if we were to modify the optimum capacity by supplementing the Kxed energy current constraint with Kxed particle number current constraint (so that, in sending a message, only a certain, Kxed number of particles are allowed in addition to a certain allowed amount of energy), there is still the problem of the statistics-dependence of the unidirectional energy current; only the entropy current is statistics-independent in the degenerate limit for unidirectional Jow. One possible resolution is to consider a more general situation in which there is a sender-receiver station pair at each end of the communications channel and the channel supports simultaneous two-way information Jow [68]. This resembles more closely the usual set-up for transport in solid state physics, as for example in the above, 1D thermal conductance; the energy current now comprises both right- and left-propagating components. Most importantly, the energy current is statistics-independent in the degenerate limit (cf., Eqs. (27) and (30)). In Ref. [68], some progress is made towards establishing the statistics-independence of such a more general optimum capacity subject to certain constraints on the signal power and particle number current. Allowing for two-way Jow avoids another problem with the conventional deKnition of the optimum capacity. In particular, for unidirectional energy Jow, there is no preferred choice of energy origin. Usually, the longitudinal band minimum is taken as the origin, although there is no reason why the transverse component of the energy should not be included as well, for example. Clearly, the right-hand side of (45) depends on the choice of energy origin which detracts from any fundamental

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signiKcance this bound might have [69]. For two-way Jow, the net energy current is the di1erence between right- and left-propagating components and so is independent of the choice of energy origin. Another possible way in which the information capacity might be modiKed within the usual, one-way information transmission set-up, is to replace the constraint on the supplied power with a constraint on the rate at which heat is generated as the information received at the channel output is erased; after the transmission of a message is completed, the sender/receiver communication system must be reset before the next message is transmitted. According to Landauer’s principle of information erasure [70,71], at least kB T ln 2 amount of heat per bit is generated when the receiver memory is erased by putting it into thermal contact with a heat bath at temperature T . This represents an unavoidable energy cost for each message transmission, whereas the power involved in the message transmission, P, need not be dissipated away and can in principle be recycled [72,73]. It would be interesting to determine whether an optimum capacity subject to the alternative constraint of given, maximum heat generation rate (due to information erasure at the receiver end) can be established and, furthermore, whether such an optimum capacity would be universal in the wider, statistics-independent sense. Note that, for the same reasons as above, such a constraint would still have to be supplemented by a constraint on the particle number current. In our view, such a modiKed deKnition for the information capacity is more natural, given the unavoidable energy cost entailed by the constraint and the fact that the constraint avoids the energy-origin ambiguity problem as well.

3. Ultrasensitive displacement detection The most direct way to probe the quantum dynamics of mechanical resonators would be to use ultrasensitive displacement transducers with fast response times. For example, in order to resolve the quantum zero-point motion of a micron-scale Si cantilever with fundamental resonant freT is required (see Eqs. (3) and (5)). quency 100 MHz, a displacement sensitivity better than 10−3 A Section 3.1 describes various displacement detection techniques relevant to (sub)micron scale mechanical resonators, while Sections 3.2 and 3.3 focus on the theory and experiment, respectively, for the SET-based displacement detector. The SET detector is singled out for extensive discussion, as it is to date the most promising scheme for achieving quantum zero-point motion sensitivities. However, only a classical analysis of the SET detector is given, with the emphasis placed on Krst establishing the necessary classical displacement sensitivity suXcient for, say, zero-point motion, given current microfabrication possibilities. Fundamental issues concerning quantum measurement, such as how to perform a quantum non-demolition measurement [74–76] using a SET-based displacement detection scheme, have yet to be addressed and so are not discussed here. The material in this section is largely a revised and updated version of Ref. [78]. 3.1. Overview of various displacement detection techniques We Krst make some generally relevant comments concerning displacement detection. Suppose that the displacement detector couples strongly to only one of the lowest, fundamental Jexural modes of the cantilever or doubly-clamped beam with a large quality factor, such that the latter can be modelled e1ectively as a single, damped harmonic oscillator. Neglecting the back reaction of the

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detector on the oscillator, then to a good approximation the oscillator’s classical equation of motion when in thermal contact with a heat bath is d 2 x m!m d x 2 x = FJuct (t) ; (46) m 2 + + m!m dt Q dt where m is the e1ective motional mass in the Jexural mode (we have dropped the subscript “e1” for notational convenience), x(t) is the fundamental mode contribution to the end-point (or mid-point) displacement of the cantilever (or beam), and the random force acting on the oscillator due to the heat bath satisKes the following (time-averaged) correlation relation:  T=2 1 2m!m kB T dt FJuct (t)FJuct (t + ) = FJuct (t)FJuct (t + ) = lim %() (47) T→∞ T −T=2 Q with FJuct (t) = 0. The simplest measured quantity which contains information about the quantum behaviour of the oscillator is the time-averaged square of the displacement:  T=2 1 2 dt x(t)2 : (48) x(t)  = lim T→∞ T −T=2 Solving Eq. (46) for the classical solution to x(t)2 :  ∞ 1 kB T 2 d!Sxm (!) = ; (49) x(t)  = 2 2 0 m!m where the mechanical oscillator displacement noise spectral density for Q1 is approximately kB T m!m Q Sxm (!) = (50)  :

!m 2 2 (! − !m ) + 2Q Solution (49) is just the well-known one which follows from the equipartition of energy. As with any ampliKer, a displacement detector will introduce noise into the measured output signal. In order to establish the displacement sensitivity of a given detector, it is convenient to express the detector noise as an equivalent input displacement noise density Sxa (!), allowing a direct comparison with the input signal. (The superscript “a” in Sxa (!) denotes “ampliKer”, to be distinguished from the mechanical oscillator signal Sxm (!)). If the measurement bandwidth is [!1 ; !2 ], then the equivalent input noise is  !2 1 d!Sxa (!) : (51) 2 !1 Suppose that the detector noise density (in the absence of a signal) is equal to the maximum signal noise density: 4QkB T : (52) Sxa (!m ) = Sxm (!m ) = 3 m!m DeKne the equivalent noise bandwidth S such that the integrated detector noise in this bandwidth equals the total signal noise:  ∞ 1 Sxa (!m )S = d! Sxm (!) ; (53) 2 0

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183

where it is assumed that Sxa (!) is approximately constant over this bandwidth. Thus, from Eqs. (49) and (52), we have for the equivalent noise bandwidth in the case of the signal being thermal motion of an oscillator: m : (54) S = 2Q In the following discussion, the quoted displacement detector   sensitivities of the various mentioned schemes are just their values for Sxa (!m ). Multiplying these by m =(2Q) then deKnes the minimum detectable rms thermal displacement for each scheme. Historically, research into ultra-sensitive displacement detection has largely been carried out within the gravity wave community [74–76]. Capacitance [77], inductance [79], SQUID-based [80] and optical interferometer [81] transducer schemes are frequently √ employed, with a capacitance transduction T Hz for a mechanical resonant frequency of scheme achieving a record sensitivity of 6 × 10−9 A= √ T Hz was obtained using a 40 kHz as long ago as 1981 [77]. Recently, a sensitivity of 2 × 10−9 A= high-Knesse optical cavity with a mirror coated on a 2 MHz mechanical resonator [82]. However, such sensitivities cannot necessarily be maintained when the transducers are scaled down to micron dimensions and below. The capacitance transducer sensitivity depends on the magnitude of the capacitance and hence plate area. With micron-sized capacitors having values below a femtofarad, parasitic capacitances arising from, e.g., the leads connecting the ampliKer to the capacitor or from the SQUID coil itself can also be a problem, e1ectively shorting out the transducer signal. Fiber-optic interferometer transducers have been demonstrated with subangstrom resolution [83], although they are limited to resolving displacements of mechanical structures on the micron-scale and above, i.e., larger than the optical Kber diameter. A technique that does not appear to su1er from similar obvious scale limitations is the so-called magnetomotive method [85,13] which involves placing the mechanical structure with metal surface electrode in a transverse magnetic Keld. The induced electromotive force is measured as the structure moves through the magnetic Keld. For √ a deterministic signal and using signal averaging techniques, T Hz for a 1 MHz, micron-sized oscillator was reported [84]. a sensitivity of around 3 × 10−5 A= The preampliKer was the dominant noise source, limiting the pre-averaged displacement detection √ T Hz. However, an analysis of the fundamental limits on the displacement sensitivity to about 10−2 A= sensitivity of the magnetomotive method has yet to be reported. Another technique that does not su1er from obvious scaling issues is the strain-sensing Keld-e1ect transistor (FET) originally developed to measure the deJection of a small cantilever [16]. The FET was fabricated from a GaAs/AlGaAs heterostructure and used the piezoelectric properties of GaAs to couple strain to the electron density, thereby obtaining an electrical response. Both a micron-scale cantilever and one an order of magnitude larger in size were considered. The vertical displacement √ T Hz, while the resolution of the A= resolution of the larger cantilever was measured to be about 10 √ T Hz, limited by the charge noise of the FET. smaller cantilever was projected to be 2 × 10−3 A= The expected increase in resolution with decreasing cantilever size is a simple consequence of the larger curvature and hence induced strain at the base of the cantilever (where the FET is located) as the cantilever length is decreased√for a given tip displacement. A similar displacement detection T Hz has in fact been demonstrated for a micron-scale, doublyscheme with sensitivity 3 × 10−3 A= clamped GaAs/AlGaAs heterostructure beam which relies on both the piezoelectric and piezoresistive e1ects [17].

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Yet another, rather straightforward displacement detection technique involves focusing a commercial scanning electron microscope (SEM) beam at a point near the edge of the mechanical resonator and using a photomultiplier to detect the scattered signal caused by the resonator moving in and out of the focal√ point of the beam [86]. The sensitivity of this technique was esT Hz, although there was signiKcant heating of the mechanical restimated to be 5 × 10−3 A= onator by the electron beam (about a one Kelvin increase in temperature), so that the method would clearly not be suitable for the investigation of the quantum dynamics of mechanical resonators. However, while the sensitivities of these various displacement detection methods are certainly impressive (and, interestingly, nearly all within an order of magnitude of each other when given in √ T Hz), they are not adequate for, say, resolving zero-point motion; considerable improvements are A= required in order to reach the quantum limit. On the other hand, there have recently been several promising proposed displacement detection schemes which commonly take advantage of the extreme sensitivity in the electronic response of certain single and few-electron devices to small changes in the local electric Keld environment. Devices which have been considered include vacuum tunnelling transducers [87–91], the single-electron transistor (SET) [92–98,1], quantum dots [99] and quantum point contacts [100]. In the following, we will focus on the SET-based displacement detection scheme, giving an analysis of the fundamental limits on the displacement sensitivity of the scheme and then outlining some open problems as well as describing recent progress √ towards realising such T Hz may be possible for a device. As we shall show, displacement sensitivities approaching 10−6 A= the example of a micron-scale cantilever with fundamental resonant frequency 100 MHz and quality T This would factor Q = 104 , corresponding to an absolute displacement sensitivity of about 10−4 A. be adequate for detecting zero-point motion. 3.2. Analysis of the fundamental noise limits on SET-based displacement detection The circuit diagram of the SET displacement detector is shown in Fig. 14. The basic principle of the device involves locating one of the gate capacitor plates of the SET on the mechanical resonator (e.g., cantilever) so that, for Kxed gate voltage bias, a mechanical displacement is converted into a polarization charge Juctuation. It is the exquisite sensitivity of the SET as detector of charge Juctuations (i.e., as an electrometer) which makes the SET an obvious choice for displacement detection. Furthermore, metal junction SETs are comparable in size to micron-scale mechanical resonators and so can be located in close proximity to each other, thus avoiding the problem of a large stray capacitance due to the wire connection between the resonator gate electrode and SET, e1ectively shorting out the gate capacitance. To gain a rough idea  of the possible displacement sensitivity, consider, for example, a realisable SET charge sensitivity Sq (we have dropped the superscript “a” for notational convenience) of √ about 10−5 e= Hz (see, e.g., Ref. [101]), static gate capacitance Cg0 =0:1 fF, plate gap d=0:1
m and gate voltage = 1 V. Determining  the equivalent input displacement noise through the approximate √ Vg  relation Sx = Sq =|9q=9x| = Sq d=(Cg0 Vg ) (where for small displacements of the cantilever gate √ √ T Hz. For a cantilever with realisable electrode, Cg ≈ Cg0 (1 − x=d)), we obtain Sx ∼ 10−5 A= fundamental Jexural frequency m = 100 MHz, quality factor Q = 104 , the minimum detectable rms T in the equivalent noise bandwidth m =(2Q). displacement is about 10−3 A

M. Blencowe / Physics Reports 395 (2004) 159 – 222 R0

185

L t

cantilever

v(t) R2,C2

_

Cs Cg

R1,C1 b _

Vg

Fig. 14. Circuit diagram of the rf-SET displacement detector. (Reproduced from Ref. [93].)

However, there is an unavoidable, large stray capacitance ∼ 0:1–1 nF on the “other side” of the SET, due to the leads contacting the SET from the room temperature ampliKer. With the usual condition on the SET that the e1ective tunnel junction resistances R1 , R2 exceed the resistance quantum h=e2 = 26 kZ, this gives an RC time constant at least of order 10
s. Thus, only input signal frequencies  ¡ 1=(2RC), i.e., less than a few kilohertz can be resolved. This is to be contrasted with the intrinsic bandwidth of the SET due to the tunnel junctions only which can be of order 10 GHz for junction capacitances C1 , C2 less than 1 fF. One way [95,96] to detect much higher frequency signals is to operate the SET as a radio-frequency mixer by exploiting the non-linearity of the drain–source current dependence on gate voltage. Adding a local oscillator signal to the input signal such that the frequency di1erence is no more than about a kilohertz will, as a result of the non-linearity, produce a resolvable output signal at the much lower di1erence frequency. By sweeping the local oscillator frequency, the resulting narrow bandwidth, frequency-di1erence outputs can be combined to build up a few hundred MHz wide bandwidth measurement of the input signal [95,96]. This of course requires that the input signal time-dependence be such that the measured, narrowband frequency-di1erence output at a given local oscillator frequency does not depend on when the measurement interval commences. This would for example be the case for mechanical thermal brownian motion, provided that each measurement interval is much longer than the mechanical damping time Q=!m . Another way to overcome the narrow bandwidth problem is to drive the SET with a monochromatic carrier microwave, fed down a coaxial cable (Fig. 14) [102]. The reJected power provides a measure of the SET’s di1erential resistance Rd . When the gate capacitor Cg is biased, mechanical motion of the cantilever is converted into di1erential resistance changes, hence modulating the reJected power. Since the cable impedance R0 = 50 ZRd , the RC time constant is now much smaller, with R0 e1ectively replacing Rd , hence considerably increasing the bandwidth. However, the large mismatch between the cable and SET impedances also gives rise to the problem of a small di1erence between the incoming and reJected power: |Pin − Pref |=Pin ∼ R0 =Rd 1. By adding an inductor L, forming a tank circuit with the stray capacitance Cs due to the metal contacts between the inductor and SET (Fig. 14), and matching the carrier wave frequency ! with the tank circuit resonant frequency !T = (LCs )−1=2 , then this power di1erence  is ampliKed by the tank circuit quality factor 1 squared: |Pin − Pref |=Pin ∼ QT2 R0 =Rd , where QT = R− L=Cs . The approximate  optimum choice for 0 the largest di1erence between incoming and reJected waves is in fact QT ∼ Rd =R0 , of order a few tens [103,104]. However, it is important that QT not be chosen too large, as this reduces the

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signal detection bandwidth. For example, with !T = 2 × 1 GHz and QT = 10 the mechanical signal frequency m . T =QT = 100 MHz. Let us now proceed with an analysis of the sensitivity of the above-described radio-frequency single-electron transistor (rf-SET) based displacement detection scheme (Fig. 14). The Krst part to derive the equivalent input displacement noise is a straightforward adaptation [93] of the rf-SET charge sensitivity analysis given in Ref. [105]. (A more in-depth charge sensitivity analysis of the rf-SET is given in Refs. [103,104]). Consider an incoming wave of the form Vin cos !t at the end of the cable. The reJected wave is Vref (t) = v(t) − Vin cos !t, where the voltage v(t) at the end of the cable satisKes the di1erential equation vLC [ s + vR ˙ 0 Cs + v = 2(1 − !2 LCs )Vin cos !t − R0 Ids (t) ;

(55)

with Ids (t) the SET drain–source current. Eq. (55) is straighforwardly derived using Kirchho1’s rules. Let us Krst suppose that there is no signal source modulating the gate voltage, nor any noise in the SET, so that Ids (t) is only driven by Vin cos !t. We wish to establish the dependence of Vref (t) on Vin cos !t, through the dependence of the former on Ids (t). Because of the non-linear I –V characteristics of the SET, v(t) in Eq. (55) will not only contain an ! frequency component,  but also higher harmonics. Setting ! = !T and substituting the Fourier decomposition v(t) = ∞ n=1 (Xn cos n!T t + Yn sin n!T t) into Eq. (55), we obtain for the Fourier coeXcients  X1 = 2 L=Cs Ids (t) sin !T tT ; (56) Y1 = −2



L=Cs Ids (t) cos !T tT ; (57) t+T where · · ·T = t dt  (· · ·)=T denotes the time average over the tank circuit oscillation period T , and we restrict the analysis to the Krst harmonic of the reJected wave (the coeXcients for the higher harmonics are small in comparison [103,104]). The current Ids (t) depends on the voltage Vds (t) across the SET, which in turn depends on v(t): 1 Vds (t) = LR− ˙ + v(t) : 0 [2Vin ! sin !T t + v(t)]

(58)

The Fourier coeXcients X1 , Y1 are found by solving iteratively Eqs. (56) and (57). However, in  the regime Rd Cs =LQT 1, we can approximate Eq. (58) as Vds (t) = 2QT Vin sin !T t. Thus the coeXcients X1 , Y1 are approximately solved for once the Ids (t) dependence on Vds (t) is known. We will also assume that the carrier frequency satisKes !|Ids |=e, the rate at which electrons tunnel through the SET. Under these conditions the current Ids (t) maintains a Kxed phase relationship with respect to Vds (t) and, from the approximate form of Vds (t) and Eq. (57), we see that the time-averaging gives Y1 = 0. Reinstating the SET noise, the voltage v(t) is now no longer periodic in T , while the coeXcient X1 (t) is still approximately deKned as above in (56), but now Juctuates in time. At the mechanical signal frequency m , the equivalent input displacement noise Sx (m ) in terms of the spectral density SX1 (m ) of Juctuations in X1 (t) is Sx (m ) = SX1 (m )=(d X1 =d x)2 ;

(59)

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187

where x denotes the displacement from equilibrium separation of the gate capacitor plates. From Eq. (56), for m T it follows approximately that 4L SX1 (m ) = SI (T ; t) sin2 !T tT ; (60) Cs where the SET drain–source current noise SI is time-dependent because of the oscillatory driving voltage. The fundamental limit on SI is given by the intrinsic shot noise due to the SET tunnelling current. Substituting Eq. (60) into (59) and using Eq. (56) and the shot noise formula SI = 2eI , which is approximately valid close to the SET tunnelling current threshold [105], Eq. (59) becomes Sx (m ) = 2e|Ids (t)| sin2 !T tT =dIds (t)=d x sin !T t2T :

(61)

In order to solve Eq. (61), we need to determine the Ids (t) dependence on Vds (t). In other words, we require a model which describes the dynamics of electrons tunnelling onto and o1 the SET island subject to the driving potential Vds (t). The electron tunnelling dynamics will depend on the position x of the cantilever through the variable gate capacitor plate gap in the presence of a Kxed, non-zero gate voltage bias Vg . The simplest description of the electron tunnelling dynamics is the so-called “orthodox” model [106], in which (a) electron energy quantization in the SET island is neglected, (b) the electron traversal time through the SET from drain to source lead is assumed negligibly small as compared with other time scales (such as the interval between subsequent electron traversals), and most importantly (c) quantum coherent traversal through the SET involving multiple, simultaneous tunnellings through the drain and source junctions are neglected. In this semiclassical model, the only quantum aspect of the electron dynamics is the process of tunnelling sequentially through the drain and then source junctions. We shall employ the orthodox model and the method of analytic solution given in Ref. [107], leaving until later a discussion of the limitations of the model. Referring to Fig. 14, and as usual deKning C9 = C1 + C2 + Cg , when the voltage amplitude across the SET, A = 2QT Vin , is small compared to the voltage e=C9 , and also the thermal energy kB T eA, then the current in the tunnelling region between stable regions of n and n + 1 excess electrons on the SET island can be well-approximated as Ids = e[b1 (n) − t1 (n)] (n) + e[b1 (n + 1) − t1 (n + 1)] (n + 1) = e[b2 (n) − t2 (n)] (n) + e[b2 (n + 1) − t2 (n + 1)] (n + 1) :

(62)

Here the probabilities (n) and (n + 1) that there are respectively n or n + 1 electrons on the island are given approximately as (n) = [t1 (n + 1) + b2 (n + 1)]=[b1 (n) + t2 (n) + t1 (n + 1) + b2 (n + 1)]

(63)

(n + 1) = [b1 (n) + t2 (n)]=[b1 (n) + t2 (n) + t1 (n + 1) + b2 (n + 1)] :

(64)

and In this approximation the tunnel current peaks are well-separated in gate voltage Vg . With respect to the SET orientation in Fig. 14, the tunnelling rates bi (ti ) from the bottom (top) across the ith junction of the SET take the form     − bi (n) = SEi− (n)=e2 Ri = 1 − e−SEi (n)=kB T (65)

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Fig. 15. Shot noise-limited displacement sensitivity as a function of gate voltage (upper, “doublet” curves) corresponding to the Krst three Ids current amplitude peaks. The current amplitude versus gate voltage is also shown (lower curve-arbitrary scale) for reference. The noise analysis assumes a symmetric rf-SET at T =30 mK (=0:01 e2 =C9 ) with junction capacitances C1 = C2 = 0:25 fF, junction resistances R1 = R2 = 50 kZ, and static gate capacitance Cg0 = 0:1 fF (corresponding to 1
m2 plate area and 0:1
m plate gap). The drain–source rf bias voltage amplitude is A = 10−4 V (=0:4 e=C9 ). (Reproduced from Ref. [93].)

and +

ti (n) = [SEi+ (n)=e2 Ri ]=[1 − e−SEi

(n)=kB T

] ;

(66)

where SE1± (n) = [ − e=2 ± (en − C2 Vds − Cg Vg )]e=C9

(67)

SE2± (n) = [ − e=2 ∓ (en + (C1 + Cg )Vds − Cg Vg )]e=C9 :

(68)

and The tunnelling rate expressions (65) and (66) result from a “golden-rule” calculation [107], where SEi+(−) (n) is the energy gained by an electron as a result of tunnelling across the ith junction from the top (bottom) with n electrons initially on the island, and the exponential factors are a consequence of the electrons in the lead and island electrodes obeying the Fermi distribution function. Unlike the analysis of Ref. [105], the gate capacitance Cg must be included explicitly and not distributed between C1 and C2 , since the mechanical displacement dependence enters only through Cg . In Fig. 15, we show the dependence of the shot noise-limited mechanical displacement sensitivity on the gate voltage Vg , ranging over the Krst few current peaks. These results assume junction capacitance and resistance values C1 = C2 = 0:25 fF and R1 = R2 = 50 kZ, and a static gate capacitance Cg = 0:1 fF. Notice that the optimum displacement sensitivity improves with increasing current peak number—equivalently gate voltage Vg . The reason for this trend is readily apparent from the rough displacement sensitivity calculation given previously above; the same charge Juctuation on a capacitor plate can be induced by a progressively smaller plate gap Juctuation as the voltage is increased across the plates; increasing the gate voltage couples the cantilever more strongly to the SET. This improvement in displacement sensitivity with gate voltage is in marked contrast to the optimized minimum detectable charge which is independent of peak number and motivates the question of how large a gate voltage can be applied to a metal junction SET. Given that electrometry is the most

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often considered SET application, for which there is no gain in charge sensitivity with increasing Vg (one current peak is as good as any other), this question has apparently received little attention in the past. The breakdown voltage is an obvious upper limit and taking a typical vacuum breakdown voltage of 108 V=m [108] gives a maximum Vg of around 10 V across a 0:1
m gap. However, unless the mechanical resonator is suXciently sti1, it is expected that before the onset of vacuum breakdown the Juctuating SET island charge due to electrons tunnelling onto and o1 the island will give rise to a back reaction on the cantilever with a displacement noise which exceeds the equivalent input displacement noise (61) above a certain gate voltage. Thus, there will be a particular gate voltage value for which the displacement sensitivity is optimum. We now give a rough estimate of this backaction displacement noise [109]; a full analysis using the orthodox theory along the lines of Ref. [110] can be found in Ref. [94]. The noise calculations below assume the simpler dc-SET (i.e., no tank circuit); the di1erences between the rf- and dc-SET noise formulae are inessential, with appropriate time-averages required in the former, resulting in only small quantitative di1erences in the noise values. We Krst require the force on the cantilever due to the SET. This can be derived using conservation of energy as follows. Suppose an external force Fext is applied to the cantilever tip, causing it to displace by an inKnitesimal distance d x. Suppose, furthermore, that the displacement occurs quasistatically, i.e., much more slowly than it takes for the charges in the SET circuit to redistribute, as determined by the RC time constant of the SET. From energy conservation, the resulting inKnitesimal work done by this external force is Fext d x = dUelectric + dUelastic − dWbattery ;

(69)

where dUelectric is the change in the electrostatic energy stored in the gate capacitor Cg and in the tunnel junction capacitors Cj , dUelastic is the change in elastic potential energy of the Jexing cantilever and dWbattery is the work done by the drain–source and gate voltages in redistributing the charges on the capacitances. Using Newton’s third law and subtracting o1 the elastic restoring force, we have for the force F exerted by the SET on the cantilever: dUelectric dWbattery + : (70) dx dx In terms of Vds , Vg and the number n of excess electrons on the SET island, the SET force on the cantilever gate electrode is

 1 1 d 2 2 − (71) (CVds + Cg Vg − ne) + Cg Vg ; F= dx 2C9 2 F = −Fext − Felastic = −

where we assume a symmetric SET with tunnel junction capacitances C1 =C2 =C, e1ective resistances R1 = R2 = R and we have neglected a possible background island charge since it will not a1ect the Knal result. At a given drain–source current peak maximum, n, Vg and Vds satisfy the condition e(n + 1=2) = Cg0 (Vg − Vds =2)

(72)

and the probabilities at any given time that there are either n or n + 1 electrons on the island are approximately equal to 12 , provided the current peaks are well-separated in gate voltage. The force noise is also a maximum at a current peak maximum; we will restrict ourselves to evaluating this force noise maximum which takes a particularly simple form. An estimate for the force noise spectral density is SF = [F(n + 1) − F(n)]2 =(Ids =e), where Ids = Vds =4R is the peak maximum drain–source

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Displacement Noise (angstrom/Hz0.5)

10

11.4MHz 45.8MHz 81.4MHz 117MHz 4

10

5

10

6

10

0

1

2

3

4

5

Gate Voltage (V) Fig. 16. The total displacement noise as a function of gate voltage for cantilever lengths 1:25, 1:5, 2 and 4
m, Kxed width 0:6
m and thickness 0:15
m, and quality factor Q = 104 . The displacement noise value at the resonant frequency of the cantilever associated with a given tunnelling current peak is determined at the gate voltage for which the current is Kxed at half its maximum value. The noise analysis assumes a symmetric dc-SET at T = 74 mK(0:01e2 =C9 ) with junction capacitance C = 100 aF, junction resistance R = 50 kZ, static gate capacitance Cg0 = 50 aF, gate capacitor gap d = 0:1
m and drain–source voltage Vds = 0:38 mV(0:6e=C9 ). (Figure courtesy of Y. Zhang.)

current. Substituting Vg and Vds for n using the above condition we obtain  2 4e3 R Cg0 (Vg − Vds =2) : SF = Vds dC9

(73)

The backaction displacement noise estimate at the fundamental Jexural frequency of the cantilever is then just given by (73) multiplied by the factor (Q=ke1 )2 , where Q is the quality factor and ke1 the e1ective spring constant of the Jexural mode. The full derivation of SF (!) using the orthodox theory yields the same result as (73) at ! = 0 and at a current peak maximum, di1ering only by an overall factor of 14 [94]. Note, however that Eq. (73) for the backaction force noise di1ers from that of Refs. [94,109]. In these latter references, it was erroneously assumed that the force is given by the gradient of the electrostatic energy stored in the gate capacitor only. This assumption is only okay when the gate capacitance is small (Cg 2C) where neglecting the electrostatic energy contributions from the other SET capacitances, as well as the work done by the power sources, gives a small error. From Eq. (73) we see that the backaction noise indeed increases with increasing gate voltage as was pointed out above. In Ref. [94], the total displacement noise is approximated by the sum of the equivalent input displacement and backaction displacement noise spectral densities, with correlations between them neglected. In Fig. 16, we show the total displacement noise as a function of

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gate voltage for Si cantilevers with the same cross sectional dimensions and a range of di1erent lengths (hence di1erent fundamental Jexural frequencies). This Kgure is modiKed from Fig. 5 in Ref. [94] using the corrected expression for the backaction force (70) and (71) to evaluate the backaction displacement noise. Note that the minimum displacement noise decreases and shifts to larger gate voltage for decreasing cantilever length. This is a direct consequence of the e1ective spring constant dependence: obviously a sti1er cantilever will be displaced less by the same backaction force. For a cantilever with resonant frequency m = 117 MHz, the minimum displacement noise √ T Hz, corresponding to an absolute displacement sensitivity of 7:5 × 10−4 A T for is 5:5 × 10−6 A= quality factor Q = 104 . If we instead choose the gate voltage such that the drain–source current is Kxed at 30% instead of half its maximum value for√a given tunnelling current peak, then the miniT Hz, corresponding to an absolute displacement mum displacement noise decreases to 4:2 × 10−6 A= − 4 T sensitivity of 5:7 × 10 A. This is close to the cantilever’s zero-point uncertainty which is about T 3:3 × 10−4 A. Thus, by appropriately tuning the SET voltages and capacitance values, it may be possible to achieve a sensitivity at or even better than the quantum zero-point displacement uncertainty limit [94]. However, care must be taken when making sensitivity predictions in the quantum regime which are based on the approximate orthodox model of the SET and classical description of the mechanical resonator. For small displacements of the mechanical resonator, |x|d, the SET works as a linear ampliKer to a good approximation and its quantum dynamics will add an irreducible amount of quantum noise to the input zero-point signal for phase insensitive detection [111,74,75]. While an analysis using the orthodox model suggests that choosing the gate voltage such that the tunnelling current is a smaller percentage of the maximum current improves the displacement sensitivity [94], clearly no meaning can be attached to such sensitivity predictions in small current regions where the quantum noise limit is violated. It is important therefore to go beyond the orthodox model of the SET to include higher order quantum corrections such as cotunnelling processes [112–114]. This will enable us to extend the displacement sensitivity analysis to the regime where the incoherent, sequential tunnelling contribution to the current is comparable to or smaller than the cotunnelling contribution, as well as consider tunnel junction resistance values comparable to or smaller than the quantum of resistance e2 =h (which may be necessary in order to have measurable currents). Indications from orthodox model analyses [94,115] suggest that quantum noise-limited detection is possible in this regime. Note that the SET behaves e1ectively as a quantum point contact in the cotunnelling-dominated regime, suggesting that quantum-limited displacement detection may also be possible with a quantum point contact-based displacement detector [100]. It is essential also that the above noise analysis be extended to describe the fully coupled, quantum dynamics of the combined SET-mechanical resonator system. While the inJuence of the resonator on the SET and backaction of the SET on the resonator have been partially analysed [93,94], the dynamics loop still needs to be closed whereby the resonator’s motion inJuences the SET which in turn acts back on the resonator and so on. Some progress has been made in Ref. [116], which describes the fully-coupled classical dynamics of SET-resonator system, while Ref. [117] describes the quantum dynamics of a single level quantum dot coupled to a mechanical resonator, viewed as a simpliKed model of the SET-resonator system. The coupled dynamics becomes especially interesting and non-trivial when the gate voltage is increased beyond the optimum operating point and longer, more Joppy resonators are used such that there is a large, Juctuating backaction force on the resonator due to the electrons tunnelling onto and o1 the SET island [116]. The resulting

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Juctuating motion of the resonator in turn a1ects the SET’s tunnelling current and can for example be detected as a peak in the current noise about the cantilever’s resonant frequency [118]. By examining the dependence of the peak height and width on the drain–source and gate voltages of the SET, information can be gained about the intrinsic island voltage noise characteristics of the SET at the resonator frequency [118,109]. This is somewhat analogous to the coupled tunnel junction-mechanical oscillator system analysed in Refs. [119] and [120]. Also of relevance is a theoretical investigation of the electromechanical noise due to an electron current transferring momentum to a suspended quantum wire through which it Jows [121–123]. The coupled quantum dynamics of the SET-resonator system Krst requires the derivation of a master equation for the quantum state density matrix in, say, the SET island number and resonator (oscillator) number basis representation. The master equation can be obtained in a systematic way from the full, microscopic Schr[odinger equation using the diagramatic method of Ref. [112]. A complementary formalism which can also be used to investigate the coupled quantum dynamics is the so-called quantum trajectories approach [124–126]. This approach will yield a description of the evolution of a single mechanical oscillator undergoing continuous measurement by the SET displacement detector. Averaging over individual SET drain–source current time traces (the measurement records) obtained within the trajectories formalism will give the ensemble-averaged current obtained from the master equation. Similarly, averaging over the position trajectories of the oscillator interacting continuously with the SET and subject to damping and noise from its environment (other than the SET), will give the oscillator’s ensemble-averaged position obtained from the master equation. One application of the trajectories approach is to active feedback control, whereby the information gained about the state of the oscillator through continuous monitoring by the SET is used to control the oscillator’s subsequent evolution so as to cool the oscillator down, for example [8]. 3.3. Experimental progress towards quantum-limited SET-based displacement detection A necessary step towards realising a rf-SET displacement detector with sensitivity at or below the quantum zero-point limit will be to Krst demonstrate a rf-SET electrometer operating at the fundamental, charge detection sensitivity limit given by the intrinsic shot noise of the SET tunnelling current. In the more traditional, low frequency dc-SETs the equivalent input charge noise is dominated by 1=f noise due to the random excitations of charge traps located in the tunnel junction dielectric, device substrate, or oxide layer covering the island [101,127–129]. At suXciently high frequencies, shot noise and ampliKer noise are expected to dominate over 1=f noise. In Ref. [130], √ a rf-SET is demonstrated in the superconducting state with a charge sensitivity of 3:2 × 10−6 e= Hz for a 2 MHz signal. The shot noise is responsible for about 60% of the total noise, with the remaining percentage due to ampliKer noise. It will also be important to investigate whether such charge sensitivities close to the shot noise limit can be maintained as the gate voltage is increased, so that we are working about a higher number current peak. As discussed earlier, this increases the coupling between the SET and gated mechanical resonator [93]. Having demonstrated a rf-SET electrometer with sensitivities close to the shot noise limit, the next step is to integrate it with a micron-scale, gated mechanical resonator such as a bridge or cantilever. The Krst experimental realization of a SET-based displacement detector, where the SET operates √ as − 5 T a radio-frequency mixer [95], is described in Ref. [96] (Fig. 17). A sensitivity of 2 × 10 A= Hz was demonstrated for a 117 MHz bridge resonator with Q = 1700 at 30 mK. This sensitivity is only

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193

Fig. 17. SEM micrograph showing the doubly-clamped GaAs beam and aluminium surface electrodes forming the SET and beam electrode. (Reproduced from Ref. [96].)

about two orders of magnitude larger than the quantum zero-point uncertainty of the resonator. The dominant noise source was the second-stage ampliKer; it was estimated that such noise could be reduced by a factor of ten with improved electronics. The Krst demonstration of √ a displacement detector based on the rf-SET [98] (Fig. 18) achieved a T Hz for a 19:7 MHz bridge resonator with Q = 3:5 × 104 at 56 mK. This sensitivity of 3:8 × 10−5 A= sensitivity is within an order of magnitude from the quantum zero-point uncertainty of the resonator (Fig. 19). Note that, while the sensitivities for the two demonstrated displacement detectors are similar, the sensitivity of the device in Ref. [98] is an order of magnitude closer to the quantum zero-point uncertainty as compared with the device in Ref. [96]. This is because the resonator in Ref. [98] has a lower frequency m and higher Q, hence narrower signal bandwidth and larger zero-point uncertainty than the resonator in Ref. [96]. The sensitivity of the rf-SET displacement detector can be improved by increasing the tank circuit quality factor (as in Ref. [130]) to its optimum, matching value QT ∼ Rd =R0 [103,104]. However, as stated earlier, this reduces the detection bandwidth and hence the resonator frequency, making it more challenging to cool to the zero-point limit. For the tank circuit of Ref. [98], T = 1:35 GHz and QT ≈ 10, giving a bandwidth of about 70 MHz. An alternative is to reduce the noise in the second-stage ampliKer [131]. To achieve better displacement sensitivities, it will also be essential to locate the mechanical resonator closer to the SET island than has been achieved in Ref. [96] (0:25
m gap) and [98] (0:7
m gap). This is so as to increase the gate capacitance, and hence the coupling strength between the metallized mechanical resonator and SET island. A possible alternative method to achieve strong coupling between the mechanical resonator and SET is to fashion the former out of a piezoelectric material. The SET can then be located directly on resonator’s surface, sensing changes in the polarization charge due to motion-induced strain in the resonator, analogously to the FET strain sensors mentioned earlier [16]. This avoids the problem of having to deKne a small gap between the resonator and separated SET. A sensitivity analysis of the rf-SET-based piezoelectric detection technique is given in Refs. [132,133].

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Fig. 18. SEM micrograph of the rf-SET displacement detector. Fig. A shows the interdigitated capacitor and square coil inductor (inset) which form a 1:35 GHz LC resonator. Fig. B shows a pair of SETs and doubly-clamped beams on the SiN membrane (dark square). Fig. C is a close-up of a SET and SiN beam with Au electrode. The tunnel junctions are at the corners, marked “J”. The Au gate to the right of the resonator controls the SET bias point. (Reproduced from Ref. [98].)

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195

Fig. 19. Position sensitivity for the rf-SET displacement detector versus gate voltage Vg . The solid line is the calculated sensitivity limited by shot noise and backaction noise. The data points are the actual, measured sensitivity, limited by the second-stage, cryogenic microwave ampliKer noise. (Reproduced from Ref. [98].)

4. Macroscopic mechanical superposition states This section describes a possible method [134] to generate and detect quantum superpositions of position states, as well as measure their subsequent decoherence, for (sub)micron scale mechanical resonators. The method uses a demonstrated, coherent quantum device called a Cooper-pair box [135–138], which is electrostatically coupled to the mechanical resonator. The existence of mechanical superposition states and their decoherence is inferred indirectly through measuring the charge state of the Cooper-pair box with an rf-SET electrometer, rather than by detecting the motion directly with an rf-SET as described in the previous section. Section 4.1 gives the background and motivations for measuring macroscopic, mechanical superposition states, while Section 4.2 describes the Cooper-pair box-based scheme for producing and detecting such states. Section 4.3 discusses the quantum entanglement properties of the Cooper box–mechanical resonator composite system. 4.1. Background and motivations If a quantum system can be in either states |< or |=, then it can also be in an arbitrary linear combination of these two states: >|< + 1|=. Standard quantum mechanics places no limits on the size of a system to which this principle of superposition applies. As Schr[odinger Krst recognized in his famous cat paradox [139], this can lead to the bizarre situation of a macroscopic object being in a superposition of macroscopically distinct states. One of the Krst demonstrations of a not-so-macroscopic “cat” state involved the generation and detection of superposition states involving a few photon Keld trapped in a microwave cavity [140]. Recent experiments [141,142] have succeeded in demonstrating indirectly the existence of macroscopic cat states in the form of superpositions of counter-rotating microampere supercurrents in superconducting quantum interference device (SQUID) rings. The fact that we do not observe such macroscopic superpositions

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under normal conditions is explained by the theory of environmentally-induced decoherence [143–145], which recognizes that all systems are open and that only a subset of system plus environment observables are accessible to measurement. A macroscopic system initially in a superposition of macroscopically distinct states will very rapidly become correlated with orthogonal states of its environment, producing an entangled state. If we perform measurements on an ensemble of such systems, ignoring the environment, then the outcomes will be indistinguishable from those that would be obtained if the system ensemble was in a classical statistical mixture of the macroscopically distinct states. As an idealised example, consider a single harmonic oscillator system with free frequency ! interacting linearly with an environment, modelled as an inKnite collection of non-interacting harmonic oscillators in thermal equilibrium at temperature T (the Caldeira–Leggett model [143]). At high temperatures, i.e., kB T ˝!, the following decoherence time estimate is obtained for an initial superposition of two coherent states whose centres are a distance Sx apart [145]: 

.T 2  d = c ; (74) Sx √ where c is the classical damping time and .T = ˝= 2mkB T is the thermal de Broglie wavelength. For, e.g., a 1 mg “dust grain” oscillator at 1 mK in an initial superposition state with Sx = 1 cm, Eq. (74) gives d =c ∼ 10−29 . Thus, even if the dust grain could be isolated from its environment to the extent that the damping time was of order the age of the universe, i.e. c ∼ 1017 s, the decoherence time would still only be about a picosecond. If we were somewhat less ambitious in considering a mechanical system with a mass much less than one milligram, but still comprising many atoms, might it be possible with currently available technology to create and detect macroscopic superposition states? Superposition states have been demonstrated for fullerene (C60 ) molecules [146] (and more recently for the biomolecule tetraphenylporphyrin (C44 H30 N4 ) and for Juorofullerene (C60 F48 ) [147]), as evidenced in the interference pattern formed by a beam of such molecules which have di1racted through a grating. While the number of atoms (sixty) involved in the superposition state is not huge, the position separation in the superposition, as determined by the di1raction grating period (100 nm), is much larger than the fullerene molecules’ typical de Broglie wavelength (∼ 10 pm). However, as we will argue in the following subsection, current technology should enable the demonstration of superposition states for much larger, micron-scale mechanical systems, albeit with much smaller position state separations. To gain a Krst idea, consider again the above decoherence time estimate (74), but now for, say, a crystalline Si cantilever with dimensions l(length) × w(width) × t(thickness) = 1:6
m × 0:1
m × 0:1
m. Such a cantilever has a fundamental Jexural frequency m ≈ 50 MHz and zero-point uncertainty Sxzp ≈ T Assuming an attainable quality factor Q = 104 , so that c = Q=m = 64
s, and a read1:4 × 10−3 A. ily attainable dilution fridge temperature T = 30 mK, the decoherence time is d ≈ 5(Sxzp =Sx)2
s. Thus, provided the position separation is no more than a few times the zero-point uncertainty, the decoherence time will be in the range 0:1 − 1
s. Crucially, this opens the door to electromechanical schemes which use available, low temperature radio frequency electronics techniques (cf. the rf-SET described in Section 3) to generate and detect superposition states. While the position separation is not large, the number of Si atoms involved in the superposition state is (∼ 109 ) and so in this sense it is a macroscopic state. Refs. [11,148] consider possible quantitative measures of the macroscopic nature of a quantum state.

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Such experiments involving micron-scale electromechanical systems would provide in two important ways an essential step along the path towards understanding how the classical world emerges from the quantum world. First, the theory of environmentally-induced decoherence needs to be tested more extensively against experiment for various macroscopic systems, including mechanical systems. While the Caldeira–Leggett model is very convenient to use and provides a direct relation between the decoherence time and classical damping time as in the above estimate (74), it is not clear whether this model applies to mechanical oscillators at low temperatures; the defect dynamics responsible for macroscopic mechanical damping and decoherence may be e1ectively that of tunnelling in a two-level system, as for other low temperature properties of amorphous solids [149,150]. And there is in fact some suggestive evidence to this e1ect for the low temperature damping behaviour in mechanical resonators [151–156]. Thus, the relevant system-environment model may be the oscillator system-spin bath model. Note that this model has yet to be investigated in any detail, in contrast with other much better understood models such as the Caldeira–Leggett, spin system-oscillator bath (spin-boson) [157] and spin system-spin bath (central spin) [158] models. It is necessary to establish under what conditions (e.g., temperature, system-bath coupling strength etc.) the oscillator system-spin bath model maps onto the Caldeira–Leggett model, and also when it does not, giving qualitatively di1erent predictions for the oscillator system damping and decoherence dynamics. Second, the theory of environmentally-induced decoherence only explains why we do not perceive macroscopic superposition states within the limited framework of the conventional Copenhagen interpretation, which views quantum mechanics as just a set of rules for predicting the results of measurements. However, most of us have the deeply rooted conviction that reality is a meaningful concept (i.e., events and processes occur “out there”, independently of observers and measuring apparatuses), at least at the macroscopic level. If this is in fact the case, then decoherence theory does not provide a full explanation of the measurement process, since it does not describe how macroscopic superposition states generated through interaction with microscopic systems “collapse” onto one of the well-deKned alternatives making up the superposition. Fundamental corrections to standard quantum dynamics must therefore show up as progressively larger systems evolve from initially prepared macroscopic superposition states, if we are to account for macroscopic reality. As an aside, one intriguing possibility is that this collapse might be induced by gravity (see, e.g., Ref. [159]); as an environment, gravity is universal, unavoidably a1ecting all systems, and the interaction strength increases with increasing system mass/energy. One way in which corrections to standard quantum dynamics might arise is through a modiKed Schr[odinger equation which allows collapse to occur dynamically [160,161]. In the Ghirardi, Rimini, Weber and Pearle (GRWP) phenomenological collapse model [160–162], the position of the system is localized by a collapse mechanism characterized by two parameters . and a; a nucleon in a superposition of two localized position states separated by a distance larger than a collapses in an average time .−1 to a wavefunction with width a centred at one of the original two positions. Crucially, the GRWP model predicts that the average collapse time scales with the number of nucleons N in the system as 1=.N . The parameters are pretty much constrained to take the values . = 10−16 s−1 and a = 100 nm by the need to agree with standard quantum mechanics at the microscopic scale and classical physics (i.e., no sustained superpositions) at the macroscopic, human scale. With these values, however, it is clear that the above proposed mechanical superposition experiments cannot test for GRWP-type corrections to the Schr[odinger equation. Even if the position separation exceeded a = 100 nm, the predicted average collapse time would be about 107 s, much longer than the environmentally-induced

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decoherence time. Nevertheless, the proposed mechanical superposition experiments are a necessary Krst step towards the goal of revealing the possible breakdown of standard quantum mechanics. A recent, general review of experimental work towards testing the limits of standard quantum mechanics is given in Ref. [12]. 4.2. Analysis of the Cooper-pair box-based mechanical superposition scheme In the following, we describe an experimentally viable electromechanical scheme [134] to produce superpositions of distinct position states and, furthermore, measure their decay due to environmentallyinduced decoherence. The scheme is partly inspired by recent optomechanical schemes [163,164,31] which propose to use optical cavities with mechanically-compliant mirrors to produce entangled states of the cavity Keld and mechanical mirror. The basic idea is to employ a two-level system (TLS) for which the following conditions can be met: 1. A strong and controllable coupling can be achieved between the TLS and fundamental mode of the mechanical resonator, such that the spin-up and spin-down states in the measurement basis of the TLS separate out the resonator position states. 2. The quantum state of the TLS can be coherently controlled to produce any desired superposition of measurement basis states. 3. The intrinsic decoherence times of the TLS superposition states due to environments other than that of the damped mechanical resonator are comparable to or larger than the oscillation period of the resonator’s fundamental mode. As e1ective TLS we will consider a Cooper-pair box which consists of a small superconducting island weakly-linked through a Josephson tunnel junction to a superconducting reservoir. Cooper box-based schemes have also been proposed for creating macroscopic quantum state superpositions in superconducting islands [165] and superconducting resonators [166], as well as possible candidates for qubits in future quantum computing devices [137,138,167,168]. Other possible electromechanical schemes worthy of investigation include coupling the mechanical resonator to a SQUID [169] or to a current-biased Josephson junction [170]. It will be of interest to compare the relative merits of these schemes. The state of the Cooper box is determined by the balance between its Coulomb charging energy, and the strength of the Cooper-pair tunnelling between the island and reservoir. Using an external gate, the Cooper box can be driven into either of two states of deKnite Cooper-pair number or a linear superposition of the two states [136,137]. The electrostatic interaction between a Cooper box and nearby capacitively-coupled mechanical resonator, such as a metallized cantilever, causes a displacement in the latter by an amount depending on which of the two charge states the Cooper box is in. When the Cooper box is prepared in a superposition of charge states, it and the cantilever become entangled and the cantilever is driven into a superposition of spatially separated states. If the coupling is strong enough, then the separation between the states in the superposition can become larger than their quantum position uncertainty, and so we can describe them as distinct. Again using external voltage gates, the degree of entanglement between the cantilever and the Cooper box after a given period of interaction (which we call the wait time) can be imprinted on the charge state of the box. For an isolated cantilever the entanglement between the cantilever and the Cooper box is

M. Blencowe / Physics Reports 395 (2004) 159 – 222 V(t)

R0

L

CJ

Cooper box

199

CT

Vgc

Cint VgSET

C gc

Cgm

CT

CgSET

mechanical resonator Vgm

Fig. 20. Circuit diagram for the coupled Cooper box-cantilever system and the rf-SET used to measure the charge state of the box. (Reproduced from Ref. [134].)

Fig. 21. Scheme showing the Cooper box and associated superconducting reservoir, cantilever, SET and the voltage gates. The metallized (aluminium) portions lie on a structural layer of Si which forms the cantilever, which is atop a sacriKcial layer of silicon oxide and then the Si substrate. Note the relative scale and layout of the di1erent components are only approximate. (Figure courtesy of K. Schwab.)

a periodic function of the wait time. However, because the cantilever is driven into a superposition of spatially separated states it will be subject to environmental decoherence which eventually destroys the periodicity in the entanglement between the cantilever and the Cooper box. Of course the Cooper box itself is also subject to environmental decoherence, but this should not prevent the decoherence rate of the cantilever being determined (as we discuss below). The charge state of the Cooper box can be measured with great sensitivity and with minimum disturbance using a rf-SET electrometer [130,171]. Probing the charge state of the box after di1erent wait times, and averaged over many di1erent runs, will give information about the periodicity in the degree of entanglement of the cantilever and the Cooper box. Furthermore, measurement of the charge state of the Cooper box after di1erent wait times will also allow the decoherence time of the cantilever due to interactions with its environment to be inferred. The circuit diagram for the system is shown in Fig. 20 and a schematic diagram of the proposed device layout is shown in Fig. 21. Note that information concerning the state of the cantilever is obtained indirectly through measuring the state of the Cooper box. While an rf-SET displacement detector could in principle be employed to directly measure the position of the cantilever (see Section 3), this would increase the complexity and hence the diXculty of realising a fully-functioning device.

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M. Blencowe / Physics Reports 395 (2004) 159 – 222 ng n+1 n+1/2 Probe stage

n 1T 4 R

τ

3 T 4 R

t

Fig. 22. Pulse sequence for manipulating the state of the box. During the Krst pulse the system evolves into a linear superposition of number states. After this pulse, the system evolves for a time, , during which no further mixing of the Cooper box states occurs and the cantilever and box become entangled. A second pulse is then applied, again causing a mixing of the box states. Finally, the charge state of the box is measured. (Reproduced from Ref. [134].)

Let us Krst focus on the dynamics of the Cooper box-coupled cantilever system, neglecting the coupling to the cantilever environment and the rf-SET electrometer. The Hamiltonian is H = 4EC %n&ˆz − 12 EJ &ˆx + ˝!m aˆ† aˆ + .(aˆ + aˆ† )&ˆz ;

(75)

where %n = ng − (n + 1=2) with ng = −(Cgc Vgc + Cgm Vgm )=2e the dimensionless, total gate charge. The control gate voltage Vgc and cantilever gate electrode voltage Vgm ranges are restricted such that   |%n| 6 12 for some chosen n, so that only Cooper charge states |n ≡ |− ≡ 01 and |n + 1 ≡   |+ ≡ 01 play a role. Thus it is natural to use spin notation where &ˆx and &ˆz are the usual Pauli matrices. The coupling constant between the box and cantilever electrode is . = −4EC nm g Sxzp =d, m V m =2e, Sx where nm = −C is the zero-point displacement uncertainty of the cantilever, and zp g g g dSxzp is the cantilever electrode-island gap. Only the in-plane fundamental Jexural mode of the cantilever, with frequency !m and operators aˆ and aˆ† , is taken into account. All other modes have a much weaker coupling to the box and will be neglected. We assume that the Josephson junction capacitance CJ Cgc and Cgm , so that the charging energy of the box EC ≈ e2 =2CJ . The scheme for the control pulse sequence is indicated in Fig. 22. Such double-pulse control sequences are commonly known as Ramsey interference experiments [172] and as we will show, allow information to be gained about the coherence of the cantilever state dynamics. It is convenient to determine the evolution of the box-cantilever system using the coherent state basis for the cantilever. At t = 0, we take as initial state |<0  = |−|>, where |> denotes a coherent state. The Krst pulse takes the box to the degeneracy point and is of duration TR =4, where TR = h=EJ is√the coherent oscillation (Rabi) period of the Cooper state. The state |<0  evolves to |, where it is assumed that !m TR 1 and the cantilever box coupling strength is such that the coherent state evolution can be neglected. Following the Krst pulse, there is a wait time, , during which the box and cantilever systems interact, resulting in an entangled state: 1 i |− () + √ e−2iEC =˝ |+|>+ () ; (76) 2 2 where we assume that EJ EC , and so neglect the Josephson tunnelling term in the evolution, and where |>± () = ei@± (>; ) |>e−i!m  ∓ ((1 − e−i!m  ), with the phase @± (>; ) = ±(2 [! − sin(!)] ± (i(=2)[>(1 − e−i! ) − >∗ (1 − ei! )] and the dimensionless coupling ( = .=˝!m . The actual degree of entanglement between the cantilever and Cooper box in the state (76) is discussed later below. The spatial separation between the cantilever states |>± () is 2((1 − cos !m )Sxzp and, thus, the

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201

Fig. 23. Qualitative behaviour of the probability density of the fundamental mode of the cantilever as a function of wait time. Initially there is a single peak, representing a coherent state, which gradually separates into two as the linear superposition of the spatially separated states |>+ () and |>− () is built up, reaching a maximum separation when  = m =2, before returning to a single peak after one period of the mode, m . (Figure courtesy of A. Armour.)

condition for the maximum separation of the states to exceed their width is: 4|(| ¿ 1. A cartoon of the evolution of the probability density of the cantilever (i.e., |x|=TR =4+ |2 with the Cooper box states traced over) is shown in Fig. 23. If the probe stage proceeds directly after the wait time, then it is not possible√ by measuring the Cooper pair number to distinguish between the entangled state and the state (1= 2)(|− + i|+) of an isolated Cooper box: in both cases, we have probability 12 that the Cooper pair number is found to be n or n + 1. However, by taking the box to the degeneracy point a second time with a pulse of duration 3TR =4, a signature of the separated cantilever states is imprinted on the Cooper pair number probabilities:   1 |− () + e−2iEC =˝ |>+ () 2   i (77) − |+ e2iEC =˝ |>− () − e−2iEC =˝ |>+ () 2 and   2 (78) P(|−) = 12 1 + cos[4EC =˝ + @+ (>; )]e−4( (1−cos !m ) : If there is no coupling between the Cooper box and cantilever (i.e., ( = 0), the second control pulse simply returns the box to its initial state |− (the Cooper state has e1ectively performed a full Rabi oscillation at the degeneracy point) provided  = 2k˝=4EC , k = 0; 1; 2; : : : : This will no longer generally be the case for non-zero coupling, however. Assuming that, before the control pulse sequence is applied, the box-cantilever system is in a thermal equilibrium state (because 4EC kB T , the box will be in its ground state |− to a good approximation), we must thermally average the above probability. This gives     2 V Pth (|−) = 12 1 + cos 4EC =˝ + 4(2 sin !m  e−4( (1−cos !m )(1+2N ) : (79) where NV = (e˝!m =kB T − 1)−1 is the thermal occupation of the cantilever mode. The cosine function leads to rapid oscillations whose magnitude is controlled by the exponential term. It is convenient to deKne the envelope of Pth (|−) as the function in Eq. (79) with the argument in the square brackets set to zero. Notice that the envelope of Eq. (79) recovers its initial value (i.e., unity) as  approaches the period m of the cantilever mode. This is a consequence of the harmonic nature of the cantilever as a measuring device for the Cooper box state; the correlations set up between the box and cantilever

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states are completely undone and the two systems are no longer entangled after an integral number of harmonic oscillation periods. This “recoherence” e1ect is discussed in Ref. [164] for a system involving a cavity Keld coupled to a movable mirror. Similar e1ects are also discussed in Refs. [165,166]. Comparing Eqs. (78) and (79), notice also that the quantum separation of cantilever states as manifested in the vanishing of the overlap amplitude >− ()|>+ () and the thermal smearing of the phase terms @+ (>; ) give rise to the same -dependence in the exponent term in Eq. (79); it is not possible just by measuring the -dependence of Pth (|−) to distinguish between the generation of superpositions of separated cantilever states and thermal smearing [173]. The conditions for the quantum state control are as follows: j ¡

h h  m ¡ cb d ; 4EC EJ

(80)

where j denotes the jitter time of the pulse sequence generator and cb d denotes the decoherence time of the Cooper box superposition states through processes other than due to the cantilever and its environment. The Krst inequality in the chain is necessary to resolve the rapid oscillations with period h=4EC in Eq. (79), and thereby measure the associated envelope function; without being able to position the pulses with suXcient temporal accuracy, the oscillations would be washed out giving a constant Pth (|−) ≈ 12 . The last inequality is necessary to observe the recoherences and the e1ects of the cantilever’s environment (which we discuss below). The middle two inequalities are not essential, their purpose being only to simplify the theoretical analysis and hence the description of the quantum dynamics. In particular, the inequality h=4EC h=EJ ensures that the ground state of the box for %n = − 12 is the Cooper pair number state |− to a good approximation and also allows us to neglect the e1ects of the Josephson tunnelling term in the Hamiltonian during the wait stage between the two applied pulses. The inequality h=EJ m allows us to neglect the e1ect of the cantilever on the Cooper box for the duration of each of the two pulses. A 1 ps jitter time is achievable. Choosing EC = 150
eV gives h=4EC ≈ 7 ps and choosing EJ = 4
eV gives h=EJ ≈ 1 ns. A fundamental Jexural frequency m = 50 MHz, giving a period m = 20 ns, is readily achievable with micron-sized cantilevers [13,14]. The most serious practical constraint arises from the decoherence of the Cooper-box itself, which if it occurs too fast will obscure the quantum dynamics of the cantilever. In the devices of Nakamura et al. [136,168,174], decoherence times of only a few ns have been achieved and are thought to be limited by background 1=f charge noise [174–177], the same noise source which dominates in  dc-SETs [101,127–129]. Also, the further we are from the bias point ng = n + 12 the shorter are the decoherence times [174]. One approach might be to modify the circuit design and/or pulse control so as to protect the Cooper box charge superposition states from dephasing due to background charge noise. Nakamura et al. [175] have demonstrated that decoherence times of the box can be extended by applying refocusing pulses. Furthermore, by operating at the bias point ng = (n + 12 ), as well as protecting the Cooper box states from Jux noise and backaction read-out noise, Vion et al. [137] have achieved decoherence times of order a 0s. However, such schemes may at the same time suppress the ability of a Cooper box charge superposition state to separate the cantilever position states, since this is e1ectively nothing other than a Cooper box state dephasing process due to a single Juctuating charge. Nevertheless, there is no fundamental reason why Cooper box decoherence times cannot be increased by an order of magnitude or so through improvements to the device

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203

fabrication procedure (including, for example, experimenting with di1erent substrate materials) just as has been possible with SETs [101,127–129]. In order that the Cooper-pair superposition state separate the cantilever coherent states by more than their width (the quantum position uncertainty), we require that the coupling strength satisKes 4|.|=˝!m ¿ 1. As mentioned earlier, a Si cantilever with dimensions l (length) × w (width)× t (thickness) = 1:6
m × 0:1
m × 0:1
m has a fundamental Jexural frequency m ≈ 50 MHz and T Assuming a cantilever electrode-Cooper island gap zero-point uncertainty Sxzp ≈ 1:4 × 10−3 A. m d = 0:1
m and gate capacitance Cgm ≈ 20 aF, the dimensionless gate charge nm g ≈ −63Vg . Substim tuting in these parameter values and EC =150
eV, we have for the separation condition: Vg ¿ 1:0 V. Such a voltage can be applied across a 0:1
m gap: it will deJect the cantilever by a much smaller distance than the gap and is well below the breakdown voltage. We now turn to consider the e1ect of the cantilever’s environment on the coupled Cooper box-cantilever dynamics. In practice, the fundamental Jexural mode of the cantilever will be coupled to a large number of microscopic degrees of freedom within the cantilever and in the substrate to which it is attached, which we refer to collectively as the environment. The coupling to the environment will progressively damp the recoherences and, thus, by measuring their suppression, we can infer the decoherence rate of the cantilever itself. We model the environment of the cantilever as a bath of independent oscillators at a Kxed temperature, T , each of which are weakly coupled to the fundamental Jexural mode. This model is equivalent to treating the cantilever mode as a damped quantum oscillator characterized by an energy damping rate parameter, 2A!m ; kB T=˝ [178]. However, it is important to bear in mind that for the reasons discussed earlier, this may not be the correct model for the environment of a micron-scale mechanical oscillator at mK temperatures. Indeed, one of the motivations for realizing the Cooper box-cantilever device is to test our models for environmentally-induced decoherence when applied to mesoscopic mechanical resonator quantum systems. When the calculation of Pth (|−) is repeated including the coupling of the cantilever to the bath oscillators we Knd (the calculation is similar to that given in Ref. [179]):   Pth (|−) = 12 1 + cos [4EC =˝ + 4(2 ’()]e−C() ; (81) where ’() is a slowly varying phase factor which depends on the properties of the cantilever. The damping of the coherent oscillations is given by  2A!m −A 4.2 (2NV + 1) A − 2 C() = 2 2 e sin(!m ) 2 2 ˝ (!m + A ) A + !m

2   2 A − !m −A + [e cos(!m ) − 1] ; (82) 2 A2 + ! m where NV = 1=(e˝!m =kB T − 1) is the thermal-averaged occupation number of the fundamental Jexural mode. Again we deKne the envelope of Pth (|−) by setting the total phase in the square brackets of Eq. (81) to zero. The energy damping rate in the model, 2A, can be estimated empirically by measuring the quality factor of the cantilever, Q, since 2A = !m =Q. Fig. 24 shows the envelope of Pth (|−) when the coupling of the cantilever to the environment is included, for Q = 1000 as a function of the quantity (2.=˝!m )2 (2NV + 1). In the presence of a Knite damping rate, the recoherences are indeed suppressed progressively as either the temperature or the cantilever-Cooper box coupling is increased. The series

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Fig. 24. Envelope of Pth (|−), including the cantilever’s environment, as a function of the wait time  for Q = 1000. The Kgures in the legend correspond to the values of the quantity 4(2 (2NV + 1). (Reproduced from Ref. [134].)

of curves shown could be obtained, for example, by setting the temperature at 30 mK and sweeping the coupling strength ( from 0:14 to 0:41. Notice that because of the predicted dependence of the decoherence rate of the cantilever on the coupling and temperature, it would be possible to separate out the e1ect of the cantilever’s environment from other contributions causing decoherence of the Cooper box, provided the latter do not dominate over the former. The Knal stage in the process is to read out the charge state of the Cooper box using the rf-SET. At the end of the control stage, the rf-SET is tuned away from the Coulomb blockade region and a non-zero drain–source voltage applied, resulting in a tunnelling current through the SET. As a result of the capacitive coupling Cint between the Cooper box and SET, the SET island voltage will be a1ected by the Cooper box island charge. Hence, the SET tunnelling current probes the Cooper box charge state. The minimum current averaging time such that the signal-to-noise ratio exceeds one is [115,138,171]: 

CJ 2 (%q)2 ; (83) measure = Cint e2 where %q is the charge sensitivity of the rf-SET. The lifetime of the initial state of the Cooper box (more accurately, the Cooper box density matrix obtained by tracing over the cantilever and SET subsystems) is fundamentally-limited due to the unavoidable rf-SET island voltage and quantum electromagnetic mode Juctuations [171] acting back on the box in concert with Josephson tunnelling. The characteristic decay time is [115,138,171]:  e 2 C 2 E 2 J int 1 − = SV ; (84) decay ˝ CJ 4EC where SV is the sum of the SET island and electromagnetic mode voltage noise is evaluated at the Cooper state oscillation frequency ! = 4EC =˝. Taking the ratio of these times, we obtain for the

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205

condition that the measurement time be shorter than the lifetime of the initial Cooper box state: 

decay 4EC 2 ˝2 = ¿1 : (85) measure EJ SV (%q)2 Using the values EC = 150
eV and EJ = 4
eV, resulting from the above state control condition, the electromagnetic-mode dominated voltage noise SV = 0:14 nV2 =Hz at 4EC =h√ = 145 GHz, and the experimentally-determined value for the rf-SET charge sensitivity %q = 6:3
e= Hz [171], we have decay =measure = 1:7 × 103 . Choosing, for example, Cint =CJ = 0:1, the respective times are in fact measure = 4 ns and decay = 7
s. Thus, provided Cint is not too small, it should be possible read out the charge state. Refs. [180,181] make considerable progress towards realizing rf-SET single-shot read-out of the charge state of a Cooper box. We have described above a scheme for entangling a micron-sized cantilever with a Cooper box and thereby driving the cantilever into a superposition of macroscopically distinct quantum states. Evidence for such quantum superposition states comes from the recoherences in the envelope of Pth (|−). This can be understood by reconsidering the system in the Copenhagen interpretation: suppose the cantilever is treated as a classical system, that is, one that cannot support superpositions. Consequently, the interaction between the cantilever and the Cooper box should lead to the “reduction of the wave function” of the Cooper box so that its state becomes an equally weighted classical mixture of the two states |+ and |−. The second pulse in Fig. 22 would not a1ect the relative probabilities of the two Cooper box states and so the Knal measurement would result in an envelope of Pth (|−) which once it reaches a value of 12 , never increases. Thus the recoherences are a signature of the quantum behaviour of the cantilever and their decay due to environmental decoherence marks the transition of the cantilever from the quantum to the classical regime. 4.3. Quantifying the Cooper box-mechanical resonator entanglement How “quantum” are the Cooper box-cantilever correlated states which are generated as a result of their interaction? It would be useful to have a measure of the quantum entanglement, especially given that the cantilever is initially in a thermal state with NV not necessarily small. For the pure state given by Eq. (76), the entanglement E, deKned as the von Neumann entropy of the reduced density matrix [182] of either the Cooper box or cantilever subsystems, is found to be   E(|
where D=e−4( (1−cos !m ) . Note that the entanglement does not depend on the coherent state parameter >. When the cantilever coherent states |>± () have negligible overlap (i.e., the position separation is much larger than the position uncertainty), we can clearly see from (86) that the entanglement E approaches one and thus the state (76) is close to being maximally entangled. However, the actual initial state of the cantilever is not a pure, coherent state, but rather a thermal mixture of coherent states (i.e., thermal equilibrium state). During the wait time after the Krst pulse in Fig. 22 when the Cooper box interacts with the cantilever, a mixed, correlated state develops:  1 2 V ˆTR =4+ = (87) d 2 >e−|>| =N |
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Fig. 25. Logarithmic negativity as a function of the wait time  over a range of temperatures kB T=˝!m = 0 to 3, corresponding to the range 0 –7 mK. Computational limitations prevented the calculation of the negativity for higher temperatures. The coupling strength is ( = 0:41. (Figure courtesy of A. Armour.)

However, it is not as clear as in the pure state case how entangled this mixed state is, if at all, for the range of coupling strengths ( and temperatures considered above; as is well-known [182], the above entropy of entanglement measure also picks up the classical correlations in mixed states. An entanglement measure which can be readily computed numerically for mixed states of bipartite systems is the so-called logarithmic negativity [183]. While this measure can be assigned an operational interpretation [184], it’s physical meaning is unclear. Nevertheless, a non-zero value of this measure does guarantee that the state is entangled (i.e., non-separable). In Fig. 25, the logarithmic negativity of the Cooper box-cantilever state (87) is plotted as a function of the wait time and for di1erent temperatures. Clearly, the Cooper box-cantilever systems become entangled. An alternative, more physical approach to verifying non-separability is to try to construct Heisenberg-like uncertainty relations involving certain linear combinations of observables of both the Cooper box and cantilever systems which necessarily hold for separable states, along the lines of the relations derived in Refs. [185,186] for bipartite, continuous variable systems. However, the spin component commutation relations are not like the position-momentum canonical commutation relations; it is not clear that such uncertainty relations will usefully generalize to 2 × ∞ bipartite systems with the mixed states considered above violating them when non-separable, as required. Some relevant generalized uncertainty relations are developed in Ref. [187].

5. Other quantum electromechanical e!ects This section gathers together discussions on several quantum electromechanical e1ects. Section 5.1 considers single phonon detection, Section 5.2 mechanical quantum squeezed states and Section 5.3 quantum tunnelling of mechanical degrees of freedom. The analysis of each e1ect is somewhat less complete than that for mechanical quantum superposition states in the previous section; further work is required to establish the experimental feasibility of each e1ect given current microfabrication possibilities.

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5.1. Single phonon detection In Sections 2.1 and 2.2, we described the theory and experiment of one-dimensional phononic heat current Jow in suspended dielectric wires. The logical next step in this line of investigation is the detection of single phonons. And looking even further ahead, it would be interesting to address the possibility of the generation, detection and manipulation of few phonon states, with a view to investigating phononic analogues of various phenomena in quantum optics. One way to detect phonons might use a device similar to that of Fig. 11 [188]. Phonons entering the central cavity via the wires are absorbed by the electron gas calorimeter, causing a sudden increase in the gas temperature which is monitored by SQUID noise thermometry, just as in the thermal conductance experiment [35,49]. A preliminary analysis of the feasibility of such a phonon detection scheme is given in Ref. [188] (see also Section 4 of Ref. [36]) assuming a GaAs suspended cavity/wire structure with doped, n+ GaAs regions in the central cavity forming the electron gas transducers. One important consideration concerns the energetics of phonon absorption. In particular, the heat capacity of the electron gas should be suXciently small that the absorbed phonon energy gives an electron gas temperature increase which is resolvable by the SQUID. Two factors working in favour of a small heat capacity are the low electron number density for n+ GaAs (as compared with that for say a thin metal Klm) and the small electron gas volume (as prescribed by the central cavity dimensions). Preliminary estimates [188] suggest that temperature sensitivities of realisable low noise SQUIDS are suXcient to detect dominant thermal phonons at temperatures down to about 10 mK. Another important consideration concerns the various relative magnitudes of the relevant scattering rates. In particular, the rate at which the electron gas reaches internal thermal equilibrium due to electron–electron scattering must be much larger than phonon absorption and emission rates by the electron gas. The phonon absorption rate should in turn be much larger than the rate at which phonons enter and leave the cavity through the wires. And the SQUID noise thermometer measurement time must be shorter than the phonon emission time. Again, preliminary estimates [188] suggest a temperature range with the correct ordering of these various rates in terms of their relative magnitudes. However, further work needs to be done, not least because the electron–phonon scattering rate estimates in Ref. [188] assumed bulk, 3D phonons, whereas for the relevant temperature range, the dominant thermal phonon wavelength can be comparable to or larger than the central cavity thickness. Other considerations which require further investigation include the backaction of the SQUID on the electron gas calorimeter, such as the e1ect of heating due to Josephson radiation, and also possible energy exchange between the electron gas and various defects in the cavity structure (or on its surface) which could possibly be confused for single phonon absorption/emission events [188]. The previous outlined phonon detection scheme is somewhat analogous to photoelectric detection. Although, unlike the photoelectric detector the gas thermometer cannot only in principle detect phonons but measure their energy as well, both schemes have in common the fact that they are demolition measurements, destroying the photon or phonon in the process. Ref. [189] proposes a quantum non-demolition (QND) type of measurement scheme, which e1ectively projects out a given, selectable phonon number eigenstate. The scheme uses the Cooper box-coupled-nanoresonator system described in Section 4. The Cooper box is biased near the degeneracy point (%n ≈ 0) so that the Josephson energy &x -term dominates over the charging energy &z -term (see Eq. (75)). The electrostatic coupling of the mechanical resonator to the Cooper box gives rise to a shift in the

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energy gap between the two Cooper box energy eigenvalues which depends on the number state of the mechanical resonator and which is strongest at the degeneracy point [190]. Suppose the initial state of the Cooper box-resonator system is given by: ˆi =

∞ 

pN |−; N −; N | ;

(88)

N =0

where |±; N  denote the energy eigenstates of the Cooper box-resonator system near the degeneracy point, and the resonator is assumed to be most generally in a mixed state which can be diagonalized with respect to the number state basis, such as for example a thermal state. Applying a microwave -pulse to the Cooper box which is tuned to the shifted energy gap for some speciKc resonator number state M will produce the following mixed state: ˆ = pM |+; M +; M | +

∞ 

pN |−; N −; N | :

(89)

N =M

Finally a current pulse is used to measure the Cooper box state according to the scheme of Vion et al. [137], projecting ˆ in (89) onto one of two possible states:  |+; M +; M |    ∞  ˆ = (90) 1   pN |−; N −; N | :   ∞ N =M pN N =M

In order to have a reasonable probability of projecting out a pure, low occupation number state from an initial thermal mixture clearly requires that kB T ∼ ˝!m , where recall !m is the fundamental frequency of the mechanical resonator. Thus, high radio-to-microwave frequency mechanical resonators are required with temperatures no higher than a few tens of mK. Furthermore, the di1erence in the energy gaps for adjacent occupation number N -values must be larger than the linewidth of the microwave signal generator in order to be able to selectively project out a given pure number state. This places a lower limit on the coupling strength between the Cooper box and mechanical resonator, which in turn limits the various parameters which determine the coupling strength, such as the resonator electrode-Cooper box island gap capacitance, gate voltage across the gap, resonator sti1ness and mass, etc. The preliminary analysis of Ref. [189] suggests that such a scheme to project onto pure phonon number states is feasible. An alternative, recently proposed QND scheme [126] to detect single phonon states in a mesoscopic mechanical resonator takes advantage of the anharmonic coupling between the Jexural modes of the resonator. The anharmonic coupling gives rise to a shift in the resonant frequency of the “read-out” mode which is proportional to the phonon occupation number of the “system” mode. A related area concerns single phonon e1ects as manifested in the source–drain I –V characteristics of certain devices. One example is the quantum mechanical single electron shuttle [24– 26] mentioned brieJy in the introduction. Single phonon e1ects have in fact been observed in a molecular-scale version of the shuttle [191]. Another example is the recent observation of single phonon-induced Coulomb blockade for single electron transport through a quantum dot embedded in a free-standing GaAs/AlGaAs membrane [192]. A similar phenomenon for a free-standing SET has also been predicted [193].

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209

5.2. Quantum squeezing of mechanical motion Another type of manifestly quantum state which can be considered for a mechanical system is a so-called quantum squeezed state: a minimum uncertainty state of an e1ectively harmonic system where the uncertainty of one of the quadrature amplitudes (see Eq. (95) below) is reduced below that of the zero-point Juctuations. Such states Krst came to prominence in the late seventies and early eighties as a means to suppress noise in optical communications [194–196] and in interferometric [197] and mechanical bar gravity wave detectors [198,74,199]. The Krst experimental demonstration of squeezed light states followed shortly thereafter [200]. Many other groups have since demonstrated squeezed light using a variety of generation and detection techniques [201]. By contrast, there has been very little experimental work on squeezed states in mechanical systems; squeezed states have been demonstrated for a single, vibrating ion [202] and possibly for crystal phonons [203]; there have also been several theoretical proposals [204–209]. It would be of great interest to try to produce squeezed states for a mechanical oscillator structure much larger than a single atom, not only to test some of the ideas developed for the detection of very weak forces such as gravity waves [75,76], but also, at a more fundamental level, to extend the domain of manifestly quantum phenomena to macroscopic mechanical systems as discussed above in Section 4. One way to squeeze a mechanical oscillator initially in a thermal state would be to use parametric pumping, characterized by a term of the form P(t)(a†2 + a2 ) in the oscillator Hamiltonian (see, e.g., Ref. [199]). The Krst demonstration of this method for classical thermomechanical noise squeezing was performed by Rugar et al. [210] using a device comprising a cantilever several hundred microns in length and a few microns in cross section. The room temperature thermal vibrational motion in the fundamental Jexural mode was parametrically squeezed in one quadrature to an e1ective temperature of about 100 K by periodically modulating the e1ective spring constant at twice the Jexural frequency. A natural question to ask is whether quantum squeezing could be achieved in a similar device. We will show [208,209] that substantial quantum squeezing can in fact be achieved using a cantilever device similar to that of Rugar et al. [210]. And as we shall see, one key property of micron-sized cantilevers which allows for the possibility of their substantial quantum squeezing is the adequately large mechanical quality factors which can be achieved [13]. Our model structure comprises a cantilever with one plate of a capacitor located on the cantilever surface and the other plate located on the substrate surface directly opposite. The classical equations of motion for the cantilever in the fundamental Jexural mode are [cf. Eq. (46)] m

d 2 x m!m d x 2 + m!m + x = Fvoltage (t) + FJuct (t) ; dt 2 Q dt

where the force on the cantilever due to the pump voltage Vp (t) applied to the capacitor is 1 dC 2 V ; Fvoltage = + 2 dx p with C0 Vp (t) C= : 1 − x(t)=d

(91)

(92)

(93)

The coordinate x denotes the displacement of the cantilever tip from the static equilibrium position (Vp = 0) with increasing x in the direction of cantilever pull-in towards the substrate surface, m

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is the cantilever e1ective mass, d is the equilibrium cantilever tip-substrate base separation, and C0 is the capacitance for equilibrium separation d. Recall that the mechanical quality factor Q and the random force term FJuct , deKned by Eq. (47), model the e1ects of the thermal environment on the Jexural mode. Substituting (92) into (91) for pump voltage having the form Vp (t)=V0 cos(!p t +@) and assuming |x|d (see justiKcation below), we obtain m

d 2 x m!m d x + + [k0 + kp (t)]x ≈ Fp (t) + FJuct (t) ; dt 2 Q dt

(94)

2 where k0 = m!m − Sk, Sk = C0 V02 =2d2 , kp (t) = −Sk cos (2!p t + 2@), and Fp (t) = kp (t)d=2. Note that the equilibrium static spring constant is shifted downwards by Sk. Thus, the resonant frequency  = 2 − Sk=m. There is also a pull-in shift in the of the cantilever is shifted downwards to !m !m 2 equilibrium position of the cantilever tip by the amount C0 V02 =(4dm!m ) and we have redeKned the origin of x to coincide with this new equilibrium position. Note that one consequence of applying the pump voltage Vp (t) across the capacitor is the sinusoidal modulation kp (t) of the spring constant. For phase @ = +=4, this modulation causes squeezing in the quadrature amplitude X1 [199,210], where

−1

  t − ! m x(t) ˙ sin !m t X1 (t) = x(t) cos !m −1

  X2 (t) = x(t) sin !m t + ! m x(t) ˙ cos !m t :

(95)

 , we obtain Pumping the cantilever from an initial thermal equilibrium state at frequency !p = !m for the quantum uncertainty in X1 [199] 

˝ QSk −1 SX12 (t → ∞) ≈ (2NV + 1) 1 + ; (96) 2 2m!m 2m!m

where recall NV = 1=(e˝!m =kB T − 1), with T the initial lattice temperature. Note that we have replaced  with ! in (96) since this causes only a small error for the parameter values to be considered !m m  − ! when setting below. On the other hand, it is important to account for the frequency shift !m m  the pump frequency !p , since the resonance width !m =Q can be smaller than this shift  for large Q. In order to have quantum squeezing, we require that the squeezing factor R = SX1 = ˝=2m!m ¡ 1,  where recall that ˝=2m!m is the zero-point uncertainty. Thus, from (96) we have
2NV + 1 ¡1 : (97) R= 2 1 + QSk=2m!m The term in the denominator of Eq. (97) is just the ratio of the maximum frequency shift (=Sk=2m!m ) to the energy relaxation rate (−1 = !m =Q). Thus, in order to have quantum squeezing (i.e., R ¡ 1), this ratio must be much larger than the thermal occupation number. For illustrative purposes, we consider a crystalline Si cantilever with mass density = 2:33 × 103 kg=m3 and the bulk value approximation for Young’s modulus: E = 1:5 × 1011 N=m3 . Expressing the various cantilever dimensions in units of microns and the initial temperature in mK, we obtain

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211

for the regime kB T ˝! R ≈ 3 × 104

T 1=2 d3=2 t ; V0 Q1=2 l

(98)

 where we have used the relations m = lwt=4, !m = 3:52 E=12 t=l2 and C0 = .j0 lw=d, with l the cantilever length, w the width, t the thickness and . a geometrical factor accounting for the fact that the capacitor plate need not cover the whole cantilever surface area. We have assumed . = 23 in (98). As an example, for the realisable values Q = 104 [13], T = 20 mK, d = 0:1
m, l = 2
m and V0 = 5 V, the squeezing factor is R ≈ 0:42. Thus, we have substantial quantum squeezing. Note that we need not start at temperatures kB T ∼ ˝!m in order to achieve quantum squeezing, a consequence of the adequately large realisable Q factors. In the analysis above, it was assumed that the cantilever tip displacement x is much smaller in magnitude than the cantilever-substrate separation d. Substituting the various above parameter values T which into the expression for the equilibrium position shift, we obtain a displacement of about 30 A is much smaller than d = 0:1
m and also the cantilever length l = 2
m. We are therefore far from snapping the cantilever and well within the range of applicability of Hooke’s force law. The  . Note that, if the frequency sinusoidal applied force gives a similarly small displacement for !p =!m  (instead of being twice this frequency), then of the applied force was resonant with the frequency !m the displacement amplitude would increase by a factor Q = 104 to tens of microns; the displacement amplitude is small because the applied force is o1-resonance, not only with the fundamental Jexural frequency, but also with the second (which is six times larger than the fundamental) and higher Jexural mode frequencies. The Casimir force can also give rise to large deJections for submicron plate separations [211,212]. Using the expression for the Casimir force between two parallel plates T for a cantilever of area A, Fcasimir = 2 ˝cA=240d4 , we obtain a negligible deJection of about 0:04 A with the above dimensions, including a width w = 1
m. In the classical squeezing analysis of Rugar et al. [210], the analogous quantity  to the squeezing factor (97) is the gain, deKned as G = |X |pump on =|X |pump o1 , where |X | = X12 + X22 . The term 2 QSk=2m!m (see Eq. (97)) also appears in their expression for G. However, there would appear to be a discrepancy: their solutions for Xi (t) break down when this term exceeds one, hence restricting their squeezing maximum (minimum gain) to 12 , whereas we have no upper bound on this term. The resolution lies in the fact that Rugar et al. assumed steady-state solutions. If this term exceeds one, as is the case for the parameter values we are considering, then X2 (t) grows exponentially without bound as t → ∞. Thus, the pumping should be terminated after the characteristic time tch for which the squeezing factor largely reaches its limiting value (97), where [199]

 2 QSk 2m!m ln tch = = : (99) 2 QSk 2m!m A related issue concerns the conversion of mechanical energy into heat, possibly warming the cantilever suXciently to take it out of the quantum squeezing regime while it is being pumped. It is not clear whether the generated heat would dissipate suXciently rapidly into the surrounding substrate to prevent this from happening; the heat dissipation rate clearly depends on the materials properties and layout of the device. Alternatively, it is reasonable to expect that heating will be negligible during the pumping stage if the pumping time is much smaller than the relaxation time:

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tch . Substituting into (99) the various chosen parameter values, we Knd tch = ≈ 0:03. Thus, based on this criterion the limiting squeezing value can be largely attained  without signiKcant heating. For the above parameter values, the zero-point uncertainty is ˝=2m!m ≈ 1:5 × 10−13 m, while the predicted squeezed uncertainty is ≈ 6 × 10−14 m. Thus, in order to directly verify quantum squeezing, a displacement detection scheme must be capable of measuring Juctuations of this small magnitude. The rf-SET detection scheme described in Section 3 can in principle attain such sensitivities. However, further analysis is required in order to establish a QND scheme based on the rf-SET for measuring uncertainties in the individual quadrature amplitudes [75,76]. An implicit assumption of our analysis is that the capacitor plate on the substrate surface opposite the cantilever is rigidly Kxed, with negligible surface amplitude Juctuations due to thermal phonon modes of the substrate as compared with fundamental Jexural mode vibrations of the cantilever tip—even in the squeezing regime. This also requires further investigation. 5.3. Quantum tunnelling of mechanical degrees of freedom Macroscopic quantum tunnelling has been demonstrated for several di1erent systems, including the Josephson tunnel junction [213,214] and the rf SQUID [215]. The theory for dissipative quantum tunnelling has also been extensively developed, both for escape from a potential well with metastable minimum into the continuum [143,216] and for coherent tunnelling in a double well with two degenerate (or nearly degenerate) minima [157]; the tunnelling is in the presence of dissipation due to coupling of the macroscopic tunnelling coordinate to a dissipative heat bath. By comparison, little attention has been directed to the possibility of observing quantum tunnelling of macroscopic mechanical degrees of freedom [217,218]. Given the recent advances in the fabrication and control of (sub)micron-scale mechanical resonators, it would be of interest to address this possibility more thoroughly. With their typically much larger quality factors than Josephson tunnel junction and SQUID devices, (sub)micron mechanical resonators could potentially a1ord the investigation of quantum tunnelling in the unexplored, very weak dissipation regime [217]. In Ref. [217], a preliminary analysis is given of the possibility to observe escape-tunnelling of the centre-of-mass coordinate of a cantilever. The 1=x attractive potential for suXciently strong electric Keld between cantilever electrode and rigid substrate electrode directly opposite combines with the approximately quadratic, elastic restoring potential of the cantilever to form a barrier with metastable minimum. Starting with zero voltage between the electrodes, the resonator initially resides in the region of the stable minimum of the elastic restoring potential. As the voltage is turned on and slowly increased, the minimum becomes less stable, with the increasing likelihood of thermal activation over the barrier and into the pulled-in state where cantilever is in contact with the substrate electrode. As necessary conditions to observe tunnelling, we require that it dominates over thermal activation, that the tunnelling rate itself not be exponentially suppressed, and that the metastable minimum and pulled-in states be practicably distinguishable. Using simple estimates, in Ref. [217] it is found that such conditions are satisKed for the example of a Si cantilever with dimensions 30 nm × 1 nm × 1 nm, a resonator electrode-substrate electrode gap of 1 nm and voltage across the gap of about 0:1 V. While cantilevers about an order of magnitude larger than this have been fabricated [219], a cantilever and gap of such small dimensions would be rather diXcult to realize and at any rate is not very macroscopic. While the quantum tunnelling rate can still dominate over the thermal activation rate for larger cantilevers with suXciently high resonant frequencies (typically

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213

Y y(x,t) -l/2

0

x

+l/2

Fig. 26. Parameter deKnitions for the beam.

a gigahertz for realisable low temperatures), the problem concerns the tunnelling rate magnitude which becomes exponentally small and unobservable as the cantilever size is increased. Such conclusions are unlikely to change even if we were to apply the more careful, relevant analysis of Ref. [216] which describes the problem of the escape-tunnelling rate of a system in thermal equilibrium and in the presence of dissipation. In Ref. [218], a brief discussion is given of the possibility to observe coherent tunnelling of the centre-of-mass coordinate of a doubly-clamped beam. The e1ective, double-well potential is achieved by applying a longitudinal compressive stress to the beam ends, suXcient to cause buckling. It is concluded that the observation of coherent tunnelling will be challenging because of the need to apply extremely small, stable compressive stresses such that the double well has only a few quantized energy levels below the central barrier height. However, while the theory developed in Ref. [157] applies only to the double well in the two-state limit, it is not immediately obvious why tunnelling cannot occur in the presence of dissipation when there are many levels below the central barrier, as would be the case for larger, more easily controllable applied stresses. While it may not be possible to observe coherent tunnelling, with the beam centre-of-mass undergoing Rabi-like oscillations, it may nevertheless be possible to observe escape-tunnelling from one well to the other. To get a better idea of the numbers, let us revisit the analysis of Ref. [218] (see also Refs. [220,221]). Consider a rectangular bar of unstressed equilibrium length l0 , width w and thickness d satisfying l0 w ¿ d, and with transverse displacements y(x; t) only in the “d”-direction (see Fig. 26). The Lagrangian is  +l=2 L[y; y] ˙ = (0=2) d xy˙ 2 − Ve [y] − Vb [y] ; (100) −l=2

where 0 = wd is the mass per unit length and l is the end-point separation. The elastic potential energy Ve due to compressive or tensile strain is Ve = (F=2l0 )(lt − l0 )2 ;

(101)

where F = Ewd is the linear modulus and E Young’s transverse modulus of the bar. For small 1 2  2 displacements the total length lt of the bar is lt = d x 1 + (y ) ≈ l + 2 d x(y ) . The bending contribution to the potential energy is  +l=2 2 Vb = (F( =2) d x(y )2 ; (102) −l=2

2

2

where ( = d =12 is the bending moment for a bar of rectangular cross section. From (100), the equations of motion for small transverse displacements of the beam follow:

  F 2 (4)     2 d x [y (x )] y = 0 ; 0y[ + F( y − Fjy − (103) 2

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where j = (l − l0 )=l0 is the strain. Assuming hinged boundary conditions at the endpoints, y(±l=2) = y (±l=2) = 0, and neglecting the anharmonic term in (103), the fundamental mode solution has the form y1 (x; t) = Y (t) cos(x=l), where Y (t) satisKes the harmonic oscillator equation of motion. Clamped boundary conditions, y(±l=2) = y (±l=2) = 0, while more relevant for actual beams [222], yield more complicated mode solutions while not changing the essential conclusions below concerning the possibility of tunnelling. Neglecting all higher modes in the decomposition of y(x; t), the Lagrangian (100) becomes L(Y; Y˙ ) = 12 m∗ Y˙ 2 − V (Y ), where m∗ = m=2 is the e1ective mass and the e1ective potential is V (Y ) = (>=2)Y 2 + (1=4)Y 4 . The coeXcients depend on the various parameters as 

2  4 Ewd(2 l >= 1+ j (104) 3 2l ( and 1=

4 Ewd : 8l3

(105)

Note from (104), that for j less than the critical strain value jc = −((=l)2 (the Euler buckling instability), the coeXcient > is negative and so V (Y ) is a double-well potential. According to Kramer’s theory [223,224], the rate for classical, thermal activation over the barrier is P ∼ exp[ − (>2 =41)=(kB T )] ;

(106)

where >2 =41 is the central barrier height measured relative to the double-well minima and where we have omitted the prefactor (as it will not be needed in the following, simple-minded analysis). On the other hand, the tunnelling rate across the barrier is [216]

  2 a P ∼ exp − dYp(Y ) ; (107) ˝ −a  of a particle with energy >2 =41 in where p(Y ) = 2m∗ [V (Y ) + >2 =41] is the classical momentum  the inverted potential −V (Y ) with turning points a = ± −>=1 and where we have again omitted a prefactor. Thus, assuming the prefactors are comparable, a necessary condition for tunnelling to occur with high probability is  2 a 1∼ dYp(Y ) ¡ (>2 =41)=(kB T ) : (108) ˝ −a  For small oscillations about the double well minima, the frequency is ! = −2>=m∗ . An estimate of the number of quantum levels below the barrier is then (>2 =41)=˝!. Consider as an example a Si beam at a temperature T =20 mK with dimensions 100 nm ×20 nm × 10 nm and assume bulk values for Young’s modulus (E=1:5×1011 N m−2 ) and the mass density ( = 2:33 × 103 kg m−3 ). According to condition (108), we Knd that the tunnelling and thermal activation rates are comparable for strain j ≈ −0:015, with the tunnelling rate exceeding the thermal activation

M. Blencowe / Physics Reports 395 (2004) 159 – 222

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rate for |j| ¡ 0:015. At this crossover value for the strain, we have (2=˝) dYp = (>2 =41)=(kB T ) = 3:8×105 , so that the tunnelling and thermal activation rates are exponentially suppressed. To increase the tunnelling rate, we must therefore decrease the strain magnitude, at the same time making sure that there are still quantum levels below the barrier. For strain j = −0:00839, the number of levels according to the above estimate is about 10; however, while tunnelling dominates over thermal activation, the tunnelling rate is still exponentially suppressed: (2=˝) dYp = 107. Furthermore, the strain is extremely close to the critical value for the onset of buckling, jc = −0:00836 and the state T spatial separation is very small: 2a = 6 A. We might therefore conclude that it is impossible to demonstrate tunnelling for mechanical beams even as small as those with the above dimensions. However, the above analysis assumes that the mechanical beam is in a state of thermal equilibrium, as does the theory developed in Refs. [143,216,157]. Is it possible to satisfy the more general condition  2 a 1∼ dYp(Y ) ¡ E=(kB T ) (109) ˝ −a for some excited state with energy E below the central barrier maximum? Consider again a beam with the above dimensions and temperature T = 20 mK. Suppose the energy is now, say, E = 3kB T . Then the action term (2=˝) dYp = 1 for strain j = −0:046. Thus, if the “particle”, which is initially in thermal equilibrium near the bottom of the well, can be excited to an energy 3kB T below the barrier maximum, then it will be ten times more likely to tunnel through the barrier (with probability close to one at the Krst attempt) than be thermally activated over it. Increasing the strain magnitude deepens the potential wells while at the same time increases the curvature of the potential at the central maximum. For this reason, the tunnelling probability for Kxed excited energy below the central maximum increases with increasing strain magnitude, while the tunnelling probability from the well minimum correspondingly decreases. The possibility to achieve tunnelling of a macroscopic system by taking it out of the well minimum where tunnelling is exponentially suppressed and into the region of the barrier maximum has in fact recently been demonstrated with a Josephson quantum ratchet system [225]. Another related phenomenon is thermally-assisted quantum tunnelling in, for example, molecular magnets [226]. Further work must be done to establish whether or not tunnelling for a macroscopic, mechanical beam excited state is feasible. Particular issues include the method of beam excitation and motion detection. A possible way to probe tunnelling is to determine which of the two stable states the beam is in before and after an impulse has been applied to the beam. By controlling the amplitude and duration of the impulse, the beam can be excited to energies of around a few kB T about the potential barrier maximum. The distribution statistics of the Knal state of the beam as a function of impulse and temperature will delineate between quantum tunnelling and thermal activation. Assuming that the Knal, stable positions are approximately the well minima, then for the above example their  spatial separation is 2 −>=1 ≈ 0:02
m, resolvable using electron microscopy. One possible way to induce suXciently precise and stable strains is to use the scheme shown in Fig. 27. The scheme is similar to that employed in, for example, Refs. [227,228] to achieve mechanically controllable, nano-sized break junctions. The high precision is a consequence of the large reduction ratio between the length of the piezo-elongated Si wafer and the length of the bending nano-sized beam. In developing the theory for thermally-activated/quantum tunnelling motion of an excited buckled beam, it will be important to include the e1ects of dissipation through the introduction of a Knite

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Fig. 27. The bending Si wafer, piezo element and nano-sized beam. The long axis of the nano-sized beam (not to scale) is aligned perpendicular to the supports.

temperature coupled environment. This will require extending the theory of Refs. [143,216,157] to non-equilibrium states. It will also be of interest to address the possibility of tunnelling from an excited state when a smaller magnitude strain is applied than in the above example, so that the action term is a few times larger than one (but still smaller than E=kB T ). The beam will then oscillate back and forth many times in one of the wells before tunnelling through and we might guess that tunnelling will likely occur provided the calculated tunnelling rate in the absence of dissipation exceeds the damping rate of the excited state. Weak dissipation (equivalently large quality factor) will clearly be a necessary condition. The actual dynamics of the buckled beam can in fact be highly non-trivial, owing to the non-linear equations of motion for a given mode and the coupling between modes. For example, beyond a certain critical strain (which is slightly larger than the critical strain for the onset of buckling) [218], the second, “S”-shaped mode conKguration lowers the potential barrier for motion between the two stable buckling conKgurations, and thus in this case the e1ective description of the beam motion as a particle in a double-well potential with Kxed central barrier would not be correct. Thus, it is important to extend the e1ective model to include the lowest few coupled modes as well. 6. Concluding remarks We have given an account of the emerging area of quantum electromechanical systems, with particular emphasis placed on addressing the possibility of demonstrating various quantum e1ects, subject to current microfabrication constraints. E1ects considered include single phonons, quantum zero-point motion and squeezed states, tunnelling of macroscopic degrees of freedom, and macroscopic superposition states. While the analysis of some of these e1ects has yet to be completed, in our view the understanding already gained, along with recent experimental progress reviewed here, point to the very real possibility of their demonstration in the near future. Having analyzed the experimental possibilities, the next stage is to investigate the quantum dynamical behaviour of some of the described systems. Exploring the interplay between the electronic and mechanical degrees of freedom of these mesoscopic devices will lead to a deeper understanding of how the quantum microscopic and classical macroscopic worlds merge.

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