# Interference Effects in a Tunable QPC-Cavity-Reflector Integrated System

Chengyu Yan<EMAIL_ADDRESS>London Centre for Nanotechnology, 17-19 Gordon
Street, London WC1H 0AH, United Kingdom
Department of Electronic and Electrical Engineering, University College
London, Torrington Place, London WC1E 7JE, United Kingdom  Sanjeev Kumar
London Centre for Nanotechnology, 17-19 Gordon Street, London WC1H 0AH, United
Kingdom
Department of Electronic and Electrical Engineering, University College
London, Torrington Place, London WC1E 7JE, United Kingdom  Michael Pepper
London Centre for Nanotechnology, 17-19 Gordon Street, London WC1H 0AH, United
Kingdom
Department of Electronic and Electrical Engineering, University College
London, Torrington Place, London WC1E 7JE, United Kingdom  Patrick See
National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW,
United Kingdom
Ian Farrer  Department of Electronic and Electrical Engineering, University of
Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom
Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 OHE, United Kingdom
David Ritchie  Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 OHE,
United Kingdom
Jonathan Griffiths  Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3
OHE, United Kingdom
Geraint Jones  Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 OHE,
United Kingdom

###### Abstract

We show experimentally how quantum interference can be produced using an
integrated quantum system comprising an arch-shaped short quantum wire (or
quantum point contact, QPC) of 1D electrons and a reflector forming an
electronic cavity. On tuning the coupling between the QPC and the electronic
cavity, fine oscillations are observed when the arch-QPC is operated in the
quasi-1D regime. These oscillations correspond to interference between the 1D
states and a state which is similar to Fabry-Perot and suppressed by a small
transverse magnetic field of $\pm$60 mT. Tuning the reflector we find a peak
in resistance which follows the behavior expected for a Fano resonance. We
suggest that this is an interesting example of a Fano resonance in an open
system which corresponds to interference at or near the Ohmic contacts due to
a directly propagating, reflected discrete path and the continuum states of
the cavity corresponding to multiple scattering. Remarkably, the Fano factor
shows an oscillatory behavior taking peaks for each fine oscillation thus
confirming coupling between the discrete and continuum states. The results
indicate that such simple quantum device could be used as building blocks to
create more complex integrated quantum circuits for possible applications
ranging from quantum information processing to realizing the fundamentals of
complex quantum systems.

††preprint: APS/123-QED

##  I Introduction

Quantum interference, one of the most remarkable effects of quantum mechanics,
arising from the wave nature of particles has led to some celebrated results
in mescoscopic system, including weak localization in two dimensional (2D)
systemsMRM81 ; RMM81 , Aharonov-Bohm oscillations in ring structuresCTR88 ;
BAS99 , sharp peaks in magnetoresistance in chaotic cavityTND95 ; BDN95  ,
etc. Among its various applications, quantum interference has been used
successfully as a tool to investigate properties of particles such as monitor
correlation and entanglement as demonstrated in Mach-Zehnder
interferometeryJCS03 ; BVD06 , Aharonov-Bohm interferometeryNOC07 ; HDQ01 ,
Hanbury Brown-Twiss interferometeryJJW07 ; KRA02 , and electronic analogue of
Hong-Ou-Mandel deviceLOY98 ; BFB13 . Among all the striking phenomenon, it is
particularly interesting to note that quantum interference is extremely
suitable for studying coupling between different quantum systemsMTM94 ; CDO15
; JMY14  which is crucial for the design of integrated quantum circuits.

A notable quantum mechanical system is a ballistic quantum wire, when short
called a quantum point contact (QPC), defined using a pair of split gatesTPA86
; DTM88 ; WHB88  on a semiconductor heterostructure. Soon after the first
observation of quantised conductance in a quasi-one-dimensional (1D) quantum
wireDTM88 ; WHB88 , a number of novel interesting observations have been made
such as the ‘0.7 structure’KNM96 , incipient Wigner latticeKTM14 ; HTM09 , and
coherent electron focusingCHB88 ; HVH89 ; CPF12 , etc. Experiments regarding
coupling between a QPC, which provides a stream of collimated 1D electrons,
and waveguideMTM94  and that between quantum dot (QD) and cavityCDO15  have
been performed, where a pronounced modulation of conductance of the system and
changing of Coulomb peaks were observed. Knowledge on the resonant
interference effects arising from the coupling of two quantum states is of
fundamental significance for the development of technology for quantum
information processing. Moreover, the Fano resonance, which arises from the
quantum interference between discrete (bound) and continuum states of a
neighboring medium resulting in an asymmetry in the resonance structureFANO61
; PP15 , is one of the definite methods for confirming quantum interference
particularly in systems with smaller length scales, and tunable electron path.
We present a system as shown in Fig. 1 realized using an integrated quantum
circuit having three components: a source of 1D electrons (emitted by a QPC),
a continuum state (an electronic cavity of 2D electrons) and a reflector (a
potential barrier to reflect the collimated electrons). Such an ensemble would
allow a beam of 1D electrons to interfere with the continuum states of the
cavity after being reflected by the potential barrier. This system was used in
our previous workCSM17  where we showed that a Fano resonance could be
observed at the 1D-2D transition regime of a QPC. In the present work we give
evidence that a bound, localized state can be established within the continuum
(2D) and there is a coupling between the bound state and the 1D states from
which it is drawn. Moreover, we show that the coupling between the 1D-2D
states enables the Fano resonance which can be tuned both electrostatically
and magnetically.

The paper from now onwards is organized in the following manner: Section II
(Experimental) covers the experimental setup, device design, and wafer
details. Section III (Results and Discussion) covers details of comprehensive
experimental results. This section is further divided into four sub-sections:
A. Coupling between the 1D-2D electrons; B. Fano resonance; C. Effect of
perpendicular magnetic field; and D. Temperature dependence.

Figure 1: Schematic of device and experiment setup. (a) The yellow-square-
blocks at the edge of mesa marked 1-6 are Ohmics contacts whereas the golden-
metallic patterns within the mesa form a pair of arch-shaped gates and a
reflector. The red dotted-lines show the possible boundary of cavity formed
between the QPC and the reflector. Excitation current is fed to Ohmic 1 while
2 is grounded; Ohmics 3 and 4 are voltage probes. (b) Differential conductance
measurement of the QPC (main plot) and reflector (inset). The series
resistance has not been removed. It should be noticed that the standard two-
terminal measurement is performed with Ohmics 1 and 5note1 . The red-dashed
line in the inset corresponds to $V_{r}$ = - 0.27 V.

##  II Experimental

Figure 2: R as a function of arch-QPC voltage for various Vr. (a) When -0.29 V
$\leq V_{r}\leq$ -0.27 V, a pronounced double-peak structure and fine
oscillations are observed. (b) When $V_{r}>$ \- 0.27 V, all the features
gradually smear out. Data have been offset vertically by 20 $\Omega$ for
clarity.

The samples consist of a pair of arch-shaped gates, with a QPC forming in the
center of the arch, and a reflector inclined at 75∘ to the current flow
direction such that the center of the QPC and the reflector are aligned as
shown in Fig. 1(a). The opening angle of the arch is 45∘ and the radius is 1.5
$\mu$m, both the length and width of the QPC embedded in the arch is 200 nm,
the width of the reflector is 300 nmCSM17 . The angle of the inclined
reflector is crucial because it allows reflected electrons to propagate
ballistically to Ohmic contact 3, while multiple scattering of others will set
up a continuum of cavity states (considering the normal 30∘ spread of the
collimated 1D electronsLAC90 ). A direct scattering of a beam of electrons
from the reflector at or near the entrance of the QPC will establish the
bound, defined, state. The scattering of electrons out of this state
interfering with cavity states will establish a Fano resonance.

The devices were fabricated on a high mobility two-dimensional electron gas
formed at the interface of GaAs/Al0.33Ga0.67As heterostructure. The electron
density (mobility) measured at 1.5 K was 1.80$\times$1011cm-2
(2.1$\times$106cm2V-1s-1) therefore both mean free path and phase coherence
length were over 10 $\mu$m which is much larger than the distance between the
QPC and reflector (1.5 $\mu$m). All the measurements were performed with the
standard lock-in technique in a cryofree dilution refrigerator with a lattice
temperature of 20 mK. For the four-terminal resistance measurement a 10 nA at
77 Hz ac current is applied while an ac voltage of 10 $\mu$V at 77 Hz is used
for the two-terminal conductance measurement. Figure 1(b) shows the
conductance trace of the QPC with well defined conductance plateaus. On the
other hand, the conductance of the reflector (inset) drops around - 0.2 V
which indicates a sharp change in the transmission probability; here Va and Vr
are the voltages applied on the arch gates forming the QPC and reflector,
respectively.

##  III Results and discussion

When the arch-QPC and reflector are operated together, collimated ballistic 1D
electronsLAC90  injected from the QPC are reflected by the potential barrier
created by the reflector and cause a voltage drop $V_{34}$ between Ohmics 3
and 4, such that

$V_{34}\propto n\times r$ $None$

where n is the population of injected electrons and r is the reflection
probabilityCSM17 .

Figure 2 shows the four-terminal resistance measured between Ohmics 3 and 4 as
a function of arch-QPC voltage for various reflector voltages Vr. It is found
that when - 0.29 V $\leq V_{r}\leq$ \- 0.27 V, a pronounced double-peak
structure is observed when arch-QPC voltage Va is around - 0.2 V, the left
peak at Va = - 0.24 V is referred to as peak I and the right peak at - 0.21 V
as peak II [Fig. 2(a)]. When making $V_{r}$ less negative, peak II weakens
gradually while the peak I remains robust. Interestingly, the position of the
double-peak structure coincides with the 1D-2D transition regime of the QPC.
In addition, fine oscillations, highlighted by the vertical dashed lines,
occur when the arch-QPC forms a quasi-1D channel. When $V_{r}>$ \- 0.27 V,
both the double-peak structure and fine oscillations weaken and eventually
smear out in this regime [Fig. 2(b)].

It is interesting to note that the fine oscillations almost align with
conductance plateaus in a similar manner to that reported in tunnelling
spectroscopy of a waveguideMTM94 ; CJ91 . However, unlike the results in
waveguide experimentMTM94 ; CJ91  where each subband contributes to a sharp
peak, conductance plateaus at - 0.6 and - 0.5 V correlate to a broad structure
or oscillation in our work. The double-peak structure in the 1D-2D transition
regime has no analogue in the waveguide structureCJ91 ; YDK92 ; YD92 . The
correlation between fine oscillations and 1D density of states (DOS) cannot
explain the strong sensitivity of the additional structures on tuning the
reflector voltage.

We can visualize our system as consisting of a dynamic electronic cavityCDO15
which is defined by the arch-gate and the reflector (the reflector is required
to define the focal point of the cavity). The multiple reflection of emitted
1D electrons between the reflector and the arch-gates give rise to a continuum
of localized states. When the electrons under the gates are depleted and thus
the cavity is switched on, the 1D states couple to the cavity states and thus
give rise to the observed non-trivial features.

The framework above can be verified by changing the coupling between the 1D
and cavity states. A strong coupling will result in the appearance of fine
oscillations while at weak coupling they will smear out.

###  III.1 Coupling between 1D-2D electrons

Figure 3: R as a function of Vr. (a) The arch-gate voltage is fixed at - 0.65
, - 0.43, - 0.36, - 0.24, - 0.21, and 0 V while sweeping Vr. Data have been
offset vertically for clarity. The right axis displays the reflector
conductance as a function of Vr as indicated by a red arrow. (b) Upper plots
show the theoretical fitting (solid line) of $\Delta$R(Vr,Va) = R(Vr,Va) -
R(Vr,0), where Va = - 0.65 and - 0.50 V, respectively; $\Delta$R follows a
well defined Fano line shape. Lower plot shows Fano factor q as a function of
Va.

To tune the coupling between the 1D and 2D cavity states, we fix the arch-QPC
voltage at -0.65, - 0.43, - 0.36 [these three voltages are the center of fine
oscillations, as indicated by the vertical lines in Fig. 2(a)], - 0.24 (Peak
I), - 0.21 (Peak II) and 0 V and sweep the reflector voltage. The result shown
in Fig. 3(a) is a direct measurement of strength of the coupling effect. When
the arch-QPC voltage is fixed the population of injected electrons n in Eq.(1)
is constant, so the measured resistance depends on the reflection probability
r only; r increases monotonically when Vr becomes more negative, therefore
resistance R should follow a similar monotonic change. The bottom two traces
in Fig. 3(a) are for arch-QPC voltages Va of 0 and - 0.21 V, respectively
(cavity is off). The resistance is initially almost constant when the
reflector voltage $V_{r}>$ \- 0.25 V. A sharp rise in resistance occurs as Vr
is made more negative so the reflection probability r increases rapidly, and
eventually saturates when r becomes unity. When the arch-QPC becomes more
negative (top four traces, the cavity turns on), $R$ initially keeps on
increasing and then decreases until it saturates, and thereby produces a
pronounced peak in the plot. It is necessary to emphasize that the reflector
voltage range (from - 0.27 to - 0.29 V) where the peak in resistance occurs
corresponds to the regime where the fine oscillations are most pronounced
[Fig. 2(a)]. The dramatic change between different traces is a clear
indication of the effect arising due to cavity formation. Once the cavity is
on, i.e. both Va and Vr are sufficiently negative, its size and therefore the
energy spacing of cavity states can be adjusted by sweeping the reflector
voltage. The strong coupling between the 1D and cavity leads to a peak in the
measurement.

The fine oscillations corresponding to interference between the waves emerging
from the 1D region and the reflected waves are observable as the phase
coherence length ($>$10 $\mu$m) is greater than the size of the device. In
this respect our observation is similar to the Fabry-Perot interference
previously observedCMH89 .

###  III.2 Fano resonance

An analysis of the line shape of peak in Fig. 3(a) clearly indicates that
after the cavity is switched on, the dynamic change in resistance
$\Delta$R(Vr, Va) = R(Vr, Va) - R(Vr, 0) (we use R(Vr, 0) to account for the
change in the reflection probability, which is always present) follows the
well known Fano resonanceFANO61 ; GGH00 ; CSM17 ,

$R=R_{0}\frac{(q+\gamma(V-V_{0}))^{2}}{1+\gamma^{2}(V-V_{0})^{2}}+R_{inc}$
$None$

where R is the measured resistance, R0 is a constant representing the
amplitude of the resonance, q is the Fano factor which decides the asymmetry
of the line shape, $\gamma$ = 20 V-1 is voltage-energy lever (estimated from
the Fermi energy and the pinch-off voltage), V0 is the arch-gate voltage at
the center of the resonance (dip), and Rinc denotes intrinsic contribution
from the background as shown in Fig. 3(b). It was noticed that the Fano factor
q takes an oscillatory behavior against the arch-QPC voltage as shown in the
lower graph in Fig. 3(b). When compared with Fig. 2, q has a peak ($\sim$ 1.9)
at the fine oscillations and remains constant ($\sim$ 0.4) elsewhere. The
increase in q is due to an increase in the resonant scattering. This result
supports the argument that the observed fine oscillations are correlated with
the Fano resonance. It is necessary to draw attention to the fact that the
Fano factor does not necessarily show a peak at a conductance plateau if there
is no fine oscillation as two conductance plateaus are well defined when - 0.6
V $<V_{a}<$ \- 0.5 V while only a single Fano factor peak is observed. This
suggests that the result does not arise simply due to the collimation of a QPC
(which will produce similar behavior at all the conductance plateaus) or an
electrostatic effect.

The energy spacing between different cavity states is small (of the order of
100 $\mu$eV, from Ref.16) due to the large size of the cavity. This indicates
one 1D-state may couple to several cavity states and could result in
relatively broad fine oscillations. On the other hand, a single broad pattern
from 2nd (Va = - 0.6 V) and 3rd (Va = - 0.5 V) plateaus or modes of the QPC
may be attributed to coupling of different modes of QPC with the degenerate
cavity states. In addition, the probability of intersubband transition in 1D
constrictionOFM93  may result in mixed states.

Most investigations of Fano Resonance in nanostructures have utilized a
discrete state, for example in a quantum dotGGH00  or coupled QPCsJMY14 . It
is clear that the system which has been studied here has significant
differences in the lack of control of the particular states undergoing
interference as well as being openRLB04 ; SJ05 ; MY05 . However, the discrete
state, forming part of the interference along with the continuum, corresponds
to a collimated electron wave between QPC and cavity that is then reflected
towards the Ohmic contact and subsequently interferes. This is essentially
identical to the case of an optical wave reflected by a sharp bend in a
waveguide which behaves as a localized stateMTM94 ; CJ91 . The momentum value
and direction required for the reflected wave to reach the contact without
further reflection are stringent, corresponding to a sharply defined state
which then can interfere with the continuum comprising scattered electrons.

###  III.3 Effect of perpendicular magnetic field

Figure 4: Effect of perpendicular magnetic field. The reflector voltage is set
to -0.28 V and the field is increased from 0 (bottom trace) to - 200 mT (top
trace) by steps of - 20 mT. It is clear that the fine oscillations disappear
around - 60 mT (marked by an arrow), peak I of the double-peak structure
weakens with increasing field while peak II gets enhanced by magnetic field
and shifts towards less negative Va as indicated by a dashed black curve.

Figure 5: Magnetoresistance of the arch-QPC-reflector assembly. (a) The
reflector is grounded or set to -0.25 V; the results show a typical Hall
voltage development as the magnetic field is swept for different QPC voltage.
Data for Vr = - 0.25 V have been offset vertically by 50 $\Omega$ for clarity.
(b) The reflector is set to - 0.3 V, QPC voltage Va is set at - 0.63 (at fine
oscillations), - 0.24 (peak I), - 0.21 (peak II) and 0 V, respectively. Data
have been offset vertically by 100 $\Omega$ for clarity. (c) Theoretical
fitting of $\Delta$R(Va,B) = R(Va,B) - R(0,B) where Va = -0.24 V, showing the
experimental data follow a Fano line shape, the Fano factor q = 0.823.

The localized states, which represent the quantization of standing waves in
the cavity, is highly dependent on the trajectory of electrons and thus
sensitive to the perpendicular magnetic field. As a consequence, the fine
oscillations arising from coupling between the cavity and 1D states should
also be field sensitiveMTM94 . In Fig. 4, we show (negative) transverse
magnetic field dependence of the resistance, which gives information on the
coupling between 1D states and cavity states. It is clear that the fine
oscillations smear out at a weak field of -60 mT as expected. The double-peak
structure (in 1D-2D transition regime) follows a more complicated trend. The
intensity of peak I gets reduced by increasing field but it survives at much
higher field compared to the fine oscillations (- 120 mT at negative field
end), on the contrary, peak II gets enhanced by the magnetic fieldCSM17 .
Similar results were obtained with positive transverse magnetic field. We note
that when peaks I and II are observed, the channel is much wider than that for
the fine structure. Consequently the restriction on the formation of the
localized state is much weaker accounting for the higher field necessary to
remove the peaks. The initial enhancement of peak II is presumably a geometric
effect.

A detailed magnetoresistance study is presented in Fig. 5. When the reflector
is grounded or set to -0.25 V, a typical Hall voltage development is seen for
both Va = 0 V where QPC is in 2D regime and Va = - 0.5 V when the quasi-1D
channel forms in the QPC, as shown in Fig. 5(a). In Fig. 5(b), a voltage of -
0.3 V is applied to the reflector. When Va is 0 and - 0.21 V (cavity is not
formed) the result is almost similar to that in Fig. 5(a), however, SdH
oscillations are seen in the large field regime which come from the
contribution of the reflected electrons. When the cavity is switched on (Va =
- 0.24 V or - 0.63 V), additional structure which is highly asymmetric against
magnetic field is observed in the small field regime up to $\pm$ 70 mT, which
is similar to the magnetic field value required to quench the fine
oscillations as shown in Fig. 4. In the large field regime a superposition of
Hall voltage and SdH oscillations dominates. The dynamics in resistance with
cavity switched on, e.g. green trace in Fig. 5(b), defined as $\Delta$R(Va,B)
= R (Va, B) - R(0, B), also resembles a Fano resonance. Figure 5(c) shows a
theoretical fitting of $\Delta$R(- 0.24 V, B) (corresponding to green trace in
Fig. 5(b)) with the Fano line shape. It was also noticed that there was a
shift of minima of R against magnetic field in Fig. 5, which is likely due to
the combined effect of inclined reflector and the negative field because both
of them guide electrons to Ohmic 3 while positive field directs electrons to
Ohmic 4 whose trajectory is then compensated by the reflector. The difference
becomes more apparent when the electrons are more ballistic and collimated
(i.e. with more negative Va).

The self-consistency between the reflector [Fig. 3(b)] and magnetic field
[Fig. 5(c)] induced Fano resonance shows that we can tune coupling between the
1D and cavity states both electrostatically and magnetically. To gain a
comprehensive understanding of the scenario, a non-Hermitian description of
the QPC-cavity hybrid system comprising of reflected wave functions into the
continuum is necessaryHI17 . This is beyond the scope of current work.

Figure 6: Temperature dependence of R. (a) The evolution of non-local
resistance at Vr= - 0.3 V against temperature; the lattice temperature
increases from 20 mK (top trace) to 1.8 K (bottom trace). Data have been
offset vertically for clarity. (b) Theoretical fitting of the height of the
peak I of the double-peak structure using Eq. (3) $\&$ (4) with p = 1 for the
fitting.

###  III.4 Temperature dependence

Temperature dependence measurement is a useful tool to investigate quantum
effect such as the origin of fine oscillations. As mentioned previously, the
energy spacing for cavity states is an order of magnitude smaller than the 1D
subband spacing in the QPC, a slightly higher temperature makes the cavity
states become a continuum while 1D subbands are still well resolved. The
thermal smearing of the cavity states in turn leads to smearing out of the
fine oscillations. Figure 6(a) shows the evolution of the fine oscillations in
the lattice temperature range of 20 mK to 1.8 K. It is clear that the fine
oscillations smear out at 1.8 K where the conductance plateaus of QPC still
persist. On the other hand, the double-peak structure is present even at 1.8
K; however, its intensity decreases against rising temperature. The double-
peak structure is a consequence of interference effect, and its intensity can
be expressed byYUC91 ,

$R\propto exp(-l/l_{\phi})$ $None$

where l is the electron propagation length and the temperature dependence of
phase coherence length $l_{\phi}$ is,

$l_{\phi}\propto\sqrt{T^{-p}}$ $None$

where $p=1$ for a 2D systemMRM81  and $p=\frac{2}{3}$ for a 1D systemPPG88 .
In our device $p=1$ gives the best fitting as shown in Fig. 6(b) which in
consistent with the fact that the cavity states are essential to observe the
quantum interferencenote1 .

##  IV Conclusion

In conclusions, we have shown the operation of an integrated quantum device
consisting of an arch-shaped QPC coupled to an electronic cavity, whose states
can be tuned using a reflector gate. We have demonstrated that it is possible
to couple and decouple the 1D states with the 2D cavity states, using either
the reflector barrier or a transverse magnetic field, resulting in the direct
observation of Fano resonance which arises from the interference between the
QPC (1D) and cavity states (2D). Present results show the promise of such
integrated quantum devices in realizing complex quantum systems to study the
1D-2D transition and a possible precursor for quantum information processing
based on modulation of 1D states, for instance a quantum analogy of amplitude-
modulation (AM) or phase-modulation (PM) in classical in classical information
processing.

The work is funded by the Engineering and Physical Sciences Research Council
(EPSRC, UK).

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