Qubit-Coupled Mechanics - IFSC

Qubit-Coupled Mechanics
Matt LaHaye – Syracuse University
São Carlos Institute of Physics- 19 April 2013

Mechanical Systems in the Quantum Regime
Develop and study mechanical quantum devices; devices which under ordinary
conditions are perfectly well-described by classical laws of physics
Devices like:
Microtoroid Resonators
Nanomechanical Beams
Cleland & Roukes
Kippenberg
Just a small subset of types of devices being explored.
*See M.Poot & H.S. van der Zant, Physics Reports 2012,
for a recent and comprehensive review of devices*
LIGO
Macroscopic
Mirrors

Mechanical Systems in the Quantum Regime
Develop and study mechanical quantum devices; devices which under ordinary
conditions are perfectly well-described by classical laws of physics
Devices like:
Microtoroid Resonators
Nanomechanical Beams
Cleland & Roukes
Kippenberg
Vibrational modes normally
‘ring’ as one would expect
for classical S.H.O.
e.g. well-defined x and p;
Continuous energy spectra
LIGO
Macroscopic
Mirrors

Mechanical Systems in the Quantum Regime
No reason that we know of why the motion of such objects shouldn’t
exhibit characteristics of quantum S.H.O. (under the right conditions)
Roukes
Cantilever in a quantum superposition
of spatially-separated states
From Schwab & Roukes,
Phys. Today 2005
m
k
x
Discrete Energy
Levels
Zero-point
fluctuations

What do We Hope to Accomplish With These Studies?
Fundamental studies of quantum mechanics
- Perform experiments to shed light on fundamental issues with QM (e.g.
quantum-classical ‘boundary’; the measurement problem)
Development of new technologies
- Quantum information
- Bio-sensing and imaging
- …
e.g.
Long-distance quantum communication
Studying new frontiers in nature
- Gravitational wave detection
- Energy transfer at the nanoscale and
non-equilbrium fluctuation theorems
e.g.
LIGO interferometer
Rabl, Lukin et al.

Hybrid Quantum Systems
Individual spins coupled to a cantilever
LaHaye (SU); Roukes & Schwab (Caltech)
Rugar (IBM, Almaden)
Bose-Einstein Condensate Coupled to Cantilever
Treutlein (Basel)
Superconducting ‘QUBIT’ coupled to nanobeam
Integrating mechanical, atomic,
optical, microwave, spin and
solid-state quantum systems
A veritable toolbox of quantum
systems at our disposal!

The Future: Quantum Machines?

Outline
• Getting to the quantum regime of mechanics
• Qubit-coupled mechanics:
An important system for manipulating/measuring
quantum states of motion
Future prospects and experiments that are in the
works at Syracuse

What does it take to get to the quantum
regime of mechanics?

What does it take to see quantum behavior of
mechanical objects?
- Extensive subject
- Lots of subtleties related to measurement, the
nature of various quantum states of motion, and to
the interaction of harmonic systems with their
Environment
- Book by Braginsky is a great place to start (began
thinking about these issues in the context of
gravitational wave detection back in the 1970’s)
*Another nice, general overview: Schwab and Roukes,
Physics Today, July 2005.*

Thermal Noise
E
Energy Spectrum
The Environment: (e.g. charge fluctuations ;
phonons from substrate, EM radiation, etc)
Energy levels only account Roukes for
KE and PE due to restoring force
PE
ħω 0
Oscillator
Embedded in
an ‘environment’
H = 1 2 k 0χ̂0 2 + p̂0 2
2m 0
KE
E = ħω 0 N + 1 2
N = 1
N = 0
N = 3
N = 2
x
Environment is
Source of random
fluctuations that ‘kick’
And damp resonator.
Nanoresonator as S.H.O.
k
m x
Environment
Temp. T

Thermal Noise
B-E: p N = NN /(N + 1) N+1
The Environment: (e.g. charge fluctuations ;
phonons from substrate, EM radiation, etc)
E
Energy Spectrum
ħω
Energy levels only account Roukes for
Rule of Thumb: KE and PE due to restoring force
Need k BT
≲ 1 PE
ħω
Oscillator
Embedded in
an ‘environment’
H = 1 2 k 0χ̂0 2 + p̂0 2
2m 0
KE
If k B T ≫ ħω
In equilibrium:
N = 1
N = 0
N = 3
N = 2
x
Environment is
Source of random
fluctuations that ‘kick’
And damp resonator.
→ E ~k B T Fluctuations ‘wash out’ discrete levels
Nanoresonator as S.H.O.
m
k
x
Environment
Temp. T

Chasing the Ground State
Cooling the environment (or, equivalently, increasing ω)
Thermal Occupation Vs. Temperature
For Several different frequencies (ω/2π)
Dashed lines are average
# of quanta N tt ≈ k B T/ħω
(solid lines: BE occupation factor)
N tt
Frequencies correspond to
typical nano/micro-mechanical
fundamental mode
‘Freeze- out’ to
Quantum regime
T (mm)
This temperature regime
Accessible with standard
Cryogenic methods

Approaching the Ground State
Cleland et al., Nature, 2010
Dilatational oscillator
ω 0
2π ~ 6 GGG!
k B T
ħω 0
~1 at ~ 300 mm!
Cleland et al. put this device on a dilution refrigerator
and cooled it down so that N tt < . 1!

Approaching the Ground State: Back-Action Cooling
Use the measurement process to cool mechanical mode!
Schwab et al. Nature 2010
Micro
‘drum head’
Teufel et al. Nature 2011
N tt = 3. 8
Coherent State Transfer
Palomaki et al. Nature 2013
N tt =0.34
Observation of zero-point fluctuations
Safavi-Naeini et al. PRL 2012
Optomechanical crystal
Chan, Alegre, et al. Nature 2011
In these examples, a mechanical mode is
coupled to EM cavity, which does work on
the mode cooling it below environment T
*See M.Poot & H.S. van der Zant, Physics Reports,
for a recent and comprehensive review *
N tt =0.85

Now that mechanical structures have been
cooled near the ground state, how do you
prepare and measure quantum states of these
structures?

Limitations of Linear Displacement Detectors
Typically, displacement transducers are linear
x
k
m
Linear
Detector
Output is linearly proportional to x
Obviously, many examples: capacitive, magnetomotive, piezoelectric, transistor current
However, linear detectors generally should drive resonator into
‘classical-like’ states, coherent states of motion
Coherent
State
|α(t)⟩ = e −iωω− α 2 (αe−iii ) n
|n⟩
n!
∞
n=0
x(t) =
ħ
mω
α cosωω

Qubit-Coupled Mechanics
First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 (2002) & Physica B 316 (2002).
Cooper-pair box (CPB) charge qubit
Nano-electromechanical resonator
+
Nakamura et al., Nature, 398 29 Apr. 1999 Cleland & Roukes, APL 69 28 Oct. 1996
electrostatic interaction
=
artificial
atom
λ
x
|2>
|1>
|0>
|n>
Harmonic oscillator
Qubit- coupled
resonator analogous
to atom-coupled
photon cavity

Schematic of Coupling Between a Cooper-Pair Box
(CPB) Charge Qubit and Nanoresonator
Flexural motion of resonator modulates CPB electrostatic energy
Pauli Matrices
V NR
d
CPB Qubit
B
J. Junctions
V g
σ z = 1 0
0 −1
nanoresonator
Aluminum
C NR
N or N+1
Charges
(Cooper-Pairs)
Silicon Nitride
σ x = 0 1
1 0
Full Hamiltonian
Interaction strength
H = E ee V g σ z
CPB
electrostatic
Energy
+ E J B σ x + ħω a † a + 1 + ħλ(a † + a) ∙ σ
2
z
CPB
Josephson
Energy
NR
displacement
CPB
charge

Schematic of Coupling Between a Cooper-Pair Box
(CPB) Charge Qubit and Nanoresonator
Flexural motion of resonator modulates CPB electrostatic energy
V NR
d
nanoresonator
Aluminum
CPB Qubit
C NR
B
N or N+1
Charges
(Cooper-Pairs)
J. Junctions
V g
Silicon Nitride
Full Hamiltonian
Full Hamiltonian is
formally analogous to
the Jaynes-Cummings
Hamiltonian in quantum
Optics/CQED
See E.K. Irish and K.C. Schwab,
PRB, 2004 for more details.
Also, Haroche/Raimond, Exploring
the Quantum for JC in general
Interaction strength
H = E ee V g σ z
CPB
electrostatic
Energy
+ E J B σ x + ħω a † a + 1 + ħλ(a † + a) ∙ σ
2
z
CPB
Josephson
Energy
NR
displacement
CPB
charge

Schematic of Coupling Between a Cooper-Pair Box
(CPB) Charge Qubit and Nanoresonator
Flexural motion of resonator modulates CPB electrostatic energy
V NR
d
CPB Qubit
B
J. Junctions
V g
Question: How can
one use this to observe
quantum properties
of the nanoresonator?
C NR
nanoresonator
Aluminum
N or N+1
Charges
(Cooper-Pairs)
Silicon Nitride
Full Hamiltonian
See E.K. Irish and K.C. Schwab,
PRB, 2004 for more details.
Also, Haroche/Raimond, Exploring
the Quantum for JC in general
Interaction strength
H = E ee V g σ z
CPB
electrostatic
Energy
+ E J B σ x + ħω a † a + 1 + ħλ(a † + a) ∙ σ
2
z
CPB
Josephson
Energy
NR
displacement
CPB
charge

The Dispersive Coupling Limit
Typically ħλ ≪ ΔE − ħω
Full Hamiltonian
H = E ee V g σ z + E J B σ x + ħω a † a + 1 2 + ħλ(a† + a) ∙ σ z
Some typical parameters
λ ~ 1 − 10 ′ s MHz
ΔE ~ 1 − 10 ′ s GHz
Coupling Strength
CPB Energy Scale
Interaction results in
simple shifts in energies of
the two systems
(Dispersive or non-resonant
Coupling limit)
where
ΔE =
ω ~ 10 ′ s − 100 ′ s MHz
E 2 2
ee + E J
Mechanical Frequency
The dispersive shift of each
system depends on the
state of the other system
*See: Irish & Schwab, PRB 68, 155311 (2003); Haroche & Raimond, Exploring the Quantum: Atoms, Cavities & Photons*

The Dispersive Coupling Limit
Interaction Leads to Shift in Energy of CPB and NEMS
Analogous To dispersive shifts in CQED
Two Dispersive Effects Occur
CPB-state-dependent
Frequency
Shift in NR
Analogous to single-atom
refractive shift in CQED
δω ± ≈ ±ħλ 2 /ΔE
Energy levels w/o interaction
N = 2
N = 1
N = 0
…
…
ħω 0
∆E
|−⟩ |+⟩
Energy levels with interaction
2
1
NR- dependent
shift in CPB
transition energy
N ≡ # Quanta in NR
Analogous
to Lamb Shift
δE ≈ ħλ 2 (2N + 1)/ΔE
Analogous
to Stark Shift
N = 2
N = 1
N = 0
…
…
ħω +
0
ħω −
∆E + δδ
|−⟩ |+⟩
1
*See: Irish & Schwab, PRB 68, 155311 (2003)
ω ± = ω + δω ±

First Realization of Qubit-Coupled Nanoresonator
NR Response with No Coupling to Qubit
CPB
Qubit

First Realization of Qubit-Coupled Nanoresonator
NR Response Coupled to Qubit in Ground State
CPB
Qubit
Expect (-) Shift:
δω − ≈ −ħλ 2 /ΔE

Nanomechanical Probe of Quantum Interference
in a CPB Qubit
From M.D. LaHaye et al., Nature 459 , 960 (2009).
Positive Frequency
Shift of Mechanics
Qubit in Excited
State on Average
Frequency Shift of
Mechanics ∆ ω /2π NR (Hz)
2.5
2.0
-400 -200 0 200 400 600
V rf
V (V) µ
(V)
Gave us confirmation that 1.5simple JC model captured the physics of this qubit-coupled
Mechanical devices (at least in the dispersive limit).
1.0
Negative Frequency
Shift of Mechanics
Qubit in Ground
State On Average
0.5
-4
-3
-3
-10.0 -8.0 -6.0 -4.0 -2.0
V
V cpb
(mV)
0 (V)
φ is a function of
Qubit parameters V rf and V 0
-4
Qubit’s State Vector
|Ψ(t)⟩ = Ψ − (t)|−⟩ + Ψ(t) + |+⟩
Ψ − (t) ∝ sin (φ)
Ψ + (t) ∝ cos (φ)

Milestone in Quantum Measurement
Nature April 1 2010
UCSB Group
Resonant Limit (∆E ≈ ω) of Jaynes-Cummings
Qubit Excited
State probability
H = ∆Eσ z + ħω a † a + 1 2 + ħλ(a† σ − + aσ + )
Qubit-Resonator Swap Quanta

The Dispersive Coupling Limit
*See: Irish & Schwab, PRB 68, 155311 (2003)
Interaction Leads to Shift in Energy of CPB and NEMS
Analogous To dispersive shifts in CQED
Two Dispersive Effects Occur
CPB-state-dependent
Frequency
Shift in NR
NR- dependent
shift in CPB
transition energy
N ≡ # Quanta in NR
χ = 2λ 2 /ΔE
Analogous to single-atom
refractive shift in CQED
δω ± ≈ ±ħχ
Analogous
to Lamb Shift
δE ≈ ħχ(2N + 1)
Analogous
to Stark Shift
Energy levels w/o interaction
N = 2
N = 1
N = 0
…
…
…
ħω 0
∆E
|−⟩ |+⟩
Energy levels with interaction
N = 2
N = 1
N = 0
∆E + 3ħχ
…
∆E + ħχ
|−⟩ |+⟩
2
1
1
0
Dispersive interaction is non-destructive: H iii ∝ σ z ⋅ a † a ⇒ H 0 , H iii = 0

The Dispersive Coupling Limit
*See: Irish & Schwab, PRB 68, 155311 (2003)
Interaction Leads to Shift in Energy of CPB and NEMS
Analogous To dispersive shifts in CQED
Two Dispersive Effects Occur
CPB-state-dependent
Frequency
Shift in NR
NR- dependent
shift in CPB
transition energy
N ≡ # Quanta in NR
χ = 2λ 2 /ΔE
Analogous to single-atom
refractive shift in CQED
δω ± ≈ ±ħχ
Analogous
to Lamb Shift
δE ≈ ħχ(2N + 1)
Analogous
to Stark Shift
Energy levels w/o interaction
N = 2
N = 1
N = 0
…
…
…
ħω 0
∆E
|−⟩ |+⟩
Energy levels with interaction
N = 2
N = 1
N = 0
∆E + 3ħχ
…
∆E + ħχ
|−⟩ |+⟩
2
1
1
0
Measurements of Stark shift just recently done in the high-N limit: Pirkklainen et al. Nature 2013

Next step: Dispersive measurements of the number-state
statistics of a mechanical mode
Perform spectroscopy on the qubit
while it is coupled to the
mechanical resonator
ω
= 300 MHz
2π
χ
= 0.6 MHz
2π
k B T~1.7ħω
Qubit Linewidth
γ qq
= 0.5 MHz
2π
Absorption spectrum of qubit
for nanoresonator in thermal
state at T = 25 mm
Absorption
Rate (A.U.)
*Calculations based on CLERK & UTAMI, PRA 75, 042302 (2007)
‘Stark’-Shifted Qubit Transitions
N = 2
N = 1
N = 0
…
N = 0
Simulation of Stark Shift
…
∆E + ħχ
|−⟩ |+⟩
2
∆E + 5ħχ
1
∆E + 3ħχ
N = 1
0.1 1.0 10
ω s − ΔE (MHz)
0
N = 2

Engineering the qubit read-out: Embed Qubit-Mechanics in
Superconducting LC Circuit
ω p
∼
50 Ω Feedline
Cryo Amp
L
C
C Q
Qubit State Changes LC Frequency - Probe By Transmission Measurement of Feedline

Engineering the qubit read-out: Embed Qubit-Mechanics in
Superconducting LC Circuit
Probe qubit state |±⟩ via ‘pull’
of LC resonator frequency
0
Feedline Transmission Simulation
Qubit Capacitance
If qubit is in |−⟩, C Q > 0
If qubit is in |+⟩, C Q < 0
S21
Amp
(dB)
-4
-8
-12
|−⟩
|+⟩
1.768 1.772 1.776 1.780 1.784
Probe Frequency - ω p /2π (GHz)

Present Status of Our Efforts at Syracuse - Lab
Dilution fridge cooled down for
first time on 11/3/11.
Base Temp. < 33 mm!
LaHaye Lab
Research Team
NSF-DMR Career Award: #1056423
Postdoc
Francisco
Rouxinol
Visiting Scientist
Seung-Bo Shim
Grad Student
Hugo Hao

Present Status of Our Efforts at Syracuse – LC
Circuit Read-out of Superconducting Qubit
Making samples at Cornell
Nanofabrication Facility (CNF) and in
the Plourde Lab at SU
Coupling
To LC
50 Ω feedline
Qubit
nanoresonator
LC Circuit
*Fabrication By: Francisco Rouxinol and Seung-Bo Shim
*Measurements By: Hugo Hao, MDL, FR & SBS
LC and Ground Plane: Niobium
Nanoresonator and Qubit: Aluminum
Signal Amplitude (nV)
100
10
1
Feedline Transmission Measurements
.5Φ 0 change
in Φ to qubit
Q i ~ 12 − 15 × 10 3
Q c ~ 300 − 350
1.938 1.944 1.950 1.956
Probe Frequency - ω p /2π (GHz)

LC Circuit Read-out of Qubit vs. Applied B-field
• LC response modulates
periodically with applied
magnetic field:
E J ∝ cos (πΦ/Φ 0 )
LC Probe Frequency (GHz)
1.96
1.95
1.94
Amplitude
0.0
-5.0
-10.0
Transmission Amplitude (dB)
LC Probe Frequency (GHz)
1.96
1.95
1.94
Phase
*fridge temperature ≲ 30 mK*
1.0
0.5
0.0
-0.5
Transmission Phase (rad.)
-5.0 5.0 15.0 25.0 35.0
Magnet Current (mA)

LC Circuit Read-out of Qubit vs. Applied B-field
• LC response modulates
periodically with applied
magnetic field:
E J ∝ cos (πΦ/Φ 0 )
• Model parameters
• E C = 1.3 GGG
• E J = 12.7 GGG
• λ/2π = 160 MMM
• ω/2π = 1.945 GGG
• General agreement with
Jaynes-Cummings Model
for qubit-coupled LC
• Model Needs Improvement
• Charge noise
• Finite temperature
LC Probe Frequency (GHz)
LC Probe Frequency (GHz)
1.96
1.95
1.94
1.96
1.95
1.94
___
…….
Amplitude
Phase
Coupled qubit & LC
Uncoupled qubit &LC
*fridge temperature ≲ 30 mK*
-1.5 0.5 0.5 1.5
Magnetic Flux (πΦ/Φ 0 )
0.0
-5.0
-10.0
1.0
0.5
0.0
-0.5
Transmission Amplitude (dB)
Transmission Phase (rad.)

Details of the UHF Flexural Nanomechanical Resonator
Close Up of Nanoresonator
Resonator Geometric Properties
- ~100 nm thick aluminum
- 200 nm width
- 1.8 micron length
- Expect in-plane f 0 ~ 300 MHz
Resonator etched using SF 6 RIE etch
Engineered 70 nm gap between
Al resonator and Al qubit electrode
COMSOL Simulation:
Qubit-Resonator Capacitance ~ 180 aF*
*Should yield strong qubit-nanoresonator
Coupling (λ/2π~10 ′ s MHz)*
*Fabrication By: Francisco Rouxinol and Seung-Bo Shim

Details of the UHF Flexural Nanomechanical Resonator
Close Up of Nanoresonator
Resonator Geometric Properties
- ~100 nm thick aluminum
- 200 nm width
- 1.8 micron length
- Expect in-plane f 0 ~ 300 MHz
COMSOL Simulation
Calculated In-plane flexural
fundamental mode: 291MHz
Signal Amplitude (nV)
0
-40
-80
Preliminary Magnetomotive Measurements
(On a 4K Probe Station)
Q~1350
0 Tesla
7 Tesla
Fit to EM
Impedance
288.7 289.0 289.3 289.6 289.9
Drive Frequency - ω/2π (MHz)
*Simulation and magnetomotive measurements by Seung-Bo Shim and Sang Goon Kim

LC Read-out of Qubit: Two-Tone Spectroscopy
10
Qubit Absorption Spectrum Vs. Magnetic Flux
January 21st Data, 10dBm MW - Simulation: Ec=1.7GHz;Ej0=11.6GHz;
15
9
10
Microwave Frequency (GHz)
Microwave Frequency (GHz)
8
7
6
5
Want to Zoom-in
Look for phonon
Stark shift, but…
Qubit transition energy
model parameters:
• E c =1.3 GHz
• E j0 =11.9 GHz
5
0
-5
-10
-15
LC
Phase
(Deg.)
4
-1.5 -1 -0.5 0 0.5 1 1.5
Flux (π)
Flux (πΦ/Φ 0 )
-20
- Resonant microwaves applied to qubit modify the dispersive shift of the LC
- Tuning flux modifies qubit transition via E J ∝ cos (πΦ/Φ 0 ) .

Two-Tone Spectroscopy – ‘Extra’ Features
10
Qubit Absorption Spectrum Vs. Magnetic Flux
January 21st Data, 10dBm MW - Simulation: Ec=1.7GHz;Ej0=11.6GHz;
15
9
10.5
10
Higher resolution spectroscopy
Microwave Frequency (GHz)
Microwave Frequency (GHz)
8
7
6
5
What is the
Extra Structure?
4
-1.5 -1 -0.5 0 0.5 1 1.5
Flux (π)
Flux (πΦ/Φ 0 )
10.0
9.5
5
0
-5
-10
-0.5 0.0
0.5
-15
-20
LC
Phase
(Deg.)
~300 MHz
~300 MHz
- ‘Extra’ structure looks like a series of avoided-level crossings
- Can’t rule out multi-photon processes , longitudinal dressed states, etc. (e.g. ~10 photons in LC)

Two-Tone Spectroscopy – ‘Extra’ Features
10
Qubit Absorption Spectrum Vs. Magnetic Flux
January 21st Data, 10dBm MW - Simulation: Ec=1.7GHz;Ej0=11.6GHz;
15
Microwave Frequency (GHz)
Microwave Frequency (GHz)
9
8
7
6
5
Working with Fred and
Amir Caldeira for model
including LC,CPB & NR
Unfortunately, sample
destroyed before fully
studied. .. We’re
making new ones now.
What is the
Extra Structure?
4
-1.5 -1 -0.5 0 0.5 1 1.5
Flux (π)
Flux (πΦ/Φ 0 )
- ‘Extra’ structure looks like a series of avoided-level crossings
- Can’t rule out multi-photon processes , longitudinal dressed states, etc. (e.g. ~10 photons in LC)
- Could it be related to the mechanical resonator? ~300 MHz spacing is suggestive
10.5
10.0
9.5
10
Higher resolution spectroscopy
5
0
-5
-10
-0.5 0.0
0.5
-15
-20
LC
Phase
(Deg.)
~300 MHz
~300 MHz

Conclusions
- Mechanics has become a new quantum technology with
many possible applications and potential for addressing
fundamental questions in quantum mechanics
- Qubit-mechanics will serve as an important test-bed for
mechanical quantum systems. Analogous to what CQED has
been for studying quantum properties of light.
Experiments at Syracuse performed with
Francisco Rouxinol & Hugo Hao - SYR
Seung-Bo Shim & Sang Goon Kim - KRISS