3. Physics of Josephson Junctions: The Voltage StateΒ Β· s e, d r-r-ut Phase -2019) AS-Chap. 3.6 - 2...
Transcript of 3. Physics of Josephson Junctions: The Voltage StateΒ Β· s e, d r-r-ut Phase -2019) AS-Chap. 3.6 - 2...
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AS-Chap. 3.6 - 1
Introduction to Chapter 6(from Chapter 3 of the lecture notes)
Quantum treatment of JJ
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AS-Chap. 3.6 - 2
Classical treatment of Josephson junctions (so far)
Phase π and charge π = πΆπ βππ
ππ‘ Purely classical variables
(π, π) are assumed to be measurable simultaneously
Dynamics Tilted washboard potential, rotating pendulum
Classical energies:
Potential energy π(π)(Josephson coupling energy / Josephson inductance)
Kinetic energy πΎ αΆπ
(Charging energy via1
2πΆπ2 =
π2
2πΆβ
ππ
ππ‘
2/ junction capacitance)
Current-phase & voltage-phase relation from macroscopic quantum model
Quantum origin
Primary macroscopic quantum effects
Second quantization
Treat (π, π) as quantum variables (commutation relations, uncertainty)
Secondary macroscopic quantum effects
3.6 Full Quantum Treatment of Josephson JunctionsSecondary Quantum Macroscopic Effects
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AS-Chap. 3.6 - 3
E
j
2πΈJ0
3.6.1 Quantum Consequences of the small Junction Capacitance
Validity of classical treatment
Classical treatment valid for πΈπ½0
βππβ
πΈπ½0
πΈπΆ
1/2β« 1 (Level spacing βͺ Potential depth)
Enter quantum regime by decreasing junction area π΄
Consider an isolated, low-damping junction, πΌ = 0 Cosine potential, depth 2πΈπ½0 Close to potential minimum
Harmonic oscillatorFrequency πp, level spacing βπp
Vacuum energyβπp
2
ββπp
2
β βπp
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AS-Chap. 3.6 - 4
Example 1Area π΄ = 10 ΞΌm2, Tunnel barrier π = 1 nm, π = 10,
π½c = 100A
cm2
πΈJ0 = 3 Γ 10β21 J
πΈJ0/β = 4500 GHz
πΆ =ππ0π΄
π= 0.9 pF
πΈπΆ = 2 Γ 10β26 J
πΈπΆ
β= 30 MHz
Classical junction
We also need π βͺ 500 mK for πBπ βͺ πΈJ0, πΈπΆ!
3.6.1 Quantum Consequences of the small Junction Capacitance
Parameters for the quantum regime
Example 2Area π΄ = 0.02 ΞΌm2
πΆ β 1 fF πΈπΆ β πΈJ0 Quantum junction
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AS-Chap. 3.6 - 5
Kinetic energy:
Total energy:
π π β 1 β cosπ Potential energyπΎ αΆπ β αΆπ2
Kinetic energy
Energy due to extra charge π on one junction electrode due to π
Consider πΈ π, αΆπ as junction Hamiltonian, rewrite kinetic energy
πΎ = π2/2π
π =β
2ππ
3.6.1 Quantum Consequences of the small Junction Capacitance
Position coordinate associated to phase π, momentum associated to charge π
Hamiltonian of a strongly underdamped junction (with ππ
ππ‘β 0)
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AS-Chap. 3.6 - 6
Canonical quantization (operator replacement)
with π =π
2π # of Cooper pairs
we get the Hamiltonian
Describes only Cooper pairs
πΈπΆ =π2
2πΆnevertheless defined as
charging energy for a single electron charge
π β‘π
2π Deviation of # of CP in electrodes from equilibrium
3.6.1 Quantum Consequences of the small Junction Capacitance
Commutation rules for the operators
Heisenberg uncertainty relation
Ξπ΄Ξπ΅ β₯1
2απ΄, π΅
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AS-Chap. 3.6 - 7
Hamiltonian in the flux basis (π =β
2ππ =
Ξ¦0
2ππ)
3.6.1 Quantum Consequences of the small Junction Capacitance
Commutator π, π = πβ
π and Q are canonically conjugate (analogous to x and p)
Circuit variables are now quantized
Superconducting quantum circuits
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AS-Chap. 3.6 - 8
Lowest energy levels localized near bottom of potential wells at ππ = 2π π
Taylor series for π π Harmonic oscillator,
Frequency πp, eigenenergies πΈπ = βπp π +1
2
Ground state: narrowly peaked wave function at π = ππ
Large fluctuations of π on electrodes since Ξπ β Ξπ β₯ 2π(small EC pairs can easily fluctuate, large Ξπ)
Small phase fluctuations ΞπNegligible Ξπ β classical treatment of phase dynamics is good approximation
3.6.2 Limiting Cases: The Phase and Charge Regime
The phase regime
βπp βͺ πΈJ0 , πΈπΆ βͺ πΈJ0 Phase π is a good quantum number!
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AS-Chap. 3.6 - 9
Hamiltonian
define π = πΈ β πΈπ½0 /πΈπΆ, π = πΈπ½0/2πΈπΆ and π§ = π/2
known from periodic potential problem in solid state physics Energy bandsGeneral solution
Bloch waves
Charge/pair number variable π is continuous (charge on capacitor!) πΉ π is not 2π-periodic
3.6.2 Limiting Cases: The Phase and Charge Regime
Mathieu equation
The phase regime
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AS-Chap. 3.6 - 10
1D problem Numerical solution straightforwardVariational approach for approximate ground state
Trial function for πΈπΆ βͺ πΈJ0
Choose π to find minimum energy:
π¬πͺ
π¬ππ= π. π
π¬π¦π’π§ = π.π π¬ππ
first order in EJ0
Emin β 0 for EC βͺ EJ0
3.6.2 Limiting Cases: The Phase and Charge Regime
The phase regime
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AS-Chap. 3.6 - 11
Tunneling coupling β exp β2πΈJ0βπΈ
βπp Very small since βπp βͺ πΈJ0
Tunneling splitting of low lying states is exponentially small
3.6.2 Limiting Cases: The Phase and Charge Regime
The phase regime
π¬πͺ
π¬ππ= π. π
π¬π¦π’π§ = π.π π¬ππ
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AS-Chap. 3.6 - 12
Kinetic energy β πΈπππ
ππ‘
2dominates
Complete delocalization of phase Wave function should approach constant value, Ξ¨ π β const. Large phase fluctuations, small charge fluctuations (π₯π β π₯π β₯ 2π)
3.6.2 Limiting Cases: The Phase and Charge Regime
βπp β« πΈJ0 , πΈπΆ β« πΈJ0 Charge π (momentum) is good quantum number
Appropriate trial function:
Hamiltonian
Approximate ground state energy
second order in EJ0
πΌ βͺ 1
The charge regime
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AS-Chap. 3.6 - 13
π¬πͺπ¬π±π
= π. π
π¬πππ = π. ππ π¬π±π
Periodic potential is weak Strong coupling between neighboring phase states Broad bands Compare to electrons moving in strong (phase regime) or weak (charge regime) periodic
potential of a crystal
3.6.2 Limiting Cases: The Phase and Charge Regime
The charge regime
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AS-Chap. 3.6 - 14
Voltage π Charge π = πΆπ, energy πΈ =π2
2πΆ
Observation of CB requires small thermal fluctuations
πΈπΆ =π2
2πΆ> ππ΅π β πΆ <
π2
2ππ΅π
πΆ β 1 fF at T = 1 K, π = 1 nm, and π = 5 π΄ β² 0.02 ΞΌm2 Small junctions!
Observation of CB requires small quantum fluctuations
Quantum fluctuations due to Heisenberg principle ΞπΈ β Ξπ‘ β₯ β Finite tunnel resistance ππ πΆ = π πΆ (decay of charge fluctuations)
Ξπ‘ = 2ππ πΆ, ΞπΈ =π2
2πΆ π β₯
β
π2= π πΎ = 24.6 kΞ© Typically satisfied
3.6.3 Coulomb and Flux Blockade
Coulomb blockade in normal metal tunnel junctions
Single electron tunneling
Charge on one electrode changes to π β π
Electrostatic energy πΈβ² =πβπ 2
2πΆ
Tunneling only allowed for πΈβ² β€ πΈ Coulomb blockade: Need π β₯ π/2 or π β₯ πCB = πc = π/2πΆ
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AS-Chap. 3.6 - 15
Coulomb blockade in superconducting tunnel junctions
For π2
2πΆ> ππ΅π, ππ (π = 2π) No flow of Cooper pairs
Threshold voltage π½ β₯ π½πͺπ© = π½π =ππ
ππͺ=
π
πͺ
Coulomb blockade Charge is fixed, phase is completely delocalized
3.6.3 Coulomb and Flux Blockade
S S2e-
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AS-Chap. 3.6 - 16
Current πΌ Flux π· = πΏ πΌ, energy πΈ = π·2/2πΏ
In presence of fluctuations we need
πΈJ0 β« πBπ (large junction area)
And π₯πΈ β π₯π‘ β₯ β with π₯t = 2ππΏ
π and π₯πΈ = 2πΈπ½0 π β€
β
(2π)2=
1
4π πΎ
3.6.3 Coulomb and Flux Blockade
Phase or flux blockade in a Josephson junction
Phase is blocked due to large πΈJ0 = π·0πΌπ/2π
πΌc takes the role of πCB Phase change of 2π equivalent to flux change of π·0
Flux blockade πΌ β₯ πΌFB = πΌc =΀Φ0 2π
πΏc
Analogy to CB πΌ β π, 2π βπ·0
2π, πΆ β πΏ
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AS-Chap. 3.6 - 17
Coherent charge statesIsland charge continuously changed by gate
Independent charge states (πΈJ0 = 0)
Parabola πΈ π = π β π β 2π 2/2πΆΞ£
Cooper pair box
3.6.4 Coherent Charge and Phase States
πΈJ0 > 0
Interaction of |πβ© and |π + 1β© at the level crossing points π = π +1
2β 2π
Avoided level crossing (anti-crossing)
Coherent superposition states πΉΒ± = πΌ π Β± π½ π + 1
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AS-Chap. 3.6 - 18
Average charge on the island as a function of the applied gate voltage Quantized in units of 2π (no coherence yet)
3.6.4 Coherent Charge and Phase States
First experimental demonstration of coherent superposition charge states by Nakamura, Pashkin, Tsai (1999, not this picture)
Coherent charge states
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AS-Chap. 3.6 - 19
Coherent phase states Interaction of two adjacent phase states Example is rf SQUID
magnetic energy
of flux π =Ξ¦0
2ππ in the ring π½πππ = π½π/π
Tunnel coupling Experimental evidence for quantum coherent superposition (Mooij et al., 1999)
3.6.4 Coherent Charge and Phase States
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AS-Chap. 3.6 - 20
Violation of conservation of energy on small time scales, obey ΞπΈ β Ξπ‘ β₯ β
Creation of virtual excitations Include Langevin force πΌπΉ with adequate statistical properties Fluctuation-dissipation theorem
πΈ π, π = energy of a quantum oscillator
Transition from βthermalβ Johnson-Nyquist noise to quantum noise:
classical limit (βπ, ππ βͺ ππ΅π):
quantum limit (βπ, ππ β« ππ΅π):
3.6.5 Quantum Fluctuations
vacuumfluctuations
occupation probabilityof oscillator (Planck distribution)
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AS-Chap. 3.6 - 21
Escape of the βphase particleβ from minimum of washboard potential by tunneling
Macroscopic, i.e., phase difference is tunneling (collective state) States easily distinguishable
Competing process
Thermal activation Low temperatures
Neglect damping
Dc-bias Term ββπΌπ
2πin Hamiltonian
Curvature at potential minimum:
(Classical) small oscillation frequency:
3.6.6 Macroscopic Quantum Tunneling
π = πΌ/πΌπ
(attempt frequency)
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AS-Chap. 3.6 - 22
Quantum mechanical treatment
Tunnel coupling of bound states to outgoing waves Continuum of states But only states corresponding to quasi-bound states have high amplitude In-well states of width Ξ = β/π (π = lifetime for escape)
Determination of wave functions
Wave matching method Exponential prefactor within WKB approximation Decay in barrier:
decay of wave function of particle with mass M and energy E
3.6.6 Macroscopic Quantum Tunneling
for π π β« πΈ0 = βπA/2
mass effective barrier height
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AS-Chap. 3.6 - 23
Constant barrier height Escape rate
Increasing bias current
π0 decreases with π π€ becomes measurable
Temperature πβ where π€tunnel = π€TA β exp βπ0
ππ΅π
for πΌ > 0: For πp β 1011 π β1
T* 100 mK
very small for π β 0
3.6.6 Macroscopic Quantum Tunneling
for πΌ β 0:
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AS-Chap. 3.6 - 24
Junction couples to the environment (e.g., Caldeira-Leggett-type heat bath)
Crossover temperature ππ΅πβ β
βπR
2πwith πR = ππ΄ 1 + πΌ2 β πΌ and πΌ β‘
1
2π NπΆπA
Strong damping πΌ β« 1 πR βͺ πA Lower πβ
Damping suppresses MQT
3.6.6 Macroscopic Quantum Tunneling
Additional topic: Effect of damping
lightly damped(small capacitance)
moderately damped(larger capacitance)
After: Martinis et al., Phys. Rev. B. 35, 4682 (1987).
πR/π
A
πΌ10310β1 10 105
1.0
010β3
0.8
0.6
0.4
0.2
Phase diffusion by MQTSee lecture notes
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AS-Chap. 3.6 - 25
Classical description only in the phase regime (large junctions): πΈπΆ βͺ πΈπ½0
For πΈπ β« πΈπ½0: quantum description (negligible damping):
Phase difference π and Cooper pair number π =π
2πare canonically conjugate variables
phase regime: Ξπ β 0 and Ξπ β βcharge regime: Ξπ β 0 and Ξπ β β
Charge regime at π = 0
Coulomb blockade Tunneling only for πCB β₯π
πΆ
Flux regime at π = 0
Flux blockade Flux motion only for πΌFB β₯Ξ¦0
2ππΏc
At πΌ < πΌc Escape out of the washboard by thermal activation or macroscopic quantum tunneling
TA-MQT crossover temperature πβ
Summary (secondary quantum macroscopic effects)