PDF (double-sided) - Physics Department, UCSB - University of ...
PDF (double-sided) - Physics Department, UCSB - University of ... PDF (double-sided) - Physics Department, UCSB - University of ...
• The energy difference between the ground and first excited state in the operating minimum as a function of flux bias: Dividing this number by will give the expected (angular) operating frequency of the qubit, i.e. the frequency with which it needs to be driven to perform operations. • The energy difference between the first and second excited levels in the operating minimum: The frequency corresponding to this transition will need to be significantly different from the operating frequency to allow for operations on the qubit without exciting it into unwanted higher levels. • The number of states in the right minimum: During measurement, the first excited state in the operating minimum (here: left minimum) will be selectively tunneled into the right minimum. There, it will end up in a level of similar energy to the one it tunneled from, i.e. fairly high up in the minimum. The rate at which the state will decay in the right minimum, and thus the rate with which the measurement “latches” the outcome, is determined by the number of the level that the state tunnels into. Higher states decay faster with a rate of approximately T 1 /n, where n is the level number. Fast decay is important to reduce the chance of the state tunneling back to the left before latching. For the calculation to yield trustable results, a few things need to be kept in mind: 46
• The x-range over which the potential is approximated needs to be large enough for the wave-functions to “comfortably” go to zero on both sides. • In a real qubit potential, the right minimum will most likely have hundreds of states at energies below the ground-state of the operating minimum. Since an n × n matrix will yield only the lowest n eigenstates, M therefore needs to have several hundred rows and columns. • The ground-state in the operating minimum will usually not be the level with the lowest overall energy if the other minimum is deeper. In the figure shown, states 9 and 11 (counting from 0) are localized mostly in the left minimum, the states above level 11 span both minima, and all other states are localized in the right minimum. Thus, it is necessary to sort the levels into the correct minimum before subtracting their energies to find transition frequencies. 3.2.2 Eigenstates of Coupled Qubit Systems It is theoretically possible to extend this method to finding the eigenstates of a system of coupled qubits. For this, the state of the second qubit is added to the Schrödinger equation, making it two-dimensional (a function of δ 1 and δ 2 ). ψn(δ r 1 , δ 2 ) and V (δ 1 , δ 2 ) would then be rewritten as vectors following a convention 47
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• The x-range over which the potential is approximated needs to be large<br />
enough for the wave-functions to “comfortably” go to zero on both sides.<br />
• In a real qubit potential, the right minimum will most likely have hundreds<br />
<strong>of</strong> states at energies below the ground-state <strong>of</strong> the operating minimum. Since<br />
an n × n matrix will yield only the lowest n eigenstates, M therefore needs<br />
to have several hundred rows and columns.<br />
• The ground-state in the operating minimum will usually not be the level<br />
with the lowest overall energy if the other minimum is deeper. In the figure<br />
shown, states 9 and 11 (counting from 0) are localized mostly in the left<br />
minimum, the states above level 11 span both minima, and all other states<br />
are localized in the right minimum. Thus, it is necessary to sort the levels<br />
into the correct minimum before subtracting their energies to find transition<br />
frequencies.<br />
3.2.2 Eigenstates <strong>of</strong> Coupled Qubit Systems<br />
It is theoretically possible to extend this method to finding the eigenstates <strong>of</strong><br />
a system <strong>of</strong> coupled qubits. For this, the state <strong>of</strong> the second qubit is added to<br />
the Schrödinger equation, making it two-dimensional (a function <strong>of</strong> δ 1 and δ 2 ).<br />
ψn(δ r 1 , δ 2 ) and V (δ 1 , δ 2 ) would then be rewritten as vectors following a convention<br />
47
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