PDF (double-sided) - Physics Department, UCSB - University of ...
PDF (double-sided) - Physics Department, UCSB - University of ... PDF (double-sided) - Physics Department, UCSB - University of ...
Decoherence consists of two parts: Energy relaxation and dephasing. These affect the two degrees of freedom of the qubit state in different ways. Energy relaxation describes the process by which the qubit loses energy and decays back to its | 0 〉-state. This decay primarily affects the θ degree of freedom of the state. It is caused by undesired coupling between the qubit and the environment which provides a path for the qubit to dissipate its energy. Dephasing captures a loss of information stored in the ϕ degree of freedom of the state. This is usually caused by magnetic flux noise that effectively applies random Z-rotations to the qubit’s state. Other noise sources, like critical current noise in the Josephson junction, can lead to the same effect, but seem to be less important [Bialczak et al., 2007]. The two decoherence mechanisms each have a timescale associated with them. T 1 captures the rate at which the qubit relaxes back into the | 0 〉-state and T ϕ captures the rate at which the qubit’s phase information is randomized. Since the decay to the | 0 〉-state also causes a loss in phase information, a new timescale T 2 is commonly defined to replace the less physical quantity T ϕ : 1 = 1 + 1 (3.67) T 2 2T 1 T ϕ Both T 1 and T ϕ are the result of classically random processes. Therefore, they do not affect the quantum state in a coherent way. Dephasing, for example, cannot be adequately described by a rotation of the state vector on the Bloch 64
sphere. Instead, it corresponds to this vector shrinking towards the Z-axis. The above described simulation formalism can therefore not be used in its presented form to capture these processes as the state’s normalization requirement |a n (t)| 2 + |a m (t)| 2 = 1 forces the state vector to remain on the surface of the Bloch sphere. To allow for the extra degree of freedom needed, it is necessary to move to the “density formalism” for describing the qubit state. This formalism adds a degree of freedom to the state by describing it as a probabilistic ensemble of pure states. Each possible pure state | A 〉 is assigned a weight w A describing the classical probability with which a randomly drawn member of the ensemble is in the state | A 〉. The weights w A fulfill: ∑ w A = 1 (3.68) A In this formalism, the state is described by a “density matrix” ρ, which takes the form: ρ = ∑ A w A | A 〉〈 A | = ∑ n w n | n 〉〈 n | (3.69) To simulate the evolution of the state, one simply needs to evolve both factors of the outer product | A 〉〈 A |: ρ(t + ∆t) = ∑ A = ∑ A w A | A(t + ∆t) 〉〈 A(t + ∆t) | w A e −iπ∆t(...) | A(t) 〉〈 A(t) | e iπ∆t(...) = e −iπ∆t(...) ρ e iπ∆t(...) (3.70) 65
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Decoherence consists <strong>of</strong> two parts: Energy relaxation and dephasing. These<br />
affect the two degrees <strong>of</strong> freedom <strong>of</strong> the qubit state in different ways.<br />
Energy relaxation describes the process by which the qubit loses energy and<br />
decays back to its | 0 〉-state. This decay primarily affects the θ degree <strong>of</strong> freedom<br />
<strong>of</strong> the state. It is caused by undesired coupling between the qubit and the<br />
environment which provides a path for the qubit to dissipate its energy.<br />
Dephasing captures a loss <strong>of</strong> information stored in the ϕ degree <strong>of</strong> freedom <strong>of</strong><br />
the state. This is usually caused by magnetic flux noise that effectively applies<br />
random Z-rotations to the qubit’s state. Other noise sources, like critical current<br />
noise in the Josephson junction, can lead to the same effect, but seem to be<br />
less important [Bialczak et al., 2007].<br />
The two decoherence mechanisms each<br />
have a timescale associated with them. T 1 captures the rate at which the qubit<br />
relaxes back into the | 0 〉-state and T ϕ captures the rate at which the qubit’s phase<br />
information is randomized. Since the decay to the | 0 〉-state also causes a loss in<br />
phase information, a new timescale T 2 is commonly defined to replace the less<br />
physical quantity T ϕ :<br />
1<br />
= 1 + 1<br />
(3.67)<br />
T 2 2T 1 T ϕ<br />
Both T 1 and T ϕ are the result <strong>of</strong> classically random processes. Therefore, they<br />
do not affect the quantum state in a coherent way.<br />
Dephasing, for example,<br />
cannot be adequately described by a rotation <strong>of</strong> the state vector on the Bloch<br />
64
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